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Convexity in complex networks

Published online by Cambridge University Press:  06 February 2018

TILEN MARC  and
LOVRO ŠUBELJ Open the ORCID record for LOVRO ŠUBELJ [Opens in a new window]
TILEN MARC
Affiliation:
Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia (e-mail: tilen.marc@imfm.si)
LOVRO ŠUBELJ
Affiliation:
University of Ljubljana, Faculty of Computer and Information Science, Ljubljana, Slovenia (e-mail: lovro.subelj@fri.uni-lj.si)
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Abstract

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Metric graph properties lie in the heart of the analysis of complex networks, while in this paper we study their convexity through mathematical definition of a convex subgraph. A subgraph is convex if every geodesic path between the nodes of the subgraph lies entirely within the subgraph. According to our perception of convexity, convex network is such in which every connected subset of nodes induces a convex subgraph. We show that convexity is an inherent property of many networks that is not present in a random graph. Most convex are spatial infrastructure networks and social collaboration graphs due to their tree-like or clique-like structure, whereas the food web is the only network studied that is truly non-convex. Core–periphery networks are regionally convex as they can be divided into a non-convex core surrounded by a convex periphery. Random graphs, however, are only locally convex meaning that any connected subgraph of size smaller than the average geodesic distance between the nodes is almost certainly convex. We present different measures of network convexity and discuss its applications in the study of networks.

Keywords

network convexityconvex subsetsconvex subgraphscore–periphery structure
Type
Research Article
Information
Network Science , Volume 6 , Issue 2 , June 2018 , pp. 176 - 203
Copyright
Copyright © Cambridge University Press 2018 

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