Article citation information:Kwatra, D., Ramachandra Rao, K., Bhatnagar, V. Novel accessibility metrics based on hierarchical decomposition of transport networks. Scientific Journal of Silesian University of Technology. Series Transport. 2023, 118, 139-160. ISSN: 0209-3324. DOI: https://doi.org/10.20858/sjsutst.2023.118.10.Divya KWATRA, Kalaga Ramachandra RAO, Vasudha BHATNAGARNOVEL ACCESSIBILITY METRICS BASED ON HIERARCHICAL DECOMPOSITION OF TRANSPORT NETWORKSSummary. Scientific analysis of public transport systems at the urban, regional, and national levels is vital in this contemporary, highly connected world. Quantifying the accessibility of nodes (locations) in a transport network is considered a holistic measure of transportation and land use and an important research area. In recent years, complex networks have been employed for modeling and analyzing the topology of transport systems and services networks. However, the design of network hierarchy-based accessibility measures has not been fully explored in transport research. Thus, we propose a set of three novel accessibility metrics based on the k-core decomposition of the transport network. Core-based accessibility metrics leverage the network topology by eliciting the hierarchy while accommodating variations like travel cost, travel time, distance, and frequency of service as edge weights. The proposed metrics quantify the accessibility of nodes at different geographical scales, ranging from local to global. We use these metrics to compute the accessibility of geographical locations connected by air transport services in India. Finally, we show that the measures are responsive to changes in the topology of the transport network by analyzing the changes in accessibility for the domestic air services network for both pre-covid and post-covid times.Keywords: integral accessibility, network topology, weighted networks, k-core decomposition, eigenvector centrality, airlines networkINTRODUCTIONAccessibility research has rapidly progressed in the last two decades ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1016/j.trd.2020.102355","ISSN":"13619209","abstract":"Since Hansen's seminal 1959 paper, measures of access have evolved with technology and through practical applications. The myriad of distinct but familially related access measures created many more options to measure places, but has sown needless confusion. This paper systematically reviews measures of access across disciplines. We categorize measures of access, covering both mainstream and innovative but less widely-used techniques; each access measure (topological, contiguous, cumulative opportunities, utility, flow-based measures) is separated into Primal and Dual measure, based on whether accessibility is represented by the level of reachable opportunities or by the travel impedance. We show access measures are unified by the same intrinsic structure consisting of travel impedance and the opportunities reachable, and that access measured by different methods are equivalent under specific assumptions on travel impedance, and the accounting of relevant opportunities. This paper overviews the use of access measures from the existing literature, and clarifies measures of access, which should help users in the selection and appropriate use of access measures for the relevant context.","author":[{"dropping-particle":"","family":"Wu","given":"Hao","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Levinson","given":"David","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Transportation Research Part D: Transport and Environment","id":"ITEM-1","issued":{"date-parts":[["2020"]]},"title":"Unifying access","type":"article-journal","volume":"83"},"uris":["http://www.mendeley.com/documents/?uuid=04481fdf-4f30-3b0e-a523-92b0fd2829b4","http://www.mendeley.com/documents/?uuid=698f78dd-0ff9-41ad-80dd-176dc3fab2a4"]}],"mendeley":{"formattedCitation":"[1]","plainTextFormattedCitation":"[1]","previouslyFormattedCitation":"(Wu & Levinson, 2020)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[1-3]. However, implementation of most of the proposed accessibility metrics is limited in practice, facing challenges in their adoption [ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1016/j.trd.2020.102319","ISSN":"13619209","abstract":"The idea that accessibility is the true goal for transportation planning goes back many decades, starting with notable writings in the 1950s, followed by a flurry of writings in the 1970s. Since then considerable academic work has gone into the development of accessibility measures, but their use in transportation planning is not yet pervasive. In this essay, I reflect on why progress toward the wholesale adoption of an accessibility-oriented approach in planning practice has been slow, review some of the promising efforts in recent years that suggest that transportation planning may be on the cusp of an accessibility era, and offer thoughts about the power of the accessibility concept as a framework for thinking about the urban environment.","author":[{"dropping-particle":"","family":"Handy","given":"Susan","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Transportation Research Part D: Transport and Environment","id":"ITEM-1","issued":{"date-parts":[["2020"]]},"title":"Is accessibility an idea whose time has finally come?","type":"article-journal","volume":"83"},"uris":["http://www.mendeley.com/documents/?uuid=78d2c427-59f4-3f2f-927a-e1d290b454b7","http://www.mendeley.com/documents/?uuid=10e7f755-d730-40f8-8e64-de21e14fa6f1"]}],"mendeley":{"formattedCitation":"[4]","plainTextFormattedCitation":"[4]","previouslyFormattedCitation":"(Handy, 2020)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}4-5]. Levine gives a critical review of the views that present obstacles in the practical implementation of accessibility metrics ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1016/j.trd.2020.102309","ISSN":"13619209","abstract":"The concept of accessibility has acquired numerous meanings along multiple dimensions during the century of its evolution. This essay argues for the salience of two dimensions: application-based and definition-based. In its application, accessibility has incorporated positive and normative dimensions which have varied in prominence over time. In its definition, accessibility has varied between a mode of evaluation incorporating measured mobility and proximity, on the one hand, and a predefined market basket of urbanist improvements to transportation and land use systems, on the other. Advocates of the accessibility shift should emphasize both the measured approach to accessibility and accessibility's normative side.","author":[{"dropping-particle":"","family":"Levine","given":"Jonathan","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Transportation Research Part D: Transport and Environment","id":"ITEM-1","issued":{"date-parts":[["2020"]]},"title":"A century of evolution of the accessibility concept","type":"article-journal","volume":"83"},"uris":["http://www.mendeley.com/documents/?uuid=fd80f6b8-beb0-3825-883d-34b9092359c4","http://www.mendeley.com/documents/?uuid=3761c6ba-04a7-4106-b499-1eaa82fa50cb"]}],"mendeley":{"formattedCitation":"[6]","plainTextFormattedCitation":"[6]","previouslyFormattedCitation":"(Levine, 2020)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[6].Traditionally, accessibility has been defined as the ease of reaching services, goods, activities, or destinations. Hansen introduced accessibility as the potential opportunity for interaction ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1080/01944365908978307","ISSN":"01944363","abstract":"An empirical examination of the residential development patterns illustrates that accessibility and the availability of vacant developable land can be used as the basis of a residential land use model. The author presents an operational definition and suggests a method for determining accessibility patterns within metropolitan areas. This is a process of distributing forecasted metropolitan population to small areas within the metropolitan region. Although the model presented is not yet sufficiently well refined for estimating purposes, the concept and the approach may be potentially useful tools for metropolitan planning purposes. © 1959 Taylor and Francis Group, LLC.","author":[{"dropping-particle":"","family":"Hansen","given":"Walter G.","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Journal of the American Planning Association","id":"ITEM-1","issue":"2","issued":{"date-parts":[["1959"]]},"title":"How Accessibility Shapes Land Use","type":"article-journal","volume":"25"},"uris":["http://www.mendeley.com/documents/?uuid=b039957f-e992-30d3-8d97-76dff7979efe","http://www.mendeley.com/documents/?uuid=7ff47dd2-013e-46b5-bc9f-5058083217f6"]}],"mendeley":{"formattedCitation":"[7]","plainTextFormattedCitation":"[7]","previouslyFormattedCitation":"(Hansen, 1959)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[7]. Since then, a plethora of accessibility definitions have been advanced in seminal papers [8-10]. Ingram defines accessibility as ”capable of being reached”, thus alluding to it as a measure of proximity between two geographical locations ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1080/09595237100185131","ISSN":"13600591","author":[{"dropping-particle":"","family":"Ingram","given":"D. R.","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Regional Studies","id":"ITEM-1","issue":"2","issued":{"date-parts":[["1971"]]},"title":"The Concept of Accessibility: A Search for an Operational Form","type":"article-journal","volume":"5"},"uris":["http://www.mendeley.com/documents/?uuid=5d5c0be6-5e08-4ac2-a0ae-133f4ae6f01d","http://www.mendeley.com/documents/?uuid=74a64b3c-abed-39bb-9cf2-63a1017c0288"]}],"mendeley":{"formattedCitation":"[9]","plainTextFormattedCitation":"[9]","previouslyFormattedCitation":"(Ingram, 1971)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[9]. Koenig asserts that accessibility is associated not only with reaching the destination but also with the quality and availability of service provided by the available transport network ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1007/BF00167128","ISSN":"00494488","abstract":"The concept of accessibility and its related indicators have been in use for a long time, with still diverging interpretations of their significance and formulation. In this paper, a review is made of various existing theoretical bases, with special emphasis on recent behavioural approaches. It is suggested that this theoretical framework now allows a better appraisal of accessibility indicators and precise recommendations are proposed for their practical formulation and use. Various examples are given, especially for disaggregate analysis where a calculation \"for a given person\" is proposed instead of the conventional calculation .\"by a given mode\". Finally, the relations between accessibility and trip rate are examined; from a study made in French cities, it is suggested that accessibility is a powerful determinant of trip rate. © 1980 Elsevier Scientific Publishing Company.","author":[{"dropping-particle":"","family":"Koenig","given":"J. G.","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Transportation","id":"ITEM-1","issue":"2","issued":{"date-parts":[["1980"]]},"title":"Indicators of urban accessibility: Theory and application","type":"article-journal","volume":"9"},"uris":["http://www.mendeley.com/documents/?uuid=271b46f7-1d1c-405b-bdf7-e1fa45edc364","http://www.mendeley.com/documents/?uuid=ba0fe758-9982-3819-8d45-2375602eb611"]}],"mendeley":{"formattedCitation":"[8]","plainTextFormattedCitation":"[8]","previouslyFormattedCitation":"(Koenig, 1980)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[8]. Geurs and Van Wee describe the accessibility of a location as a reflection of the spatial organization and quality of the transport system that enables an individual’s connectivity to the location ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1016/j.jtrangeo.2003.10.005","ISSN":"09666923","abstract":"A review of accessibility measures is presented for assessing the usability of these measures in evaluations of land-use and transport strategies and developments. Accessibility measures are reviewed using a broad range of relevant criteria, including theoretical basis, interpretability and communicability, and data requirements of the measures. Accessibility impacts of land-use and transport strategies are often evaluated using accessibility measures, which researchers and policy makers can easily operationalise and interpret, such as travelling speed, but which generally do not satisfy theoretical criteria. More complex and disaggregated accessibility measures, however, increase complexity and the effort for calculations and the difficulty of interpretation. The current practice can be much improved by operationalising more advanced location-based and utility-based accessibility measures that are still relatively easy to interpret for researchers and policy makers, and can be computed with state-of-the-practice data and/or land-use and transport models. Research directions towards theoretically more advanced accessibility measures point towards the inclusion of individual's spatial-temporal constraints and feedback mechanisms between accessibility, land-use and travel behaviour. Furthermore, there is a need for theoretical and empirical research on relationships between accessibility, option values and non-user benefits, and the measurement of different components of accessibility. © 2003 Elsevier Ltd. All rights reserved.","author":[{"dropping-particle":"","family":"Geurs","given":"Karst T.","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Wee","given":"Bert","non-dropping-particle":"van","parse-names":false,"suffix":""}],"container-title":"Journal of Transport Geography","id":"ITEM-1","issue":"2","issued":{"date-parts":[["2004"]]},"title":"Accessibility evaluation of land-use and transport strategies: Review and research directions","type":"article-journal","volume":"12"},"uris":["http://www.mendeley.com/documents/?uuid=9c8de474-6f72-4d17-a1b5-4d6997685157","http://www.mendeley.com/documents/?uuid=f705f022-78a6-363b-8829-68acb5d03e82"]}],"mendeley":{"formattedCitation":"[3]","plainTextFormattedCitation":"[3]","previouslyFormattedCitation":"(Geurs & van Wee, 2004)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[3].Given diverse definitions, accessibility is often assessed by considering the measures that evaluate the availability and quality of a transport service. Accessibility can either be relative or integral. The categorization is based on the degree of connectivity of the location with another location or all possible locations in the network. Relative accessibility is asymmetric, yet it is the simplest measure of accessibility. Integral accessibility of a location is the aggregation of its relative accessibility to all other locations in the network. Koenig highlighted two dimensions for studying accessibility in transport networks, namely transport services and activity ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1007/BF00167128","ISSN":"00494488","abstract":"The concept of accessibility and its related indicators have been in use for a long time, with still diverging interpretations of their significance and formulation. In this paper, a review is made of various existing theoretical bases, with special emphasis on recent behavioural approaches. It is suggested that this theoretical framework now allows a better appraisal of accessibility indicators and precise recommendations are proposed for their practical formulation and use. Various examples are given, especially for disaggregate analysis where a calculation \"for a given person\" is proposed instead of the conventional calculation .