Convexity in complex networks

TL;DR

This paper explores the concept of convexity in complex networks, defining convex subgraphs based on geodesic paths, and analyzes how various real-world networks exhibit convexity properties unlike random graphs.

Contribution

It introduces a mathematical definition of convexity in networks, analyzes its presence in different network types, and discusses measures and applications of network convexity.

Findings

Convexity is common in spatial infrastructure and social networks.

Food webs are predominantly non-convex.

Random graphs are only locally convex.

Abstract

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.