Path Evaluation and Centralities in Weighted Graphs - An Axiomatic Approach

We study the problem of extending the classic centrality measures to weighted graphs. Unfortunately, in the existing extensions, paths in the graph are evaluated solely based on their weights, which is a restrictive and undesirable assumption for a variety of settings. Given this, we define a notion of the path evaluation function that assesses a path between two nodes by looking not only on the sum of edge weights, but also on the number of intermediaries. Using an axiomatic approach, we propose three classes of path evaluation functions. Building upon this analysis, we present the first systematic study how classic centrality measures can be extended to weighted graphs while taking into account an arbitrary path evaluation function. As an application, we use the newly-defined measures to identify the most well-linked districts in a sample public transport network.

[1]  Adnan Aijaz,et al.  2015 IEEE Global Communications Conference (GLOBECOM) , 2013, GLOBECOM 2015.

[2]  O. Bagasra,et al.  Proceedings of the National Academy of Sciences , 1914, Science.

[3]  Gert Sabidussi,et al.  The centrality index of a graph , 1966 .

[4]  Melissa E. Libertus,et al.  Comment on "Log or Linear? Distinct Intuitions of the Number Scale in Western and Amazonian Indigene Cultures" , 2009, Science.

[5]  Piotr Faliszewski,et al.  Multiwinner Voting: A New Challenge for Social Choice Theory , 2017 .

[6]  Vasileios Karyotis,et al.  Hyperbolic Embedding for Efficient Computation of Path Centralities and Adaptive Routing in Large-Scale Complex Commodity Networks , 2017, IEEE Transactions on Network Science and Engineering.

[7]  Oskar Skibski,et al.  Axiomatic Characterization of Game-Theoretic Centrality , 2018, J. Artif. Intell. Res..

[8]  Aravaipa Canyon Basin,et al.  Volume 3 , 2012, Journal of Diabetes Investigation.

[9]  L. Freeman.  Centrality in social networks conceptual clarification , 1978 .

[10]  U. Brandes.  A faster algorithm for betweenness centrality , 2001 .

[11]  R. Rosenfeld.  Nature , 2009, Otolaryngology--head and neck surgery : official journal of American Academy of Otolaryngology-Head and Neck Surgery.

[12]  Chris Arney.  Network Analysis: Methodological Foundations , 2012 .

[13]  John Skvoretz,et al.  Node centrality in weighted networks: Generalizing degree and shortest paths , 2010, Soc. Networks.

[14]  M. Newman,et al.  Scientific collaboration networks. II. Shortest paths, weighted networks, and centrality. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  Diego Noble,et al.  An Analysis of Centrality Measures for Complex and Social Networks , 2016, 2016 IEEE Global Communications Conference (GLOBECOM).

[16]  L. da F. Costa,et al.  Characterization of complex networks: A survey of measurements , 2005, cond-mat/0505185.

[17]  Manuj Garg,et al.  Axiomatic Foundations of Centrality in Networks , 2009 .

[18]  Annamaria Conte,et al.  A New Weighted Degree Centrality Measure: The Application in an Animal Disease Epidemic , 2016, PloS one.

[19]  Ee-Peng Lim,et al.  Measuring Centralities for Transportation Networks Beyond Structures , 2015, Applications of Social Media and Social Network Analysis.