\"by a given mode\". Finally, the relations between accessibility and trip rate are examined; from a study made in French cities, it is suggested that accessibility is a powerful determinant of trip rate. © 1980 Elsevier Scientific Publishing Company.","author":[{"dropping-particle":"","family":"Koenig","given":"J. G.","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Transportation","id":"ITEM-1","issue":"2","issued":{"date-parts":[["1980"]]},"title":"Indicators of urban accessibility: Theory and application","type":"article-journal","volume":"9"},"uris":["http://www.mendeley.com/documents/?uuid=271b46f7-1d1c-405b-bdf7-e1fa45edc364","http://www.mendeley.com/documents/?uuid=ba0fe758-9982-3819-8d45-2375602eb611"]}],"mendeley":{"formattedCitation":"[8]","plainTextFormattedCitation":"[8]","previouslyFormattedCitation":"(Koenig, 1980)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[8]. The ease of traveling from one location to another in terms of distance, time, or cost reflects the transport service dimension of accessibility. Activity dimension, on the other hand, is measured in terms of distribution and amount of attractive activities at a location, including stores, offices, residences, etc. Handy distinguished between local and regional accessibility based on the distance between a given location and activity ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"abstract":"The question of how alternative forms of development affect travel patterns has recently been the focus of a heated debate, much of which centers on the effects of suburbanization in particular. The concept of accessibility provides an important tool for resolving this question. By measuring both the accessibility to activity within the community, or \"local\" accessibility, and the accessibility to regional centers of activity from that community, or \"regional\" accessibility, the structure of a community is more fully characterized. The research summarized uses the concepts of local and regional accessibility to test the implications for shopping travel of alternative forms of development in a case study of the San Francisco Bay Area. The results show that higher levels of both local and regional accessibility are associated with lower average shopping distances but are not associated with differences in shopping frequency. As a result, higher levels of both local and regional accessibility are associated with less total shopping travel. However, the effect of high levels of local accessibility is greatest when regional accessibility is low and vice versa. These findings suggest that policies should be directed toward enhancing both types of accessibility, but that the effects may work against each other to some degree. Copyright","author":[{"dropping-particle":"","family":"Handy","given":"Susan","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"University of California Transprtation Center","id":"ITEM-1","issued":{"date-parts":[["1993"]]},"title":"Regional Versus Local Accessibility : Implications for Nonwork Travel","type":"article-journal"},"uris":["http://www.mendeley.com/documents/?uuid=c9759733-ec6d-3d39-96a2-d5d683458624","http://www.mendeley.com/documents/?uuid=d81e9302-bb26-4b69-bf31-d4094c94614d"]}],"mendeley":{"formattedCitation":"[11]","plainTextFormattedCitation":"[11]","previouslyFormattedCitation":"(Handy, 1993)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[11].Geurs and Van Wee classified accessibility measures into four categories based on location, utility, people, and infrastructure ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1016/j.jtrangeo.2003.10.005","ISSN":"09666923","abstract":"A review of accessibility measures is presented for assessing the usability of these measures in evaluations of land-use and transport strategies and developments. Accessibility measures are reviewed using a broad range of relevant criteria, including theoretical basis, interpretability and communicability, and data requirements of the measures. Accessibility impacts of land-use and transport strategies are often evaluated using accessibility measures, which researchers and policy makers can easily operationalise and interpret, such as travelling speed, but which generally do not satisfy theoretical criteria. More complex and disaggregated accessibility measures, however, increase complexity and the effort for calculations and the difficulty of interpretation. The current practice can be much improved by operationalising more advanced location-based and utility-based accessibility measures that are still relatively easy to interpret for researchers and policy makers, and can be computed with state-of-the-practice data and/or land-use and transport models. Research directions towards theoretically more advanced accessibility measures point towards the inclusion of individual's spatial-temporal constraints and feedback mechanisms between accessibility, land-use and travel behaviour. Furthermore, there is a need for theoretical and empirical research on relationships between accessibility, option values and non-user benefits, and the measurement of different components of accessibility. © 2003 Elsevier Ltd. All rights reserved.","author":[{"dropping-particle":"","family":"Geurs","given":"Karst T.","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Wee","given":"Bert","non-dropping-particle":"van","parse-names":false,"suffix":""}],"container-title":"Journal of Transport Geography","id":"ITEM-1","issue":"2","issued":{"date-parts":[["2004"]]},"title":"Accessibility evaluation of land-use and transport strategies: Review and research directions","type":"article-journal","volume":"12"},"uris":["http://www.mendeley.com/documents/?uuid=9c8de474-6f72-4d17-a1b5-4d6997685157","http://www.mendeley.com/documents/?uuid=f705f022-78a6-363b-8829-68acb5d03e82"]}],"mendeley":{"formattedCitation":"[3]","plainTextFormattedCitation":"[3]","previouslyFormattedCitation":"(Geurs & van Wee, 2004)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[3]. Location-based measures describe the level of accessibility to spatially distributed specific activities (for example, jobs within a particular region) [ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1111/gean.12220","ISSN":"15384632","abstract":"Accessibility, measuring the ease of reaching potential destinations, is increasingly being considered as an effective indicator to evaluate the performance of transport and land use interactions. Primal accessibility, a generalization of the first accessibility formulation proposed by Hansen, has been widely used in many studies and demonstrated to be a reliable tool for project, program, and policy evaluation. The dual of accessibility, measuring the time required to reach a given number of opportunities, is less often considered but can be used for optimization in location covering-type problems. This article, hence, clarifies the definitions of primal and dual access, and applies both measures to the Minneapolis—St. Paul metropolitan area for auto and transit to demonstrate their practicality as a metropolitan-level measurement. We explore the correlations and differences between the primal and dual access to better understand the relative strengths of the measures. It is found that, as with primal accessibility, dual accessibility is an efficient approach to evaluate accessibility, which is straightforward to calculate and to explain to policy-makers and the public.","author":[{"dropping-particle":"","family":"Cui","given":"Mengying","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Levinson","given":"David","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Geographical Analysis","id":"ITEM-1","issue":"3","issued":{"date-parts":[["2020"]]},"title":"Primal and Dual Access","type":"article-journal","volume":"52"},"uris":["http://www.mendeley.com/documents/?uuid=8f640bb7-c60a-4e20-ad84-ea8ee9e53aaf","http://www.mendeley.com/documents/?uuid=8ac90146-e845-3c76-aa1f-c2605e7aaeb2"]}],"mendeley":{"formattedCitation":"[12]","plainTextFormattedCitation":"[12]","previouslyFormattedCitation":"(Cui & Levinson, 2020)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}12, ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1016/j.jtrangeo.2020.102747","ISSN":"09666923","abstract":"Uncertainties in travel times due to traffic congestion and delay are risks for drivers and public transit users. To avoid undesired consequences such as losing jobs or missing medical appointments, people can manage the risks of missing on-time arrivals to destinations using different strategies, including leaving earlier to create a safety margin and choosing routes that have more reliable rather than fastest travel times. This research develops a general analytical framework for measuring accessibility considering automobile or public transit travelers' heterogeneous strategies for dealing with travel time uncertainty. To represent different safety margin plans, we use effective travel time (expected time + safety margin), given specified on-time arrival probabilities. Heterogeneity in routing strategy is addressed using different Pareto-optimal routes with two main criteria: faster travel time vs. higher reliability. Based on various safety margin and routing strategy combinations, we examine how accessibility changes under varying safety margin plans and routing strategies. Also, we define and measure robust accessibility: geographic regions that are accessible regardless of the safety margin planning and routing strategy. Robust accessibility can provide a conservative and reasonable view of accessibility under travel time uncertainty. To demonstrate the applicability of the methods, we carry out an empirical study on measuring the impacts of new transit service on healthcare accessibility in a deprived neighborhood in Columbus, Ohio, USA.","author":[{"dropping-particle":"","family":"Lee","given":"Jinhyung","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Miller","given":"Harvey J.","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Journal of Transport Geography","id":"ITEM-1","issued":{"date-parts":[["2020"]]},"title":"Robust accessibility: Measuring accessibility based on travelers' heterogeneous strategies for managing travel time uncertainty","type":"article-journal","volume":"86"},"uris":["http://www.mendeley.com/documents/?uuid=ba9564c0-cfee-3320-bbaa-93d9904da06f","http://www.mendeley.com/documents/?uuid=6ffa4b24-67cf-47dc-93b5-3363f947dcaa"]}],"mendeley":{"formattedCitation":"[13]","plainTextFormattedCitation":"[13]","previouslyFormattedCitation":"(Lee & Miller, 2020)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}13]. Utility-based measures typically incorporate individual traveler preferences and activities at a location for quantifying accessibility. Hellervik et al. considered such activities as attractor variables for quantifying accessibility ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1177/2399808318812888","ISSN":"23998091","abstract":"The fact that accessibility shapes the geographic distribution of activity needs to be addressed in any long-term policy and planning for urban systems. One major problem is that current accessibility measures rely on the identification and quantification of attractions in the system. We propose that it is possible to devise a network centrality measure that bypasses this reliance and predicts the distribution of urban activity directly from the structure of the infrastructure networks over which interactions take place. From a basis of spatial interaction modelling and eigenvector centrality measures, we develop what we call a preferential centrality measure that recursively and self-consistently integrates activity, attraction and accessibility. Derived from the same logic as Google’s PageRank algorithm, we may describe its operation by drawing a parallel: Google’s PageRank algorithm ranks the importance of networked documents without the need to perform any analysis of their contents. Instead it considers the topological structure of the network and piggybacks thereby on contextualised and deep evaluation of documents by the myriad distributed agents that constructed the network. We do the same thing with regard to networked geographical zones. Our approach opens up new applications of modelling and promises to alleviate a host of recalcitrant problems, associated with integrated modelling, and the need for large volumes of socio-economic data. We present an initial validation of our proposed measure by using land taxation values in the Gothenburg municipality as an empirical proxy of urban activity. The resulting measure shows a promising correlation with the taxation values.","author":[{"dropping-particle":"","family":"Hellervik","given":"Alexander","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Nilsson","given":"Leonard","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Andersson","given":"Claes","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Environment and Planning B: Urban Analytics and City Science","id":"ITEM-1","issue":"7","issued":{"date-parts":[["2019"]]},"title":"Preferential centrality – A new measure unifying urban activity, attraction and accessibility","type":"article-journal","volume":"46"},"uris":["http://www.mendeley.com/documents/?uuid=99fe50f5-45bf-3d40-9f1d-18f729a88cf8","http://www.mendeley.com/documents/?uuid=c1b1207d-aa65-40eb-bee1-162f90a54708"]}],"mendeley":{"formattedCitation":"[14]","plainTextFormattedCitation":"[14]","previouslyFormattedCitation":"(Hellervik, Nilsson, & Andersson, 2019)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[14]. People-based measures consider people’s behavior by focusing on their space and time and the use of places based on the activities they are interested in ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1016/j.apgeog.2010.05.006","ISSN":"01436228","abstract":"In this paper a GIS toolkit (http://cartogis.ugent.be/sta_toolkit) is described that is able to assess the accessibility of urban opportunities to one or multiple persons. Based on information about the transportation system, urban opportunities and individual activity schedules, the toolkit provides a dynamic and animated view of the activity locations that are accessible to a person or group during the course of the day. By developing the toolkit we seek to contribute not only to the literature about space-time accessibility analysis and activity-based travel demand analysis but also to the dissemination of time geographical concepts to wider audiences. To achieve the latter, we have implemented the toolkit as a stand-alone application that is independent of commercial software and the platform used. Query results can easily be exported to Google™ Earth/Maps as well as conventional GIS software. As an illustrative example, a representative day-to-day rendezvous scenario was set up to demonstrate how meeting possibilities are shaped by various individual spatial and temporal constraints. © 2010 Elsevier Ltd.","author":[{"dropping-particle":"","family":"Neutens","given":"Tijs","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Versichele","given":"Mathias","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Schwanen","given":"Tim","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Applied Geography","id":"ITEM-1","issue":"4","issued":{"date-parts":[["2010"]]},"title":"Arranging place and time: A GIS toolkit to assess person-based accessibility of urban opportunities","type":"article-journal","volume":"30"},"uris":["http://www.mendeley.com/documents/?uuid=55dde9d3-770d-3ab3-97b8-8111c596dc4a","http://www.mendeley.com/documents/?uuid=bc9ead5c-e2bf-4c81-8e09-77ef479b26d3"]}],"mendeley":{"formattedCitation":"[15]","plainTextFormattedCitation":"[15]","previouslyFormattedCitation":"(Neutens, Versichele, & Schwanen, 2010)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[15]. Infrastructure-based measures use travel speed and level of congestion between pairs of locations to quantify accessibility between them ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.4337/9781781000106.00021","author":[{"dropping-particle":"","family":"Manaugh","given":"Kevin","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"El-Geneidy","given":"Ahmed","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Accessibility Analysis and Transport Planning: Challenges for Europe and North America","id":"ITEM-1","issued":{"date-parts":[["2012"]]},"title":"Who Benefits from New Transportation Infrastructure? Using Accessibility Measures to Evaluate Social Equity in Transit Provision","type":"article-journal"},"uris":["http://www.mendeley.com/documents/?uuid=b06b37e7-dcce-4fac-bb6f-f4ba3ef5c061"]}],"mendeley":{"formattedCitation":"[16]","plainTextFormattedCitation":"[16]","previouslyFormattedCitation":"(Manaugh & El-Geneidy, 2012)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[16]. Handy argued that in addition to the abovementioned measures, one should focus on the reachability of location and not on the means to reach it while quantifying accessibility ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1108/9780080460550-007","abstract":"Access to Destination s D.M. Levinson and K.J. Krizek (editors) © 2005 Elsevier Ltd. All rights reserved. 13 1 CHAPTER 7 PLANNING FOR ACCESSIBILITY : IN THEORY AND IN PRACTIC E Susan Handy, University of California, Davis INTRODUCTIO N Pick up a transportation plan for a major metropolitan area in the US and you are likely t o find improved mobility and accessibility highlighted as goals. The 2020 Regiona l Transportation Plan for the Austin region, for example, stated that \"The primary goal of th e CAMPO 2020 Plan is to provide an acceptable level of mobility and accessibility for th e region's residents with the least detrimental effects.\" The 2020 Regional Transportation Pla n for the Chicago region aimed to \"provide an integrated and coordinated transportation syste m that maximizes accessibility and includes a variety of mobility options that serve the needs o f residents and businesses in the region.\" Such statements are likely influenced by th e Transportation Equity Act for the 21st Century, commonly known as TEA-21, whic h established seven \"planning factors\" for consideration in the planning process, includin g \"increase the accessibility and mobility options available to people and for freight.","author":[{"dropping-particle":"","family":"Handy","given":"Susan","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Access to Destinations","id":"ITEM-1","issued":{"date-parts":[["2005"]]},"title":"Planning for Accessibility: In Theory and in Practice","type":"chapter"},"uris":["http://www.mendeley.com/documents/?uuid=bdc9e073-6aa8-3165-ab28-d3d08900db9d","http://www.mendeley.com/documents/?uuid=b112d05c-c2cd-4c74-9491-7063757e2b5c"]}],"mendeley":{"formattedCitation":"[17]","plainTextFormattedCitation":"[17]","previouslyFormattedCitation":"(Handy, 2005)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[17]. Affirming the argument, Wu and Levinson emphasized that distance between locations is nonexistent unless they are connected by a network and invigorates the study of accessibility measures based on the topology of the transport network ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1016/j.trd.2020.102355","ISSN":"13619209","abstract":"Since Hansen's seminal 1959 paper, measures of access have evolved with technology and through practical applications. The myriad of distinct but familially related access measures created many more options to measure places, but has sown needless confusion. This paper systematically reviews measures of access across disciplines. We categorize measures of access, covering both mainstream and innovative but less widely-used techniques; each access measure (topological, contiguous, cumulative opportunities, utility, flow-based measures) is separated into Primal and Dual measure, based on whether accessibility is represented by the level of reachable opportunities or by the travel impedance. We show access measures are unified by the same intrinsic structure consisting of travel impedance and the opportunities reachable, and that access measured by different methods are equivalent under specific assumptions on travel impedance, and the accounting of relevant opportunities. This paper overviews the use of access measures from the existing literature, and clarifies measures of access, which should help users in the selection and appropriate use of access measures for the relevant context.","author":[{"dropping-particle":"","family":"Wu","given":"Hao","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Levinson","given":"David","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Transportation Research Part D: Transport and Environment","id":"ITEM-1","issued":{"date-parts":[["2020"]]},"title":"Unifying access","type":"article-journal","volume":"83"},"uris":["http://www.mendeley.com/documents/?uuid=698f78dd-0ff9-41ad-80dd-176dc3fab2a4","http://www.mendeley.com/documents/?uuid=04481fdf-4f30-3b0e-a523-92b0fd2829b4"]}],"mendeley":{"formattedCitation":"[1]","plainTextFormattedCitation":"[1]","previouslyFormattedCitation":"(Wu & Levinson, 2020)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[1].It is well established that network topology augments transport and activity dimension and provides a holistic picture of the accessibility of a geographic location. It defines the physical connectivity between two geographical locations and is known to influence accessibility significantly [18-20]. Hierarchy is a well-recognized topological characteristic that reveals the organization of a network. However, its potential has been recently realized for the analysis of transport networks[21-24].Motivation and contributionAccording to Koenig, transport services and activities are the two dimensions of transport networks ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1007/BF00167128","ISSN":"00494488","abstract":"The concept of accessibility and its related indicators have been in use for a long time, with still diverging interpretations of their significance and formulation. In this paper, a review is made of various existing theoretical bases, with special emphasis on recent behavioural approaches. It is suggested that this theoretical framework now allows a better appraisal of accessibility indicators and precise recommendations are proposed for their practical formulation and use. Various examples are given, especially for disaggregate analysis where a calculation \"for a given person\" is proposed instead of the conventional calculation .\"by a given mode\". Finally, the relations between accessibility and trip rate are examined; from a study made in French cities, it is suggested that accessibility is a powerful determinant of trip rate. © 1980 Elsevier Scientific Publishing Company.","author":[{"dropping-particle":"","family":"Koenig","given":"J. G.","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Transportation","id":"ITEM-1","issue":"2","issued":{"date-parts":[["1980"]]},"title":"Indicators of urban accessibility: Theory and application","type":"article-journal","volume":"9"},"uris":["http://www.mendeley.com/documents/?uuid=271b46f7-1d1c-405b-bdf7-e1fa45edc364","http://www.mendeley.com/documents/?uuid=ba0fe758-9982-3819-8d45-2375602eb611"]}],"mendeley":{"formattedCitation":"[8]","plainTextFormattedCitation":"[8]","previouslyFormattedCitation":"(Koenig, 1980)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[8]. We advocate the study of accessibility measures in transport networks along the third dimension, that is, network topology. Our argument is grounded on several existing studies emphasizing that connectivity in transport networks influences accessibility [ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1016/j.trd.2020.102355","ISSN":"13619209","abstract":"Since Hansen's seminal 1959 paper, measures of access have evolved with technology and through practical applications. The myriad of distinct but familially related access measures created many more options to measure places, but has sown needless confusion. This paper systematically reviews measures of access across disciplines. We categorize measures of access, covering both mainstream and innovative but less widely-used techniques; each access measure (topological, contiguous, cumulative opportunities, utility, flow-based measures) is separated into Primal and Dual measure, based on whether accessibility is represented by the level of reachable opportunities or by the travel impedance. We show access measures are unified by the same intrinsic structure consisting of travel impedance and the opportunities reachable, and that access measured by different methods are equivalent under specific assumptions on travel impedance, and the accounting of relevant opportunities. This paper overviews the use of access measures from the existing literature, and clarifies measures of access, which should help users in the selection and appropriate use of access measures for the relevant context.","author":[{"dropping-particle":"","family":"Wu","given":"Hao","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Levinson","given":"David","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Transportation Research Part D: Transport and Environment","id":"ITEM-1","issued":{"date-parts":[["2020"]]},"title":"Unifying access","type":"article-journal","volume":"83"},"uris":["http://www.mendeley.com/documents/?uuid=698f78dd-0ff9-41ad-80dd-176dc3fab2a4","http://www.mendeley.com/documents/?uuid=04481fdf-4f30-3b0e-a523-92b0fd2829b4"]}],"mendeley":{"formattedCitation":"[1]","plainTextFormattedCitation":"[1]","previouslyFormattedCitation":"(Wu & Levinson, 2020)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}1, ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1016/j.jtrangeo.2017.06.002","ISSN":"09666923","abstract":"The structure of transport networks is the outcome of a large number of infrastructure investment decisions taken over a long time span. Network indicators are widely used for characterizing transport network topology and its performance as well as provide insights on possible developments. Little is known however on how rail bound public transport networks and their network indicators have evolved into their current form. This study conducts a longitudinal analysis of the topological evolution of a multimodal rail network by investigating the dynamics of its topology for the case of Stockholm in 1950–2025. The starting year marks the opening of the metro system while the end year is set to mark the completion of the current development plan. Based on a compilation of network topology and service properties, a year-on-year analysis of changes in global network efficiency and directness as well as local nodal centrality were conducted. Changes in network topology exhibit smooth long-term technological and spatial trends as well as the signature of top-down planning interventions. Stockholm rail network evolution is characterized by contraction and stagnation periods followed by network extensions and is currently undergoing a considerable densification, marking a shift from peripheral attachment to preferential attachment.","author":[{"dropping-particle":"","family":"Cats","given":"Oded","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Journal of Transport Geography","id":"ITEM-1","issued":{"date-parts":[["2017"]]},"title":"Topological evolution of a metropolitan rail transport network: The case of Stockholm","type":"article-journal","volume":"62"},"uris":["http://www.mendeley.com/documents/?uuid=8114927a-85e6-45b9-a515-8a9617a0adb4","http://www.mendeley.com/documents/?uuid=2e0c5fc9-0c42-3f7c-aae6-a7d80449492f"]}],"mendeley":{"formattedCitation":"[19]","plainTextFormattedCitation":"[19]","previouslyFormattedCitation":"(Cats, 2017)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}19, ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.3390/su11195404","ISSN":"20711050","abstract":"Many cities have integrated their public transportation modes to provide increased accessibility and reduced commute times. However, current transport network topology studies have focused on unimodal networks. Therefore, it is of significant interest for policymakers to examine the topology of integrated public transportation networks and to assess strategies for improving them. The objective of this study was to discuss a comprehensive analysis of an integrated public transportation network using graph theory, compare its characteristics to unimodal networks, and draw insights for improving their performance. Results demonstrate pertinent information concerning the structural composition of the Seoul Metropolitan Area's (SMA) public transportation network. Despite the integration, the spatial configuration of the network was found to have low fault tolerance. However, the highly agglomerated community structure validated the robustness of integrated networks. Network centrality measures confirmed that integration improves connectivity and spatial accessibility to suburbs within the city. The study found that the SMA's current public transportation network possesses structural defects that need to be addressed to improve its resilience and performance. Based on the outcomes of this study, the strategic creation or relocation of stations, and the construction of more links, is imperative for the enhancement of mobility.","author":[{"dropping-particle":"","family":"Hong","given":"Jungyeol","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Tamakloe","given":"Reuben","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Lee","given":"Soobeom","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Park","given":"Dongjoo","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Sustainability (Switzerland)","id":"ITEM-1","issue":"19","issued":{"date-parts":[["2019"]]},"title":"Exploring the topological characteristics of complex public transportation networks: Focus on variations in both single and integrated systems in the Seoul Metropolitan Area","type":"article-journal","volume":"11"},"uris":["http://www.mendeley.com/documents/?uuid=487500a9-59bd-4f17-902d-511effa1d4e5","http://www.mendeley.com/documents/?uuid=28a81e6c-98d2-3dc5-9808-40288dc13c0e"]}],"mendeley":{"formattedCitation":"[20]","plainTextFormattedCitation":"[20]","previouslyFormattedCitation":"(Hong et al., 2019)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}20]. We advance this view further and assert that the accessibility of a geographic location is additionally impacted by its position in the hierarchical decomposition of the transport network.We propose novel accessibility metrics based on an inherent hierarchy of nodes in the network topology revealed by −core decomposition. Since the algorithm for -core decomposition is linear in the number of edges, accessibility for large networks (for example, nationwide, continent-wide or global transport networks) can be computed efficiently. The measures are flexible and proficient at capturing the combined effect of opportunities and reachability. They can also accommodate attributes like travel cost, travel time, distance, frequency of service, etc., that arise naturally in the analysis of transport networks. These attributes can be assigned as edge weights in the network to yield an objective accessibility metric. When new services are added or existing ones are withdrawn, the accessibility of nodes is impacted. The proposed metrics are responsive to changes in the topology of the transport network.The proposed core-based accessibility measures consider the density of connections in the immediate neighborhood, two-hop neighborhood, that is, region, and the entire network. The intuition that density-differential of transport connections for a geographical location is an essential determinant of accessibility is well captured by core-based accessibility metrics. To the best of the authors’ knowledge, this is the first attempt to design accessibility metrics that can be computed and interpreted at multiple granularity levels while exploring the topological hierarchy of the transport network.The rest of the paper is organized as follows. Section 2 presents the background, explains the -core decomposition of a complex network, and graph-theoretic measures related to accessibility research. Then in Section 3, we propose and explain the core-based accessibility measures. We present applications of the proposed accessibility measures on the test network in Section 4 and a case study to compare the accessibility of Indian airports connected by domestic airlines in the pre-covid and post-covid times in Section 5. Finally, we conclude the paper.BACKGROUND AND RELATED WORKSRecently, complex networks have attracted serious attention for the analysis of urban and national level public transport systems. A complex network is an abstract mathematical structure , where is the set of nodes and is the set of edges. There exists an edge between , if and only if they are directly related. The definition of relation is specific to the application. The network is represented by an adjacency matrix , where if , else . In case the relation between two nodes is directional, the adjacency matrix is asymmetric. Further, if the strength of the relation is quantified, denotes the weight of the edge . Unless specified, we assume a network to be undirected and unweighted.Example 1: We model the transport system as a simple, weighted and undirected toy network , as shown in REF _Ref114533435 \h \* MERGEFORMAT Fig. 1, where is the set of 11 geographical locations. The edge weight represents the connection strength between and (distance, time to travel, or cost of travel etc.). REF _Ref111541465 \h \* MERGEFORMAT Tab. 1 shows the weighted and symmetric adjacency matrix for . Fig. SEQ Figure \* ARABIC 1. Toy transport network with 11 nodes and 15 edgesTab. SEQ Table \* ARABIC 1Weighted adjacency matrix for toy network in REF _Ref114533435 \h \* MERGEFORMAT Fig. 1Node000080000000010744012000010011300300007110702004084370170000004001700000000020000000012300000700000000070050004000000000000000500Topology-based accessibility measures The topology of a network reveals the patterns of connectivity between geographical locations. Researchers have employed degree, closeness and betweenness centrality measures to quantify the accessibility in transport networks [ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1016/j.jtrangeo.2017.06.002","ISSN":"09666923","abstract":"The structure of transport networks is the outcome of a large number of infrastructure investment decisions taken over a long time span. Network indicators are widely used for characterizing transport network topology and its performance as well as provide insights on possible developments. Little is known however on how rail bound public transport networks and their network indicators have evolved into their current form. This study conducts a longitudinal analysis of the topological evolution of a multimodal rail network by investigating the dynamics of its topology for the case of Stockholm in 1950–2025. The starting year marks the opening of the metro system while the end year is set to mark the completion of the current development plan. Based on a compilation of network topology and service properties, a year-on-year analysis of changes in global network efficiency and directness as well as local nodal centrality were conducted. Changes in network topology exhibit smooth long-term technological and spatial trends as well as the signature of top-down planning interventions. Stockholm rail network evolution is characterized by contraction and stagnation periods followed by network extensions and is currently undergoing a considerable densification, marking a shift from peripheral attachment to preferential attachment.","author":[{"dropping-particle":"","family":"Cats","given":"Oded","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Journal of Transport Geography","id":"ITEM-1","issued":{"date-parts":[["2017"]]},"title":"Topological evolution of a metropolitan rail transport network: The case of Stockholm","type":"article-journal","volume":"62"},"uris":["http://www.mendeley.com/documents/?uuid=8114927a-85e6-45b9-a515-8a9617a0adb4"]}],"mendeley":{"formattedCitation":"[19]","plainTextFormattedCitation":"[19]","previouslyFormattedCitation":"(Cats, 2017)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}19, ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.3390/su11195404","ISSN":"20711050","abstract":"Many cities have integrated their public transportation modes to provide increased accessibility and reduced commute times. However, current transport network topology studies have focused on unimodal networks. Therefore, it is of significant interest for policymakers to examine the topology of integrated public transportation networks and to assess strategies for improving them. The objective of this study was to discuss a comprehensive analysis of an integrated public transportation network using graph theory, compare its characteristics to unimodal networks, and draw insights for improving their performance. Results demonstrate pertinent information concerning the structural composition of the Seoul Metropolitan Area's (SMA) public transportation network. Despite the integration, the spatial configuration of the network was found to have low fault tolerance. However, the highly agglomerated community structure validated the robustness of integrated networks. Network centrality measures confirmed that integration improves connectivity and spatial accessibility to suburbs within the city. The study found that the SMA's current public transportation network possesses structural defects that need to be addressed to improve its resilience and performance. Based on the outcomes of this study, the strategic creation or relocation of stations, and the construction of more links, is imperative for the enhancement of mobility.","author":[{"dropping-particle":"","family":"Hong","given":"Jungyeol","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Tamakloe","given":"Reuben","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Lee","given":"Soobeom","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Park","given":"Dongjoo","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Sustainability (Switzerland)","id":"ITEM-1","issue":"19","issued":{"date-parts":[["2019"]]},"title":"Exploring the topological characteristics of complex public transportation networks: Focus on variations in both single and integrated systems in the Seoul Metropolitan Area","type":"article-journal","volume":"11"},"uris":["http://www.mendeley.com/documents/?uuid=487500a9-59bd-4f17-902d-511effa1d4e5"]}],"mendeley":{"formattedCitation":"[20]","plainTextFormattedCitation":"[20]","previouslyFormattedCitation":"(Hong et al., 2019)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}20, ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1016/j.jtrangeo.2010.08.012","ISSN":"09666923","abstract":"This paper uses a complex network approach to examine the network structure and nodal centrality of individual cities in the air transport network of China (ATNC). Measures for overall network structure include degree distribution, average path length and clustering coefficient. Centrality metrics for individual cities are degree, closeness and betweenness, representing a node's location advantage as being directly connected to others, being accessible to others, and being the intermediary between others, respectively. Results indicate that the ATNC has a cumulative degree distribution captured by an exponential function, and displays some small-world (SW) network properties with an average path length of 2.23 and a clustering coefficient of 0.69. All three centrality indices are highly correlated with socio-economic indicators of cities such as air passenger volume, population, and gross regional domestic product (GRDP). This confirms that centrality captures a crucial aspect of location advantage in the ATNC and has important implications in shaping the spatial pattern of economic activities. Most small and low-degree airports are directly connected to the largest cities with the best centrality and bypass their regional centers, and therefore sub-networks in the ATNC are less developed except for Kunming in the southwest and Urumchi in the northwest because of their strategic locations for geographic and political reasons. The ANTC is relatively young, and not as efficient and well-developed as that of the US. © 2010 Elsevier Ltd.","author":[{"dropping-particle":"","family":"Wang","given":"Jiaoe","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Mo","given":"Huihui","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Wang","given":"Fahui","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Jin","given":"Fengjun","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Journal of Transport Geography","id":"ITEM-1","issue":"4","issued":{"date-parts":[["2011"]]},"title":"Exploring the network structure and nodal centrality of China's air transport network: A complex network approach","type":"article-journal","volume":"19"},"uris":["http://www.mendeley.com/documents/?uuid=729f9b78-ab27-400b-8dcd-20e55fa969b8"]}],"mendeley":{"formattedCitation":"[25]","plainTextFormattedCitation":"[25]","previouslyFormattedCitation":"(J. Wang, Mo, Wang, & Jin, 2011)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}25-27].Degree has been used to determine important nodes in transport networks ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1063/1.4982546","ISSN":"15517616","abstract":"Based on the actual data of urban transport in Guangzhou, 19,150 bus stations in Guangzhou (as of 2014) are selected as nodes. Based on the theory of complex network, the network model of Guangzhou urban transport is constructed. By analyzing the degree centrality index, betweenness centrality index and closeness centrality index of nodes in the network, the level of centrality of each node in the network is studied. From a different point of view to determine the hub node of Guangzhou urban transport network, corresponding to the city's key sites and major transfer sites. The reliability of the network is determined by the stability of some key nodes (transport hub station). The research of network node centralization can provide a theoretical basis for the rational allocation of urban transport network sites and public transport system planning.","author":[{"dropping-particle":"","family":"Wang","given":"Kui","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Fu","given":"Xiufen","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"AIP Conference Proceedings","id":"ITEM-1","issued":{"date-parts":[["2017"]]},"title":"Research on centrality of urban transport network nodes","type":"paper-conference","volume":"1839"},"uris":["http://www.mendeley.com/documents/?uuid=d5c5f2ee-3789-4cd3-8e9e-7610e64994cd"]}],"mendeley":{"formattedCitation":"[26]","plainTextFormattedCitation":"[26]","previouslyFormattedCitation":"(K. Wang & Fu, 2017)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[26]. Since the degree is a local property of the node, it indicates connectivity in the immediate neighborhood of a geographical location. However, it may not be an accurate predictor of accessibility as it does not indicate the extent of connectivity of neighbor nodes. For instance, in REF _Ref114533435 \h \* MERGEFORMAT Fig. 1, nodesand have the same degree butis more accessible due to its connections with better connected nodes (and) in the network than.Closeness centrality of a node is calculated as the reciprocal of the sum of the shortest path length from node to other nodes in the network, as given below. ( SEQ ( \* ARABIC 1)The measure indicates the importance of a node by quantifying its closeness to other nodes in the network. Nodes with higher scores are easier to reach than other nodes in the network and thus reflect higher accessibility. Betweenness centrality measures the extent to which a node lies on paths between other nodes. It is computed as the sum of the fraction of shortest paths between all pairs of nodes passing through vi, as shown below. ( SEQ ( \* ARABIC 2)Here, is the number of the shortest paths between the nodes passing through node, andis the number of paths between nodes and in the network.Betweenness centrality is useful to identify nodes that serve as a bridge or transfer points to other nodes in the network. Removal of these nodes from the network may disrupt reachability. Betweenness centrality is augmented with other measurable attributes of the geographical location to quantify betweenness-accessibility ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1016/j.jtrangeo.2020.102680","ISSN":"09666923","abstract":"Accessibility is a central concept in transport geography research that has been described as a holistic measure of transportation and land use systems. This concept has numerous implementations, but virtually all share the way accessibility is measured as an attribute of pairs of origins and destinations. Borrowing from concepts in network science, in this paper we propose a new centrality measure called betweenness-accessibility. This measure couples the familiar betweenness indicator of social networks with the geographical concept of accessibility. Betweenness-accessibility is useful to estimate the impacts of accessibility on networks as potential for interaction is reflected on them. The new measure is illustrated using a reproducible example. In addition, an application to vulnerability analysis in the city of Zurich, Switzerland, provides an empirical case study to showcase the usefulness of betweenness-accessibility.","author":[{"dropping-particle":"","family":"Sarlas","given":"Georgios","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Páez","given":"Antonio","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Axhausen","given":"Kay W","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Journal of Transport Geography","id":"ITEM-1","issued":{"date-parts":[["2020"]]},"title":"Betweenness-accessibility: Estimating impacts of accessibility on networks","type":"article-journal","volume":"84"},"uris":["http://www.mendeley.com/documents/?uuid=641a4bbf-687b-4581-9143-24673296077f"]}],"mendeley":{"formattedCitation":"[27]","plainTextFormattedCitation":"[27]","previouslyFormattedCitation":"(Sarlas et al., 2020)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[27]. Betweenness centrality of terminal nodes is zero, due to which the betweenness-accessibility of such nodes may be zero.Hierarchical decomposition: preliminariesComplex networks can be decomposed into a hierarchy of sub-networks based on the connectivity of nodes, such that the densest sub-network is at the top. Hierarchy is recognized as an important principle of organization in complex networks and is observed in all real-life networks ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1103/PhysRevE.67.026112","ISSN":"1063651X","abstract":"Many real networks in nature and society share two generic properties: they are scale-free and they display a high degree of clustering. We show that these two features are the consequence of a hierarchical organization, implying that small groups of nodes organize in a hierarchical manner into increasingly large groups, while maintaining a scale-free topology. In hierarchical networks, the degree of clustering characterizing the different groups follows a strict scaling law, which can be used to identify the presence of a hierarchical organization in real networks. We find that several real networks, such as the Worldwideweb, actor network, the Internet at the domain level, and the semantic web obey this scaling law, indicating that hierarchy is a fundamental characteristic of many complex systems. © 2003 The American Physical Society.","author":[{"dropping-particle":"","family":"Ravasz","given":"Erzsébet","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Barabási","given":"Albert László","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics","id":"ITEM-1","issue":"2","issued":{"date-parts":[["2003"]]},"title":"Hierarchical organization in complex networks","type":"article-journal","volume":"67"},"uris":["http://www.mendeley.com/documents/?uuid=afe2602d-d89d-4c65-9b4e-a6263c092190"]}],"mendeley":{"formattedCitation":"[28]","plainTextFormattedCitation":"[28]","previouslyFormattedCitation":"(Ravasz & Barabási, 2003)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[28]. Examples of hierarchy span from social networks [ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1103/PhysRevE.76.046118","ISSN":"15393755","abstract":"Complex networks emerge under different conditions including design (i.e., top-down decisions) through simple rules of growth and evolution. Such rules are typically local when dealing with biological systems and most social webs. An important deviation from such a scenario is provided by groups, collectives of agents engaged in technology development, such as open-source communities. Here we analyze their network structure, showing that it defines a complex weighted network with scaling laws at different levels, as measured by looking at e-mail exchanges. We also present a simple model of network growth involving nonlocal rules based on betweenness centrality. Our weighted network analysis suggests that a well-defined interplay between the overall goals of the community and the underlying hierarchical organization play a key role in shaping its dynamics. © 2007 The American Physical Society.","author":[{"dropping-particle":"","family":"Valverde","given":"Sergi","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"V","family":"Solé","given":"Ricard","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Physical Review E - Statistical, Nonlinear, and Soft Matter Physics","id":"ITEM-1","issue":"4","issued":{"date-parts":[["2007"]]},"title":"Self-organization versus hierarchy in open-source social networks","type":"article-journal","volume":"76"},"uris":["http://www.mendeley.com/documents/?uuid=55b1ccde-0707-4fc3-bcc8-8f9abc3e07ed"]}],"mendeley":{"formattedCitation":"[29]","plainTextFormattedCitation":"[29]","previouslyFormattedCitation":"(Valverde & Solé, 2007)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[29], ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1109/TKDE.2016.2546243","ISSN":"10414347","abstract":"Social hierarchy (i.e., pyramid structure of societies) is a fundamental concept in sociology and social network analysis. The importance of social hierarchy in a social network is that the topological structure of the social hierarchy is essential in both shaping the nature of social interactions between individuals and unfolding the structure of the social networks. The social hierarchy found in a social network can be utilized to improve the accuracy of link prediction, provide better query results, rank web pages, and study information flow and spread in complex networks. In this paper, we model a social network as a directed graph G, and consider the social hierarchy as DAG (directed acyclic graph) of G, denoted as GD. By DAG, all the vertices in G can be partitioned into different levels, the vertices at the same level represent a disjoint group in the social hierarchy, and all the edges in DAG follow one direction. The main issue we study in this paper is how to find DAG GD in G. The approach we take is to find GD by removing all possible cycles from G such that G = u(G) ∪ GD, where u(G) is a maximum Eulerian subgraph which contains all possible cycles. We give the reasons for doing so, investigate the properties of GD found, and discuss the applications. In addition, we develop a novel two-phase algorithm, called Greedy-&-Refine, which greedily computes an Eulerian subgraph and then refines this greedy solution to find the maximum Eulerian subgraph. We give a bound between the greedy solution and the optimal. The quality of our greedy approach is high. We conduct comprehensive experimental studies over 14 real-world datasets. The results show that our algorithms are at least two orders of magnitude faster than the baseline algorithm.","author":[{"dropping-particle":"","family":"Lu","given":"Can","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Yu","given":"Jeffrey Xu","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Li","given":"Rong Hua","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Wei","given":"Hao","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"IEEE Transactions on Knowledge and Data Engineering","id":"ITEM-1","issue":"8","issued":{"date-parts":[["2016"]]},"title":"Exploring hierarchies in online social networks","type":"article-journal","volume":"28"},"uris":["http://www.mendeley.com/documents/?uuid=aa64365a-b3b6-4080-bffb-2cdbba22e1d0"]}],"mendeley":{"formattedCitation":"[30]","plainTextFormattedCitation":"[30]","previouslyFormattedCitation":"(Lu, Yu, Li, & Wei, 2016)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[30]], biological networks [ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1186/1471-2105-13-250","ISSN":"14712105","abstract":"Background: Transcription factors regulate numerous cellular processes by controlling the rate of production of each gene. The regulatory relations are modeled using transcriptional regulatory networks. Recent studies have shown that such networks have an underlying hierarchical organization. We consider the problem of discovering the underlying hierarchy in transcriptional regulatory networks.Results: We first transform this problem to a mixed integer programming problem. We then use existing tools to solve the resulting problem. For larger networks this strategy does not work due to rapid increase in running time and space usage. We use divide and conquer strategy for such networks. We use our method to analyze the transcriptional regulatory networks of E. coli, H. sapiens and S. cerevisiae.Conclusions: Our experiments demonstrate that: (i) Our method gives statistically better results than three existing state of the art methods; (ii) Our method is robust against errors in the data and (iii) Our method's performance is not affected by the different topologies in the data. © 2012 Gülsoy et al.; licensee BioMed Central Ltd.","author":[{"dropping-particle":"","family":"Gülsoy","given":"Günhan","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Bandhyopadhyay","given":"Nirmalya","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Kahveci","given":"Tamer","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"BMC Bioinformatics","id":"ITEM-1","issue":"1","issued":{"date-parts":[["2012"]]},"title":"HIDEN: Hierarchical decomposition of regulatory networks","type":"article-journal","volume":"13"},"uris":["http://www.mendeley.com/documents/?uuid=cd44590d-afe4-473d-9dcb-3eb97722774c"]}],"mendeley":{"formattedCitation":"[31]","plainTextFormattedCitation":"[31]","previouslyFormattedCitation":"(Gülsoy, Bandhyopadhyay, & Kahveci, 2012)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}31, ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1371/journal.pcbi.1004829","ISSN":"15537358","abstract":"Hierarchical organization—the recursive composition of sub-modules—is ubiquitous in biological networks, including neural, metabolic, ecological, and genetic regulatory networks, and in human-made systems, such as large organizations and the Internet. To date, most research on hierarchy in networks has been limited to quantifying this property. However, an open, important question in evolutionary biology is why hierarchical organization evolves in the first place. It has recently been shown that modularity evolves because of the presence of a cost for network connections. Here we investigate whether such connection costs also tend to cause a hierarchical organization of such modules. In computational simulations, we find that networks without a connection cost do not evolve to be hierarchical, even when the task has a hierarchical structure. However, with a connection cost, networks evolve to be both modular and hierarchical, and these networks exhibit higher overall performance and evolvability (i.e. faster adaptation to new environments). Additional analyses confirm that hierarchy independently improves adaptability after controlling for modularity. Overall, our results suggest that the same force–the cost of connections–promotes the evolution of both hierarchy and modularity, and that these properties are important drivers of network performance and adaptability. In addition to shedding light on the emergence of hierarchy across the many domains in which it appears, these findings will also accelerate future research into evolving more complex, intelligent computational brains in the fields of artificial intelligence and robotics.","author":[{"dropping-particle":"","family":"Mengistu","given":"Henok","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Huizinga","given":"Joost","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Mouret","given":"Jean Baptiste","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Clune","given":"Jeff","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"PLoS Computational Biology","id":"ITEM-1","issue":"6","issued":{"date-parts":[["2016"]]},"title":"The Evolutionary Origins of Hierarchy","type":"article-journal","volume":"12"},"uris":["http://www.mendeley.com/documents/?uuid=c63d8390-f41f-4499-b951-a17cecbf7a14"]}],"mendeley":{"formattedCitation":"[32]","plainTextFormattedCitation":"[32]","previouslyFormattedCitation":"(Mengistu, Huizinga, Mouret, & Clune, 2016)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}32], technological networks ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1016/j.jmsy.2018.03.002","ISSN":"02786125","abstract":"Manufacturing systems are becoming increasingly complex as more advanced and emerging technologies are integrated into the factory floor to yield new processes or increase the efficiency of existing processes. As greater complexity is formed across the factory, new relationships are often generated that can lead to advanced capabilities, yet produce unforeseen faults and failures. Industrial robot arm work cells within the manufacturing environment present increasing complexity, emergent technologies, new relationships, and unpredicted faults/failures. To maintain required levels of productivity, process quality, and asset availability, manufacturers must reconcile this complexity to understand how the health degradation of constituent physical elements and functional tasks impact one another through the monitoring of critical informative measures and metrics. This article presents the initial efforts in developing a novel hierarchical decomposition methodology. The innovation in this method is that it provides the manufacturer with sufficient discretion to physically deconstruct their system and functionally decompose their process to user-defined levels based upon desired monitoring, maintenance, and control levels. This enables the manufacturer to specify relationships within and across the physical, functional, and information domains to identify impactful health degradations without having to know all possible failure modes. The hierarchical decomposition methodology will advance the state of the art in terms of improving machine health by highlighting how health degradations propagate through the relationship network prior to a piece of equipment compromising the productivity or quality of a process. The first two steps of the methodology, physical decomposition and functional decomposition, are defined in detail and applied to a multi-robot work cell use case.","author":[{"dropping-particle":"","family":"Weiss","given":"Brian A","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Sharp","given":"Michael","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Klinger","given":"Alexander","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Journal of Manufacturing Systems","id":"ITEM-1","issued":{"date-parts":[["2018"]]},"title":"Developing a hierarchical decomposition methodology to increase manufacturing process and equipment health awareness","type":"article-journal","volume":"48"},"uris":["http://www.mendeley.com/documents/?uuid=578f0b6a-ef4d-4b31-882b-9042879e1129"]}],"mendeley":{"formattedCitation":"[33]","plainTextFormattedCitation":"[33]","previouslyFormattedCitation":"(Weiss, Sharp, & Klinger, 2018)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[33], and urban systems ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"author":[{"dropping-particle":"","family":"Raimbault","given":"Juste","non-dropping-particle":"","parse-names":false,"suffix":""}],"id":"ITEM-1","issued":{"date-parts":[["2020"]]},"title":"Hierarchy and co-evolution processes in urban systems","type":"article"},"uris":["http://www.mendeley.com/documents/?uuid=9106adb7-c8d2-44f2-a290-9fea971cb6bf"]}],"mendeley":{"formattedCitation":"[34]","plainTextFormattedCitation":"[34]","previouslyFormattedCitation":"(Raimbault, 2020)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[34] to transport networks [ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1016/j.tre.2016.03.009","ISSN":"13665545","abstract":"This paper encapsulates the Chinese Airline Network (CAN) into multi-layer infrastructures via the \"k-core decomposition\" method. The network is divided into three layers: Core layer, containing airports of provincial capital cities, is densely connected and sustains most flight flow; Bridge layer, consisting of airports in Tier 2 and Tier 3 cities, mainly connects two other layers; and Periphery layer, comprising airports of remote areas, sustains little flight flow. Moreover, it is unveiled that CAN stays the most robust when low-degree nodes or high flight flow links are removed, which is similar to the Worldwide Airline Network (WAN), albeit less redundant.","author":[{"dropping-particle":"","family":"Du","given":"Wen Bo","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Zhou","given":"Xing Lian","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Lordan","given":"Oriol","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Wang","given":"Zhen","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Zhao","given":"Chen","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Zhu","given":"Yan Bo","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Transportation Research Part E: Logistics and Transportation Review","id":"ITEM-1","issued":{"date-parts":[["2016"]]},"title":"Analysis of the Chinese Airline Network as multi-layer networks","type":"article-journal","volume":"89"},"uris":["http://www.mendeley.com/documents/?uuid=48d9415b-9e8c-41b9-b81d-c1e1096b96d7"]}],"mendeley":{"formattedCitation":"[22]","plainTextFormattedCitation":"[22]","previouslyFormattedCitation":"(Du et al., 2016)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}22-27], ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1103/PhysRevE.90.032816","ISSN":"15502376","abstract":"We generalize the theory of k-core percolation on complex networks to k-core percolation on multiplex networks, where k≡(k1,k2,...,kM). Multiplex networks can be defined as networks with vertices of one kind but M different types of edges, representing different types of interactions. For such networks, the k-core is defined as the largest subgraph in which each vertex has at least ki edges of each type, i=1,2,...,M. We derive self-consistency equations to obtain the birth points of the k-cores and their relative sizes for uncorrelated multiplex networks with an arbitrary degree distribution. To clarify our general results, we consider in detail multiplex networks with edges of two types and solve the equations in the particular case of Erdos-Rényi and scale-free multiplex networks. We find hybrid phase transitions at the emergence points of k-cores except the (1,1)-core for which the transition is continuous. We apply the k-core decomposition algorithm to air-transportation multiplex networks, composed of two layers, and obtain the size of (k1,k2)-cores.","author":[{"dropping-particle":"","family":"Azimi-Tafreshi","given":"N","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Gómez-Gardeñes","given":"J","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Dorogovtsev","given":"S N","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Physical Review E - Statistical, Nonlinear, and Soft Matter Physics","id":"ITEM-1","issue":"3","issued":{"date-parts":[["2014"]]},"title":"K-core percolation on multiplex networks","type":"article-journal","volume":"90"},"uris":["http://www.mendeley.com/documents/?uuid=2b8db888-dc5e-4e27-8cc4-094d41fae640"]}],"mendeley":{"formattedCitation":"[35]","plainTextFormattedCitation":"[35]","previouslyFormattedCitation":"(Azimi-Tafreshi, Gómez-Gardeñes, & Dorogovtsev, 2014)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[35]. We unravel the hierarchy of nodes (geographical locations) using the -core decomposition to quantify their accessibility.Seidman proposed the -core method to decompose a complex network into a hierarchy of maximally connected sub-networks ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1016/0378-8733(83)90028-X","ISBN":"0378-8733","ISSN":"03788733","PMID":"18429079","abstract":"Social network researchers have long sought measures of network cohesion, Density has often been used for this purpose, despite its generally admitted deficiencies. An approach to network cohesion is proposed that is based on minimum degree and which produces a sequence of subgraphs of gradually increasing cohesion. The approach also associates with any network measures of local density which promise to be useful both in characterizing network structures and in comparing networks. © 1983.","author":[{"dropping-particle":"","family":"Seidman","given":"Stephen B.","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Social Networks","id":"ITEM-1","issue":"3","issued":{"date-parts":[["1983"]]},"page":"269-287","title":"Network structure and minimum degree","type":"article-journal","volume":"5"},"uris":["http://www.mendeley.com/documents/?uuid=b17cae78-e03a-4bdf-909f-88529092359c"]}],"mendeley":{"formattedCitation":"[36]","plainTextFormattedCitation":"[36]","previouslyFormattedCitation":"(Seidman, 1983)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[36]. The decomposition gives rise to a laminar family such that . Here, is the sub-network induced by the nodes with core values greater than or equal to . Seidman defines the-core decomposition of a network as follows ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1016/0378-8733(83)90028-X","ISBN":"0378-8733","ISSN":"03788733","PMID":"18429079","abstract":"Social network researchers have long sought measures of network cohesion, Density has often been used for this purpose, despite its generally admitted deficiencies. An approach to network cohesion is proposed that is based on minimum degree and which produces a sequence of subgraphs of gradually increasing cohesion. The approach also associates with any network measures of local density which promise to be useful both in characterizing network structures and in comparing networks. © 1983.","author":[{"dropping-particle":"","family":"Seidman","given":"Stephen B.","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Social Networks","id":"ITEM-1","issue":"3","issued":{"date-parts":[["1983"]]},"page":"269-287","title":"Network structure and minimum degree","type":"article-journal","volume":"5"},"uris":["http://www.mendeley.com/documents/?uuid=b17cae78-e03a-4bdf-909f-88529092359c"]}],"mendeley":{"formattedCitation":"[36]","plainTextFormattedCitation":"[36]","previouslyFormattedCitation":"(Seidman, 1983)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[36].Definition 1: -core: Given a simple, undirected, unweighted network , -core is the largest sub-network of induced by the subset of nodes and , if and only if the degree of each node in is greater than or equal to . Thus, coreness of a node is defined as: ( SEQ ( \* ARABIC 3)The -core of is obtained by recursively removing all nodes with degrees less than until the remaining nodes in the network have degrees at least . It is established that the -core of a network is unique and connected ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"author":[{"dropping-particle":"","family":"Bickle","given":"Allan","non-dropping-particle":"","parse-names":false,"suffix":""}],"id":"ITEM-1","issued":{"date-parts":[["2010"]]},"number-of-pages":"209","publisher":"Western Michigan University","title":"The k-cores of a graph","type":"book"},"uris":["http://www.mendeley.com/documents/?uuid=d89ab31e-280f-4124-9f9c-1e34598e32d1"]}],"mendeley":{"formattedCitation":"[37]","plainTextFormattedCitation":"[37]","previouslyFormattedCitation":"(Bickle, 2010)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[37].Batagelj and Zaversnik proposed a linear time algorithm for -core decomposition ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"abstract":"The structure of large networks can be revealed by partitioning them to smaller parts, which are easier to handle. One of such decompositions is based on $k$--cores, proposed in 1983 by Seidman. In the paper an efficient, $O(m)$, $m$ is the number of lines, algorithm for determining the cores decomposition of a given network is presented.","author":[{"dropping-particle":"","family":"Batagelj","given":"Vladimir","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Zaversnik","given":"M.","non-dropping-particle":"","parse-names":false,"suffix":""}],"id":"ITEM-1","issued":{"date-parts":[["2003"]]},"page":"1-10","title":"An O(m) Algorithm for Cores Decomposition of Networks","type":"article-journal"},"uris":["http://www.mendeley.com/documents/?uuid=fcab2d4e-7e8d-421a-9f07-cd6d1b5fccf3"]}],"mendeley":{"formattedCitation":"[38]","plainTextFormattedCitation":"[38]","previouslyFormattedCitation":"(Batagelj & Zaversnik, 2003)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[38], which makes it very attractive for analyzing large sparse networks common in diverse domains. The algorithm is available in well-known R and Python packages.Example 2: REF _Ref111394225 \h \* MERGEFORMAT Fig. 2 shows the -core decomposition of the toy network shown in REF _Ref114533435 \h \* MERGEFORMAT Fig. 1. The -core sub-networks are extracted through a peeling process. As minimum degree of the nodes is one, core value is assigned to the nodes , and they are removed from the network. Removal of node reduces the degree of to one and hence is also assigned core value 1. Next, nodes with degree 2 are assigned the core value 2 and are removed from the network . Proceeding in this way, nodes with degree 3 are assigned the core value 3. After removing these nodes, we get a null network. Thus, we have maximum core value for .Fig. SEQ Figure \* ARABIC 2. Core decomposition of the toy network shown in REF _Ref114533435 \h \* MERGEFORMAT Fig. 1 reveals three level hierarchy (ovals) in the network. Nodes marked in the same color have the same core valueHierarchical decomposition of transport networkThe emergence of hierarchy in transport networks is well documented by Yerra and Levinson ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1007/s00168-005-0230-4","ISSN":"05701864","abstract":"A transportation network is a complex system that exhibits the properties of self-organization and emergence. Previous research in dynamics related to transportation networks focuses on traffic assignment or traffic management. This research concentrates on the dynamics of the orientation of major roads in a network and abstractly models these dynamics to understand the basic properties of transportation networks. A model is developed to capture the dynamics that leads to a hierarchical arrangement of roads for a given network structure and land use distribution. Localized investment rules, wherein revenue produced by traffic on a link is invested for that link's own development, are employed. Under reasonable parameters, these investment rules, coupled with traveler behavior, and underlying network topology result in the emergence of a hierarchical pattern. Hypothetical networks subject to certain conditions are tested with this model to explore their network properties. Though hierarchies seem to be designed by planners and engineers, the results show that they are intrinsic properties of networks. Also, the results show that roads, specific routes with continuous attributes, are emergent properties of transportation networks. © Springer-Verlag 2005.","author":[{"dropping-particle":"","family":"Yerra","given":"Bhanu M.","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Levinson","given":"David M.","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Annals of Regional Science","id":"ITEM-1","issue":"3","issued":{"date-parts":[["2005"]]},"title":"The emergence of hierarchy in transportation networks","type":"article-journal","volume":"39"},"uris":["http://www.mendeley.com/documents/?uuid=b8a2b20b-e501-474a-b87e-e5d601e6982b"]}],"mendeley":{"formattedCitation":"[39]","plainTextFormattedCitation":"[39]","previouslyFormattedCitation":"(Yerra & Levinson, 2005)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[39]. Boguna et al. argued that the core of a network plays a significant role in enhancing the navigability of networks ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1038/nphys1130","ISSN":"17452481","abstract":"Routing information through networks is a universal phenomenon in both natural and man-made complex systems. When each node has full knowledge of the global network connectivity, finding short communication paths is merely a matter of distributed computation. However, in many real networks, nodes communicate efficiently even without such global intelligence. Here, we show that the peculiar structural characteristics of many complex networks support efficient communication without global knowledge. We also describe a general mechanism that explains this connection between network structure and function. This mechanism relies on the presence of a metric space hidden behind an observable network. Our findings suggest that real networks in nature have underlying metric spaces that remain undiscovered. Their discovery should have practical applications in a wide range of areas where networks are used to model complex systems.","author":[{"dropping-particle":"","family":"Bogũá","given":"Marián","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Krioukov","given":"Dmitri","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Claffy","given":"K C","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Nature Physics","id":"ITEM-1","issue":"1","issued":{"date-parts":[["2009"]]},"title":"Navigability of complex networks","type":"article-journal","volume":"5"},"uris":["http://www.mendeley.com/documents/?uuid=41590aeb-c4be-4867-a4a8-c64b31b17030"]}],"mendeley":{"formattedCitation":"[40]","plainTextFormattedCitation":"[40]","previouslyFormattedCitation":"(Bogũá, Krioukov, & Claffy, 2009)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[40]. Wuellner et al. used core decomposition to analyze the seven largest airlines in the US ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1103/PhysRevE.82.056101","ISSN":"15393755","abstract":"The air transportation network, a fundamental component of critical infrastructure, is formed from a collection of individual air carriers, each one with a methodically designed and engineered network structure. We analyze the individual structures of the seven largest passenger carriers in the USA and find that networks with dense interconnectivity, as quantified by large k cores for high values of k, are extremely resilient to both targeted removal of airports (nodes) and random removal of flight paths (edges). Such networks stay connected and incur minimal increase in an heuristic travel time despite removal of a majority of nodes or edges. Similar results are obtained for targeted removal based on either node degree or centrality. We introduce network rewiring schemes that boost resilience to different levels of perturbation while preserving total number of flight and gate requirements. Recent studies have focused on the asymptotic optimality of hub-and-spoke spatial networks under normal operating conditions, yet our results indicate that point-to-point architectures can be much more resilient to perturbations. © 2010 The American Physical Society.","author":[{"dropping-particle":"","family":"Wuellner","given":"Daniel R","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Roy","given":"Soumen","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"D'Souza","given":"Raissa M","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Physical Review E - Statistical, Nonlinear, and Soft Matter Physics","id":"ITEM-1","issue":"5","issued":{"date-parts":[["2010"]]},"title":"Resilience and rewiring of the passenger airline networks in the United States","type":"article-journal","volume":"82"},"uris":["http://www.mendeley.com/documents/?uuid=46ab419b-b5ca-4b59-a847-a480a577583d"]}],"mendeley":{"formattedCitation":"[41]","plainTextFormattedCitation":"[41]","previouslyFormattedCitation":"(Wuellner, Roy, & D’Souza, 2010)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[41]. They observed that nodes belonging to the highest k-core are the hub nodes and the most viable transfer points ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1103/PhysRevE.82.056101","ISSN":"15393755","abstract":"The air transportation network, a fundamental component of critical infrastructure, is formed from a collection of individual air carriers, each one with a methodically designed and engineered network structure. We analyze the individual structures of the seven largest passenger carriers in the USA and find that networks with dense interconnectivity, as quantified by large k cores for high values of k, are extremely resilient to both targeted removal of airports (nodes) and random removal of flight paths (edges). Such networks stay connected and incur minimal increase in an heuristic travel time despite removal of a majority of nodes or edges. Similar results are obtained for targeted removal based on either node degree or centrality. We introduce network rewiring schemes that boost resilience to different levels of perturbation while preserving total number of flight and gate requirements. Recent studies have focused on the asymptotic optimality of hub-and-spoke spatial networks under normal operating conditions, yet our results indicate that point-to-point architectures can be much more resilient to perturbations. © 2010 The American Physical Society.","author":[{"dropping-particle":"","family":"Wuellner","given":"Daniel R","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Roy","given":"Soumen","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"D'Souza","given":"Raissa M","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Physical Review E - Statistical, Nonlinear, and Soft Matter Physics","id":"ITEM-1","issue":"5","issued":{"date-parts":[["2010"]]},"title":"Resilience and rewiring of the passenger airline networks in the United States","type":"article-journal","volume":"82"},"uris":["http://www.mendeley.com/documents/?uuid=46ab419b-b5ca-4b59-a847-a480a577583d"]}],"mendeley":{"formattedCitation":"[41]","plainTextFormattedCitation":"[41]","previouslyFormattedCitation":"(Wuellner et al., 2010)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[41]. Thus, the authors concluded that nodes with higher core values are more resilient to attacks on nodes and edges of a network. Azimi-Tafreshi et al. used k-core decomposition to compare the topological structures of various low-cost and major airlines in the country ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1103/PhysRevE.90.032816","ISSN":"15502376","abstract":"We generalize the theory of k-core percolation on complex networks to k-core percolation on multiplex networks, where k≡(k1,k2,...,kM). Multiplex networks can be defined as networks with vertices of one kind but M different types of edges, representing different types of interactions. For such networks, the k-core is defined as the largest subgraph in which each vertex has at least ki edges of each type, i=1,2,...,M. We derive self-consistency equations to obtain the birth points of the k-cores and their relative sizes for uncorrelated multiplex networks with an arbitrary degree distribution. To clarify our general results, we consider in detail multiplex networks with edges of two types and solve the equations in the particular case of Erdos-Rényi and scale-free multiplex networks. We find hybrid phase transitions at the emergence points of k-cores except the (1,1)-core for which the transition is continuous. We apply the k-core decomposition algorithm to air-transportation multiplex networks, composed of two layers, and obtain the size of (k1,k2)-cores.","author":[{"dropping-particle":"","family":"Azimi-Tafreshi","given":"N","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Gómez-Gardeñes","given":"J","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Dorogovtsev","given":"S N","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Physical Review E - Statistical, Nonlinear, and Soft Matter Physics","id":"ITEM-1","issue":"3","issued":{"date-parts":[["2014"]]},"title":"K-core percolation on multiplex networks","type":"article-journal","volume":"90"},"uris":["http://www.mendeley.com/documents/?uuid=2b8db888-dc5e-4e27-8cc4-094d41fae640"]}],"mendeley":{"formattedCitation":"[35]","plainTextFormattedCitation":"[35]","previouslyFormattedCitation":"(Azimi-Tafreshi et al., 2014)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[35].Du et al. decomposed the Chinese airlines network into three layers: core, bridge, and periphery ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1016/j.tre.2016.03.009","ISSN":"13665545","abstract":"This paper encapsulates the Chinese Airline Network (CAN) into multi-layer infrastructures via the \"k-core decomposition\" method. The network is divided into three layers: Core layer, containing airports of provincial capital cities, is densely connected and sustains most flight flow; Bridge layer, consisting of airports in Tier 2 and Tier 3 cities, mainly connects two other layers; and Periphery layer, comprising airports of remote areas, sustains little flight flow. Moreover, it is unveiled that CAN stays the most robust when low-degree nodes or high flight flow links are removed, which is similar to the Worldwide Airline Network (WAN), albeit less redundant.","author":[{"dropping-particle":"","family":"Du","given":"Wen Bo","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Zhou","given":"Xing Lian","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Lordan","given":"Oriol","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Wang","given":"Zhen","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Zhao","given":"Chen","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Zhu","given":"Yan Bo","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Transportation Research Part E: Logistics and Transportation Review","id":"ITEM-1","issued":{"date-parts":[["2016"]]},"title":"Analysis of the Chinese Airline Network as multi-layer networks","type":"article-journal","volume":"89"},"uris":["http://www.mendeley.com/documents/?uuid=48d9415b-9e8c-41b9-b81d-c1e1096b96d7"]}],"mendeley":{"formattedCitation":"[22]","plainTextFormattedCitation":"[22]","previouslyFormattedCitation":"(Du et al., 2016)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[22]. They concluded that the core layer is densely connected and sustains the maximum flow of flights connecting airports of capital cities. And the bridge layer consists of airports that connects two other layers. Further, the periphery layer is sparsely connected and sustains little flight flow connecting airports in remote areas. Dai et al. explored the evolution of the topological and multilayered structure of the Southeast Asian air transport network from 1979 to 2012 ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1016/j.jtrangeo.2018.02.010","ISSN":"09666923","abstract":"This paper presents a novel approach to investigating and understanding the evolving structure of the Southeast Asian air transport network (SAAN) over the period 1979–2012. Our approach captures the main topological and spatial changes from a complex network perspective. We find that the SAAN combines a relatively stable topological structure with a changing multilayered geographical structure. Statistical analysis indicates that the SAAN is a scale-free network with an increasing number of hub cities and has been characterized by small-world properties since 1996. Furthermore, the SAAN exhibits a recently intensified disassortative mixing pattern, suggesting an increasing dependence of small cities on hub-and-spoke configuration for better accessibility. A decomposition analysis is used to disaggregate the SAAN into a hierarchical core-bridge-periphery structure. The core layer consists of capital cities, the most economic vibrant secondary cities, and tourist destinations. This core layer is also densely interconnected with its center of gravity moving towards the north. The periphery layer, comprised of cities in remote areas, sustains a low significance with declining internal connectivity despite a rising number of cities being connected. The bridge layer lies in between both extremes, and is characterized by a high volatility over time. The connections and passengers between different layers increase, especially those between core and bridge after 1996. In our discussion, we trace these changes back to a series of socio-economic and politico-institutional dynamics in Southeast Asia.","author":[{"dropping-particle":"","family":"Dai","given":"Liang","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Derudder","given":"Ben","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Liu","given":"Xingjian","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Journal of Transport Geography","id":"ITEM-1","issued":{"date-parts":[["2018"]]},"title":"The evolving structure of the Southeast Asian air transport network through the lens of complex networks, 1979–2012","type":"article-journal","volume":"68"},"uris":["http://www.mendeley.com/documents/?uuid=42c6d929-dcc4-4d15-997a-11fefc2f1b67"]}],"mendeley":{"formattedCitation":"[23]","plainTextFormattedCitation":"[23]","previouslyFormattedCitation":"(Dai et al., 2018)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[23]. They decomposed the network into three layers based on the k-core values of the nodes. The periphery layer has nodes with core value one, the core layer has nodes with the maximum core value, and the bridge layer has nodes with intermediate core values. The authors concluded that the multilayered structure of the Southeast Asian air network has significantly evolved and is less mature and integrated than its EU counterpart.3. HIERARCHY-BASED ACCESSIBILITY METRICSWe propose to use the core decomposition of a network to quantify the accessibility of nodes in the transport network, advancing the ideas proposed in [ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1016/j.tre.2016.03.009","ISSN":"13665545","abstract":"This paper encapsulates the Chinese Airline Network (CAN) into multi-layer infrastructures via the \"k-core decomposition\" method. The network is divided into three layers: Core layer, containing airports of provincial capital cities, is densely connected and sustains most flight flow; Bridge layer, consisting of airports in Tier 2 and Tier 3 cities, mainly connects two other layers; and Periphery layer, comprising airports of remote areas, sustains little flight flow. Moreover, it is unveiled that CAN stays the most robust when low-degree nodes or high flight flow links are removed, which is similar to the Worldwide Airline Network (WAN), albeit less redundant.","author":[{"dropping-particle":"","family":"Du","given":"Wen Bo","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Zhou","given":"Xing Lian","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Lordan","given":"Oriol","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Wang","given":"Zhen","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Zhao","given":"Chen","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Zhu","given":"Yan Bo","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Transportation Research Part E: Logistics and Transportation Review","id":"ITEM-1","issued":{"date-parts":[["2016"]]},"title":"Analysis of the Chinese Airline Network as multi-layer networks","type":"article-journal","volume":"89"},"uris":["http://www.mendeley.com/documents/?uuid=48d9415b-9e8c-41b9-b81d-c1e1096b96d7"]}],"mendeley":{"formattedCitation":"[22]","plainTextFormattedCitation":"[22]","previouslyFormattedCitation":"(Du et al., 2016)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}22], ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1103/PhysRevE.90.032816","ISSN":"15502376","abstract":"We generalize the theory of k-core percolation on complex networks to k-core percolation on multiplex networks, where k≡(k1,k2,...,kM). Multiplex networks can be defined as networks with vertices of one kind but M different types of edges, representing different types of interactions. For such networks, the k-core is defined as the largest subgraph in which each vertex has at least ki edges of each type, i=1,2,...,M. We derive self-consistency equations to obtain the birth points of the k-cores and their relative sizes for uncorrelated multiplex networks with an arbitrary degree distribution. To clarify our general results, we consider in detail multiplex networks with edges of two types and solve the equations in the particular case of Erdos-Rényi and scale-free multiplex networks. We find hybrid phase transitions at the emergence points of k-cores except the (1,1)-core for which the transition is continuous. We apply the k-core decomposition algorithm to air-transportation multiplex networks, composed of two layers, and obtain the size of (k1,k2)-cores.","author":[{"dropping-particle":"","family":"Azimi-Tafreshi","given":"N","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Gómez-Gardeñes","given":"J","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Dorogovtsev","given":"S N","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Physical Review E - Statistical, Nonlinear, and Soft Matter Physics","id":"ITEM-1","issue":"3","issued":{"date-parts":[["2014"]]},"title":"K-core percolation on multiplex networks","type":"article-journal","volume":"90"},"uris":["http://www.mendeley.com/documents/?uuid=2b8db888-dc5e-4e27-8cc4-094d41fae640"]}],"mendeley":{"formattedCitation":"[35]","plainTextFormattedCitation":"[35]","previouslyFormattedCitation":"(Azimi-Tafreshi et al., 2014)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[35], ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1038/nphys1130","ISSN":"17452481","abstract":"Routing information through networks is a universal phenomenon in both natural and man-made complex systems. When each node has full knowledge of the global network connectivity, finding short communication paths is merely a matter of distributed computation. However, in many real networks, nodes communicate efficiently even without such global intelligence. Here, we show that the peculiar structural characteristics of many complex networks support efficient communication without global knowledge. We also describe a general mechanism that explains this connection between network structure and function. This mechanism relies on the presence of a metric space hidden behind an observable network. Our findings suggest that real networks in nature have underlying metric spaces that remain undiscovered. Their discovery should have practical applications in a wide range of areas where networks are used to model complex systems.","author":[{"dropping-particle":"","family":"Bogũá","given":"Marián","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Krioukov","given":"Dmitri","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Claffy","given":"K C","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Nature Physics","id":"ITEM-1","issue":"1","issued":{"date-parts":[["2009"]]},"title":"Navigability of complex networks","type":"article-journal","volume":"5"},"uris":["http://www.mendeley.com/documents/?uuid=41590aeb-c4be-4867-a4a8-c64b31b17030"]}],"mendeley":{"formattedCitation":"[40]","plainTextFormattedCitation":"[40]","previouslyFormattedCitation":"(Bogũá et al., 2009)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[40], ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1103/PhysRevE.82.056101","ISSN":"15393755","abstract":"The air transportation network, a fundamental component of critical infrastructure, is formed from a collection of individual air carriers, each one with a methodically designed and engineered network structure. We analyze the individual structures of the seven largest passenger carriers in the USA and find that networks with dense interconnectivity, as quantified by large k cores for high values of k, are extremely resilient to both targeted removal of airports (nodes) and random removal of flight paths (edges). Such networks stay connected and incur minimal increase in an heuristic travel time despite removal of a majority of nodes or edges. Similar results are obtained for targeted removal based on either node degree or centrality. We introduce network rewiring schemes that boost resilience to different levels of perturbation while preserving total number of flight and gate requirements. Recent studies have focused on the asymptotic optimality of hub-and-spoke spatial networks under normal operating conditions, yet our results indicate that point-to-point architectures can be much more resilient to perturbations. © 2010 The American Physical Society.","author":[{"dropping-particle":"","family":"Wuellner","given":"Daniel R","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Roy","given":"Soumen","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"D'Souza","given":"Raissa M","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Physical Review E - Statistical, Nonlinear, and Soft Matter Physics","id":"ITEM-1","issue":"5","issued":{"date-parts":[["2010"]]},"title":"Resilience and rewiring of the passenger airline networks in the United States","type":"article-journal","volume":"82"},"uris":["http://www.mendeley.com/documents/?uuid=46ab419b-b5ca-4b59-a847-a480a577583d"]}],"mendeley":{"formattedCitation":"[41]","plainTextFormattedCitation":"[41]","previouslyFormattedCitation":"(Wuellner et al., 2010)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[41]. The underlying intuition is that better-connected nodes in higher cores of the network are more advantaged. Existing topology-based accessibility measures (Betweenness accessibility ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1016/j.jtrangeo.2020.102680","ISSN":"09666923","abstract":"Accessibility is a central concept in transport geography research that has been described as a holistic measure of transportation and land use systems. This concept has numerous implementations, but virtually all share the way accessibility is measured as an attribute of pairs of origins and destinations. Borrowing from concepts in network science, in this paper we propose a new centrality measure called betweenness-accessibility. This measure couples the familiar betweenness indicator of social networks with the geographical concept of accessibility. Betweenness-accessibility is useful to estimate the impacts of accessibility on networks as potential for interaction is reflected on them. The new measure is illustrated using a reproducible example. In addition, an application to vulnerability analysis in the city of Zurich, Switzerland, provides an empirical case study to showcase the usefulness of betweenness-accessibility.","author":[{"dropping-particle":"","family":"Sarlas","given":"Georgios","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Páez","given":"Antonio","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Axhausen","given":"Kay W","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Journal of Transport Geography","id":"ITEM-1","issued":{"date-parts":[["2020"]]},"title":"Betweenness-accessibility: Estimating impacts of accessibility on networks","type":"article-journal","volume":"84"},"uris":["http://www.mendeley.com/documents/?uuid=641a4bbf-687b-4581-9143-24673296077f"]}],"mendeley":{"formattedCitation":"[27]","plainTextFormattedCitation":"[27]","previouslyFormattedCitation":"(Sarlas et al., 2020)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[27]) do not acknowledge the advantage that accrues due to linkage with better accessible nodes. Hence, we propose three novel core-based accessibility metrics for unweighted and weighted networks in this section.3.1Accessibility metrics for unweighted networksWe define and introduce the mathematical formulations for the proposed three topology-based metrics for unweighted networks, highlighting the role of connectivity in quantifying the accessibility of nodes in transport networks.Core AccessibilityCore accessibility is a measure that provides a snapshot of similar connectivity patterns of nodes in the network. A node with core value k is connected to at least similarly connected nodes. Thus, the core value gives a lower bound to the connectivity of a node.Definition 2: Core Accessibility (CA): Given a simple, unweighted and undirected network, core accessibilityof a node is defined as the core value of in core decomposition of . ( SEQ ( \* ARABIC 4)Core accessibility partitions the network into multiple layers (levels), generalizing the three-layered representation suggested by ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1016/j.tre.2016.03.009","ISSN":"13665545","abstract":"This paper encapsulates the Chinese Airline Network (CAN) into multi-layer infrastructures via the \"k-core decomposition\" method. The network is divided into three layers: Core layer, containing airports of provincial capital cities, is densely connected and sustains most flight flow; Bridge layer, consisting of airports in Tier 2 and Tier 3 cities, mainly connects two other layers; and Periphery layer, comprising airports of remote areas, sustains little flight flow. Moreover, it is unveiled that CAN stays the most robust when low-degree nodes or high flight flow links are removed, which is similar to the Worldwide Airline Network (WAN), albeit less redundant.","author":[{"dropping-particle":"","family":"Du","given":"Wen Bo","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Zhou","given":"Xing Lian","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Lordan","given":"Oriol","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Wang","given":"Zhen","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Zhao","given":"Chen","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Zhu","given":"Yan Bo","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Transportation Research Part E: Logistics and Transportation Review","id":"ITEM-1","issued":{"date-parts":[["2016"]]},"title":"Analysis of the Chinese Airline Network as multi-layer networks","type":"article-journal","volume":"89"},"uris":["http://www.mendeley.com/documents/?uuid=48d9415b-9e8c-41b9-b81d-c1e1096b96d7"]}],"mendeley":{"formattedCitation":"[22]","plainTextFormattedCitation":"[22]","previouslyFormattedCitation":"(Du et al., 2016)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[22]. Maximum core accessibility in the network indicates the levels of hierarchy in the network. Accordingly, it can be used to compare navigability in the network as suggested in ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1038/nphys1130","ISSN":"17452481","abstract":"Routing information through networks is a universal phenomenon in both natural and man-made complex systems. When each node has full knowledge of the global network connectivity, finding short communication paths is merely a matter of distributed computation. However, in many real networks, nodes communicate efficiently even without such global intelligence. Here, we show that the peculiar structural characteristics of many complex networks support efficient communication without global knowledge. We also describe a general mechanism that explains this connection between network structure and function. This mechanism relies on the presence of a metric space hidden behind an observable network. Our findings suggest that real networks in nature have underlying metric spaces that remain undiscovered. Their discovery should have practical applications in a wide range of areas where networks are used to model complex systems.","author":[{"dropping-particle":"","family":"Bogũá","given":"Marián","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Krioukov","given":"Dmitri","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Claffy","given":"K C","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Nature Physics","id":"ITEM-1","issue":"1","issued":{"date-parts":[["2009"]]},"title":"Navigability of complex networks","type":"article-journal","volume":"5"},"uris":["http://www.mendeley.com/documents/?uuid=41590aeb-c4be-4867-a4a8-c64b31b17030"]}],"mendeley":{"formattedCitation":"[40]","plainTextFormattedCitation":"[40]","previouslyFormattedCitation":"(Bogũá et al., 2009)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[40]. Since core accessibility indicates minimum connectivity, it groups multiple locations at the same hierarchical level. Consequently, core accessibility is a coarse metric (Section 4). We define below an advanced metric for perceptive comparison of accessibility of geographic locations at the regional level.Local Core AccessibilityConnectivity in the immediate neighborhood of the node plays an influential role in determining its accessibility at the local level. Since the accessibility of a node (location) is impacted by that of its neighbors, a node stands to gain an advantage if it is connected to neighbor nodes with higher accessibility. Local core accessibility measure considers the density of connections in the two-hop neighborhood of a node by giving due consideration to the core accessibility of all its neighbors.Definition 3: Local Core Accessibility (LCA): Given a simple, unweighted and undirected network , local core accessibilityof node is defined as the sum of the core accessibility values of neighbor nodes of . ( SEQ ( \* ARABIC 5)Here, denotes the set of neighbors of . Local core accessibility serves as a metric for quantifying regional accessibility of geographical locations. Aggregating core accessibility of neighbors empowers the LCA to expose finer differences in the accessibility of geographical locations within a region. Hence, it is particularly useful in discriminating between the local accessibility of nodes with same core accessibility.Network-wide Core AccessibilityIntuitively, the accessibility of a node at the global level is affected by the accessibility of its neighbors, which in turn are impacted by the accessibility of their neighbors, and so on. The nodes connected to neighbors with higher regional accessibility are advantaged compared to those with neighbors having lower regional accessibility. This intuition is captured by the Eigen Vector centrality of a node in a network which quantifies the importance of a node recursively while considering the topology of the network ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1080/0022250X.1972.9989806","ISSN":"15455874","author":[{"dropping-particle":"","family":"Bonacich","given":"Phillip","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"The Journal of Mathematical Sociology","id":"ITEM-1","issue":"1","issued":{"date-parts":[["1972"]]},"title":"Factoring and weighting approaches to status scores and clique identification","type":"article-journal","volume":"2"},"uris":["http://www.mendeley.com/documents/?uuid=8d1a747e-f863-4d42-a35d-a6afb1c97cdb"]}],"mendeley":{"formattedCitation":"[42]","plainTextFormattedCitation":"[42]","previouslyFormattedCitation":"(Bonacich, 1972)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[42].Network-wide core accessibility quantifies the accessibility of a node at the global level and is computed recursively by considering the local core accessibility of its neighbors. Consider matrix that consolidates local core accessibility of all pairs of nodes in as follows. ( SEQ ( \* ARABIC 6)Element in connotes the accessibility advantage that node gets because of node. It is noteworthy that for a node pair, the accessibility advantage for from may be different from that for from . The difference arises due to the variation in their respective neighborhood networks. Retention of the accessibility differences among nodes due to their location in the overall topology of the network renders matrix asymmetric. Lack of symmetry also empowers matrix to expose subtle differences between the accessibility of nodes while considering their position in the hierarchy prevalent in the network. Example 3 in Appendix D demonstrates the construction of matrix for the toy network given in REF _Ref114533435 \h \* MERGEFORMAT Fig. 1.L can be considered an adjacency matrix of a weighted and directed graph with nodes at an abstract level. Eigenvector centrality of the nodes in this network connotes network-wide core accessibility. Let vector of size denote the network-wide core accessibility of all nodes in . We initialize to all ones assuming that all nodes are initially equally accessible over the entire transport network. Subsequently, network-wide accessibility of all nodes is computed iteratively using L (Definition 4).Definition 4: Network-wide Core Accessibility (NCA): Let L be the matrix of pairwise local core accessibility values as computed in Equation (6). Network-wide accessibility , of a node is computed iteratively as: ( SEQ ( \* ARABIC 7)until converges.NCA provides a global (network-wide) view of the accessibility of locations and thus can be classified as an Integral accessibility measure as defined by Ingram ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1080/09595237100185131","ISSN":"13600591","author":[{"dropping-particle":"","family":"Ingram","given":"D. R.","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Regional Studies","id":"ITEM-1","issue":"2","issued":{"date-parts":[["1971"]]},"title":"The Concept of Accessibility: A Search for an Operational Form","type":"article-journal","volume":"5"},"uris":["http://www.mendeley.com/documents/?uuid=5d5c0be6-5e08-4ac2-a0ae-133f4ae6f01d"]}],"mendeley":{"formattedCitation":"[9]","plainTextFormattedCitation":"[9]","previouslyFormattedCitation":"(Ingram, 1971)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[9]. Further, NCA is more discerning than LCA because it ensconces the location of the node in the overall topology of the network.We present applications of core-based accessibility measures for unweighted transport networks in Section 4.Accessibility metrics for weighted networksAccessibility measures proposed in the previous sub-section highlight the role of connectivity in quantifying the accessibility of nodes in transport networks. However, these measures ignore several important aspects of real-world transport networks like the cost of travel, transport capacity, travel time, distance, transport modes, service frequencies, etc. These attributes of connectivity between two nodes can be naturally modeled as edge weights in the transport network and have a substantial impact on the quantitative assessment of the accessibility of locations.Given a simple, undirected and weighted network , where is the weight matrix representing the weight of edges, as per the attribute(s) chosen for study. Matrix could represent any of the attributes of connectivity mentioned above or a function of these attributes. For example, a product of travel cost and capacity denotes revenue earned by the services company. Weighted degree of node is the sum of weights of edges incident on it. Normalized weighted degree of is defined as follows. ( SEQ ( \* ARABIC 8)Please note that isolated nodes in the network have normalized weighted degree zero.Abiding by the fact that the weights of the edges impact the accessibility of nodes, it is now feasible to differentiate their accessibility considering their normalized weighted degree. We, thus, extend the definitions of the three accessibility indicators given in Section 3.1 to propose their weighted versions, namely weighted core accessibility, weighted local core accessibility, and weighted network-wide core accessibility.Definition 5: Weighted Core Accessibility (wCA) of a node is defined as the product of its normalized weighted degree () and its core accessibility. ( SEQ ( \* ARABIC 9)Definition 6: Weighted Local Core Accessibility (wLCA) of a node is defined as the product of its normalized weighted degree and its local core accessibility. ( SEQ ( \* ARABIC 10)Definition 7: Weighted Network-wide Core Accessibility (wNCA) of a node u ∈ V (G) is defined as the product of its normalized weighted degree and its network-wide core accessibility. ( SEQ ( \* ARABIC 11)We present a case study in Section 5 to demonstrate the advantages of the proposed weighted metrics.4. APPLICATION OF CORE-BASED ACCESSIBILITY METRICS FOR UNWEIGHTED NETWORKWe consider an unweighted and undirected test network of domestic air connectivity by Air India between 36 states and union territories of India that are coded with two letters. Two states are connected if and only if two cities in the states are connected by a flight. We compute the proposed core-based accessibility metrics for this network, rank the nodes, and explain the significance of each metric.The k-core decomposition of the test network unfolds the hierarchy that provides insight into the different layers contributing to the network connectivity ( REF _Ref111395350 \h \* MERGEFORMAT Błąd! Nie można odnaleźć źródła odwołania.). Nodes with higher core values ( = 5, cyan blue) belong to denser regions of the network compared to those in lower cores. Edges marked in black connect the nodes in the same core, and edges marked in blue connect the nodes in different cores. Node SK is disconnected in the network as no Air India flight is connecting it to any state, and hence its core value is zero in this network. Throughout this paper, we use this test network to explain the relevant concepts and measures. Comparing core accessibilityBased on the premise that the position of a node in the hierarchical decomposition of the network determines its accessibility in the network, we compare the accessibility of different Indian states and union territories serviced by Air India. We compare the core accessibility of locations in the transport network shown in REF _Ref111395350 \h \* MERGEFORMAT Błąd! Nie można odnaleźć źródła odwołania.. Core decomposition of the network reveals five levels of hierarchy, indicating that states can be ranked from 1 to 5 based on their connectivity patterns. Nodes belonging to highest core are ranked one and have maximum core accessibility in the network. REF _Ref111650030 \h \* MERGEFORMAT Tab. 2 shows the ranks of connected nodes in the test network.Limitations of Core Accessibility: As evident in REF _Ref111650030 \h \* MERGEFORMAT Tab. 2, core accessibility (CA) bunches multiple locations together and assigns the same rank to them. In other words, it cannot discriminate between the relative importance of nodes in the same core. For example, AS and BR are ranked three, as shown in Table 2, even though AS is better connected than BR, as shown in REF _Ref111395350 \h \* MERGEFORMAT Błąd! Nie można odnaleźć źródła odwołania.. It is noteworthy that core accessibility cannot bring out this difference between similarly ranked locations.Fig. SEQ Figure \* ARABIC 3. Core Decomposition of unweighted and undirected test network reveals five levels of hierarchy in the network. Nodes with the same color belong to the same coreTab. SEQ Table \* ARABIC 2Ranks of nodes in the test network according to their core accessibility (CA)Rank12345Nodes in the test networkAP, DL, KA, MH, TS, TN, WBGA, KL, ODAN, AS, BR, GJ, MP, PY, UPAR, CG, CH, HP, HR, JK, LA, MN, MZ, NL, PB, RJDD, JH, LD, ML, TR, UK4.2 Comparing accessibility at the regional levelThe local core accessibility (LCA) measure quantifies the ease of reaching a location (node) within a region by aggregating core accessibility of its neighbor nodes. LCA captures finer distinctions between nodes with the same core accessibility values by considering the location of its neighbors in the network hierarchy. Nodes with the same core accessibility may have different LCA values due to differences in the core accessibility of their respective neighbors. Analyzing nodes AS and BR, ranked equally for core accessibility measure, have different local core accessibility values. Note that in Table 3a, AS is ranked 10, while BR does not appear among the top 15 nodes. Thus, the LCA measure can distinguish among the nodes ranked equally by the core accessibility measure.(a) Neighbor of node AS with their respective CA values(b) Neighbor of node OD with their respective CA valuesFig. SEQ Figure \* ARABIC 4. Core accessibility values of neighbor of nodes AS and OD as in the test networkConsider the seven highest-ranked nodes (AP, DL, KA, MH, TS, TN, and WB), according to their core accessibility in REF _Ref111650030 \h \* MERGEFORMAT Tab. 2. These nodes are ranked uniquely by the LCA measure, as shown in Table 3a. It is interesting to note that though AS has a lower core accessibility than OD, it has a higher LCA value. This is because AS is connected to nodes with higher core accessibility than OD, as shown in Figures 4a and b.(a) Neighbors of GJ with their respective LCA values(b) Neighbors of AN with their respective LCA valuesFig. SEQ Figure \* ARABIC 5. LCA values of neighbors of GJ and AN as in the test networkLimitations of Local Core Accessibility: Though the LCA computes the accessibility of a node by giving due consideration to its neighborhood, it is deficient in discerning the network-wide accessibility of nodes. Two nodes with the same local core accessibility may have different patterns of network-wide connectivity. For example, in Table 3a, AN and GJ in the test network ( REF _Ref111395350 \h \* MERGEFORMAT Błąd! Nie można odnaleźć źródła odwołania.) have the same local core accessibility. But neighbor nodes of GJ have higher regional accessibility than the neighbors of AN. This is illustrated in Figure 5a, showing the neighbors of GJ, and Figure 5b, showing the neighbors of AN along with their local core accessibility values. The LCA is unable to discriminate the nodes on this ground and thus ranks them equally.Tab. SEQ Table \* ARABIC 3Ranks of top 15 nodes of the test network in REF _Ref111395350 \h \* MERGEFORMAT Błąd! Nie można odnaleźć źródła odwołania. for their (a) local core accessibility and (b) network-wide core accessibility) valuesNodeRankDL711MH582TN483WB424KA415TS366AP327KL288GA229AS2110OD2011MP1812UP1613AN1514GJ1514PB1415(b) valuesNodeRankDL11MH0.81562TN0.63853KA0.5394WB0.48225TS0.44756AP0.35987KL0.27328OD0.17199GA0.155210AS0.131711MP0.129812GJ0.103713UP0.098214PB0.080615Comparing accessibility at the network levelTransport planners and service providers often need a global (network-wide) view of the accessibility of locations. NCA measure further teases out the differences in accessibility from the perspective of the overall topology of the network. Consider nodes GJ and AN, ranked equally (14) by the LCA measure, as shown in Table 3a. They are assigned different network-wide accessibility values (Table 3b). GJ is ranked 13 by the NCA measure, but AN does not appear among the top15 nodes. Thus, NCA discriminates among the nodes ranked equally by the LCA measure. Critical observations, as elaborated below, provide additional insights into the advantages of network-wide accessibility.Table 3 displays the top 15 nodes ranked according to their network-wide core accessibility values. Consider nodes WB and KA, which ranked four and five, respectively, for LCA values (Table 3a). Interestingly, despite KA and WB being connected to a few common neighbor nodes (AS, DL, MH, TN, and TS), KA is ranked higher than WB according to network-wide core accessibility ( REF _Ref111486386 \h \* MERGEFORMAT Fig. 6). This is due to more connections of KA to nodes with higher regional accessibility (GA, AP, and KL), as shown in REF _Ref111486386 \h \* MERGEFORMAT Fig. 6a. On the other hand, WB is connected to more nodes with relatively lower regional accessibility (AR, BR, AN, ML, MZ, NL, and TR). This can be observed from the neighborhood network of WB, as shown in REF _Ref111486386 \h \* MERGEFORMAT Fig. 6b.(a) Neighborhood of node KA with their respective LCA values(b) Neighborhood of node WB with their respective LCA valuesFig. SEQ Figure \* ARABIC 6. Neighborhood of KA and WB as in the test network5. COMPARING ACCESSIBILITY IN WEIGHTED AND UNWEIGHTED NETWORKSWe first demonstrate the advantages of the weighted version of accessibility metrics over their unweighted counterparts using the monotonicity of the metrics as a quality measure. Subsequently, we present a case study demonstrating the application of the proposed metrics to study temporal changes in transport networks. We use the proposed metrics to compare the accessibility of Indian airports in pre-and post-covid times. The data for both experiments are sourced from the Directorate General of Civil Aviation. The code for the metrics is written in R (64 bits, v 4.0.3).5.1Assessing the quality of metricsIt is important to assess the discriminatory power of the metrics as all values are eventually transformed to rank for practical utility. A metric that assigns the same rank to many nodes is less preferable to a metric that can tease out the differences better and has fewer nodes with the same rank.We use the monotonicity function to assess the fidelity of our metrics quantitatively ADDIN CSL_CITATION {"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1016/j.physa.2013.10.047","ISSN":"03784371","abstract":"Identifying influential spreaders is an important issue in understanding the dynamics of information diffusion in complex networks. The k-shell index, which is the topological location of a node in a network, is a more efficient measure at capturing the spreading ability of a node than are the degree and betweenness centralities. However, the k-shell decomposition fails to yield the monotonic ranking of spreaders because it assigns too many nodes with the same k-shell index. In this paper, we propose a novel measure, coreness centrality, to estimate the spreading influence of a node in a network using the k-shell indices of its neighbors. Our experimental results on both real and artificial networks, compared with an epidemic spreading model, show that the proposed method can quantify the node influence more accurately and provide a more monotonic ranking list than other ranking methods. © 2013 Elsevier B.V. All rights reserved.","author":[{"dropping-particle":"","family":"Bae","given":"Joonhyun","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Kim","given":"Sangwook","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"Physica A: Statistical Mechanics and its Applications","id":"ITEM-1","issued":{"date-parts":[["2014"]]},"title":"Identifying and ranking influential spreaders in complex networks by neighborhood coreness","type":"article-journal","volume":"395"},"uris":["http://www.mendeley.com/documents/?uuid=ff9e495a-1220-4976-82c5-20cae8ff7abf"]}],"mendeley":{"formattedCitation":"[43]","plainTextFormattedCitation":"[43]","previouslyFormattedCitation":"(Bae & Kim, 2014)"},"properties":{"noteIndex":0},"schema":"https://github.com/citation-style-language/schema/raw/master/csl-citation.json"}[43]. We rank the nodes according to their accessibility values. The node with the highest accessibility is ranked one. The nodes with equal accessibility are assigned the same ranks. Let be a vector of the ranks assigned tonodes of the graph , with as the number of ties in the ranks. Monotonicityof ranks is defined as given below. ( SEQ ( \* ARABIC 12)Monotonicity of the metrics is one if each node in the network is assigned a unique rank and zero when all nodes are assigned the same rank. We compute the monotonicity of each metric to ensure its efficacy. 5.2Analyzing the accessibility of a service providerWe construct a simple, undirected, weighted network for the winter schedule 2020 (winter20) of Air Asia, a domestic airline. Then, we compute the weekly frequency of each incoming and outgoing flight, excluding the flights scheduled only for a specific day. Thereafter, we consider the minimum frequency of incoming/outgoing flights between pairs of nodes (airports) in the network as the edge weight. The network is shown in REF _Ref111579692 \h \* MERGEFORMAT Błąd! Nie można odnaleźć źródła odwołania. in Appendix B. REF _Ref111579735 \h \* MERGEFORMAT Tab. 4 shows the degree (column Degree), normalized weighted degree (column ), and ranks of nodes (airports) assigned as per the proposed unweighted and weighted versions of the three-accessibility metrics. The nodes are sorted by their normalized weighted degree (column ). We compare the ranks assigned to the airports, presenting a detailed analysis below.We observe that nodes belonging to the same core (column Core) are ranked equally for core accessibility (column CA). But when the weighted core accessibility is computed, the ranks are distinct. For example, BLR, DEL, BOM, CCU, MAA, and HYD belong to core five and are ranked one according to their core accessibility. However, BLR is ranked highest according to its weighted core accessibility (column wCA) due to the maximum number of flights (158) connecting it to other airports in the network. Since core accessibility bunches multiple nodes together, assigning them the same rank, it has the least monotonicity (0.57), showing that it is a coarse metric. However, when weights are considered, monotonicity increases significantly to 0.98. Note that COK and JAI are differently accessible due to the difference in the accessibility of their respective neighbors. However, despite its high monotonicity, wCA is unable to reflect this difference and assigns the same rank to the two nodes.LCA, which is designed to reveal the effect of accessibility of its neighbors, can discern between COK and JAI (column LCA in REF _Ref111579735 \h \* MERGEFORMAT Tab. 4). However, the LCA assigns the same rank to multiple locations (AMD, BBI, IXB, IXR, and PNQ), each of which is ranked distinctly by the wLCA metric. BBI has the highest weighted local core accessibility value due to the highest number of connecting flights (31) among these nodes. Note that wLCA ranks IDR and VTZ equally, which is reflected in its monotonicity value, which is less than the perfect score of one. Lower monotonicity of LCA compared to wCA should not be considered a drawback, as they both quantify accessibility at different granularity levels and have different purposes.Tab. SEQ Table \* ARABIC 4Ranks of airports connected by Air Asia flight (winter schedule 2020), according to the proposed core-based accessibility measuresNodesDegreeCoreCAwCALCAwLCANCAwNCABLR170.18465111111DEL160.17295122222BOM120.095133333CCU80.07135145454MAA90.06315154545HYD60.05025166666GOI40.04674279798GAU40.044433910101010BBI30.0374331011111111IXR30.0362331111121112COK50.03045188989JAI60.03045187877IXB30.028331211131313AMD30.0234331311141214PNQ30.0222331411151415SXR10.021151613171819IDR20.0164241512161516VTZ20.0164241512161717IMF10.0082151814191920IXC20.007241712181618Monotonicity---0.570.980.870.990.991As mentioned in Section 4.2, the LCA does not capture the recursive impact of accessibility of the neighbor nodes. Intuitively, a node connected to neighbors with high local accessibility should have better network-wide accessibility. We observe that VTZ and IDR are ranked equally for all metrics except NCA. IDR and VTZ have a common neighbor, BLR. But the difference in their rank is attributed to the difference in the local accessibility of their other neighbor, DEL (rank 2) and MAA (rank 4), respectively. This example shows that NCA is a more discerning metric than the LCA.Observing the column NCA in REF _Ref111579735 \h \* MERGEFORMAT Tab. 4, we find that BBI and IXR are ranked equally as they are connected to the same nodes in the network. But BBI is ranked higher by weighted network-wide core accessibility (column wNCA) due to more connecting flights. We observe that the wNCA metric has the highest monotonicity (one) and is the most discerning. Since wNCA depends on the wLCA as well as its connectivity in the network (captured by eigenvector centrality), we believe that wNCA will generate unique rankings. However, in the unlikely case of duplicate ranks of two nodes, both are considered to be equally accessible.5.3Temporal changes in accessibilityTemporal changes in accessibility provide valuable inputs to the stakeholders of the transport industry. Transport service networks expand (shrink) over time due to addition (reduction) in destinations or increase (decrease) in frequencies of services. The influence of these changes on the accessibility of locations in the network is often not obvious. We demonstrate the utility of weighted metrics to compare and reveal the changes in the accessibility of Indian airports at two different points in time. We use the domestic airlines network for the following two schedules obtained from the Directorate General of Civil Aviation NOTEREF _Ref112772660 \h \* MERGEFORMAT 9.Pre-covid times: winter schedule 2019 (winter19) for Air Asia, Air India, Alliance Air, Deccan, Go Air, Heritage, Indigo, Pawan Hans, Spicejet, Star Air, Truejet, and Vistara airlinesPost-covid times: winter schedule 2020 (winter20) for Air Asia, Air India, Alliance Air, Go Air, Indigo, Pawan Hans, Spicejet, Star Air, Truejet, and Vistara airlinesFor constructing the weighted network for all domestic air carriers, we first construct the network for each airline as described in Section 5.2. Thereafter, we sum up the weights of the edges, where the edge weight connotes the minimum frequency of incoming/outgoing flights between pairs of nodes in the network over all air carriers. REF _Ref111599297 \h \* MERGEFORMAT Fig. 7 displays the comparison of wLCA and wNCA values of Indian airports in the pre-and post-covid times. The airports are sorted on weighted accessibility values for pre-covid times. REF _Ref111599297 \h \* MERGEFORMAT Fig. 7a reveals that the regional accessibility of the top 10 airports has reduced in post-covid times, as expected. This is attributed to the drop in the number of flights to and from these nodes in the winter20 schedule. REF _Ref111599297 \h \* MERGEFORMAT Fig. 7b shows the regional accessibility of the bottom 10 airports. It can be seen that six out of the ten least accessible airports are still not accessible (wLCA is zero) due to the non-resumption of flight services. Surprisingly, the regional accessibility of four airports, namely SXV, LUH, IXP, and TEZ, has improved post-covid. The decrease in the overall connectivity in the network led to this improvement. The increase in the frequency of flights between TEZ and IXP in the post-covid schedule further amplifies their wLCA scores. REF _Ref111599297 \h \* MERGEFORMAT Fig. 7c shows the change in network-wide accessibility for the top 10 airports in the two periods. The figure reveals the holistic view of the accessibility of domestic airports at two points in time. The resumption of flights in the post-covid schedule has almost restored the accessibility of the top ten airports. For example, we observe a slight improvement in the network-wide accessibility of BLR and HYD, despite a fall in their regional accessibility. The regional accessibility of BLR and HYD reduces post-covid due to the reduction in the flights connecting them in the network. The increase in their network-wide accessibility is attributed to the increase in the accessibility of their neighbors due to the increase in their connectivity with other nodes in the network. However, network-wide accessibility for six of the bottom ten airports is yet to resume ( REF _Ref111599297 \h \* MERGEFORMAT Fig. 7d). IXP, IXT, and TEZ are exceptions with an increase in their network-wide accessibility post-covid due to the increased frequency of flights.wLCA values of the top 10 Indian airportswLCA values of the bottom 10 Indian airportswNCA values of the top 10 Indian airportswLCA values of the bottom 10 Indian airportsFig. SEQ Figure \* ARABIC 7. Comparison of local and network-wide accessibility of airports in pre-covid (winter19) and post-covid (winter20) times.Airports are sorted by their local and network-wide accessibility in winter19 times6. CONCLUSIONIn this paper, we propose novel accessibility metrics based on hierarchical decomposition of a transport network. The metrics gauge the accessibility of a geographical location (node in the network) at different granularity levels and are scalable. Different attributes of accessibility, like travel cost, travel time, distance, quality of service, etc., can be quantified as edge weights establishing the versatility of the proposed metrics.We use the k-core decomposition of the network to elicit the hierarchical position of a node and use it in three different forms to quantify core accessibility (coarse level), local core accessibility (regional level), and network-wide core accessibility (network level). Weighted versions of the metrics capture finer differences between the accessibility of locations that are similarly connected.Furthermore, we demonstrate the applications of the proposed metric using test networks of Indian Airlines. We also present a case study of the Air Asia network to elucidate the utility, relative advantages, and disadvantages of the proposed metrics. Finally, we show that the measures are responsive to changes in the topology of the transport network by presenting an analysis of changes in accessibility for the domestic air services network for both pre-covid and post-covid times.ReferencesADDIN Mendeley Bibliography CSL_BIBLIOGRAPHY Wu H., D. Levinson. 2020. “Unifying access”. Transp. Res. Part D Transp. Environ. 83.Levinson D., H. Wu. 2020. “Towards a general theory of access.”. J. Transp. Land Use 13(1).Geurs K.T., B. van Wee. 2004. “Accessibility evaluation of land-use and transport strategies: Review and research directions”. J. Transp. Geogr. 12(2).Handy S. 2020. “Is accessibility an idea whose time has finally come?” Transp. Res. Part D Transp. Environ. 83.El-Geneidy A., D. Levinson. 2022. “Making accessibility work in practice”. Transport Reviews 42(2).Levine J. 2020. “A century of evolution of the accessibility concept”. Transp. Res. Part D Transp. Environ. 83.Hansen W.G. 1959. “How Accessibility Shapes Land Use”. J. Am. 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D’Souza. 2010. “Resilience and rewiring of the passenger airline networks in the United States”. Phys. Rev. E - Stat. Nonlinear, Soft Matter Phys. 82(5).Bonacich P. 1972. “Factoring and weighting approaches to status scores and clique identification”. J. Math. Sociol. 2(1).Bae J., S. Kim. 2014. “Identifying and ranking influential spreaders in complex networks by neighborhood coreness”. Phys. A Stat. Mech. its Appl. 395.Mohammed J. Zaki, Wagner Meira Jr. 2014. Data Mining and Analysis: Fundamental Concepts and Algorithms. Cambridge University Press.Received 06.11.2022; accepted in revised form 19.01.2023Scientific Journal of Silesian University of Technology. Series Transport is licensed under a Creative Commons Attribution 4.0 International License PAGE 160D. Kwatra, K. Ramachandra Rao, V. Bhatnagar Department of Computer Science, Hansraj College, University of Delhi, Delhi, India. Email: divya@hrc.du.ac.in. ORCID: 0000-0002-7688-5293 Department of Civil Engineering, and Transportation Research and Injury Prevention Centre (TRIPC), Indian Institute of Technology Delhi, New Delhi, India. Email: rrkalaga@civil.iitd.ac.in. ORCID: 0000-0002-7229-519X Department of Computer Science, University of Delhi, Delhi, India. Email: vbhatnagar@cs.du.ac.in. ORCID: 0000-0002-9706-9340 igraph library (https://igraph.org/r/). NetworkX package (https://networkx.org/). See Appendix C for formal definition and example. See REF _Ref111601588 \h \* MERGEFORMAT Błąd! Nie można odnaleźć źródła odwołania. in Appendix A for codes of states and union territories. Note that node SK might be connected by some other airline. https://www.dgca.gov.in. In the interest of reproducibility, the code will be made public after the notification. Schedule during the period Oct'19 - Mar'20. Schedule during the period Oct'20 - Mar'21.Scientific Journal of Silesian University of Technology. Series TransportZeszyty Naukowe Politechniki Śląskiej. 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