Core–periphery models for graphs based on their δ-hyperbolicity: An example using biological networks

Abstract

Hyperbolicity is a global property of graphs that measures how close their structures are to trees in terms of their distances. It embeds multiple properties that facilitate solving several problems that found to be hard in the general graph form. In this paper, we investigate the hyperbolicity of graphs not only by considering Gromov's notion of -hyperbolicity but also by analyzing its relationship to other graph's parameters. This new perspective allows us to classify graphs with respect to their hyperbolicity and to show that many biological networks are hyperbolic. Then we introduce the eccentricity-based bending property which we exploit to identify the core vertices of a graph by proposing two models: the maximum-peak model and the minimum cover set model. In this extended version of the paper, we include some new theorems, as well as proofs of the theorems proposed in the conference paper. Also, we present the algorithms we used for each of the proposed core identification models, and we provide more analysis, explanations, and examples.

Fast Approximation of Centrality and Distances in Hyperbolic Graphs

“…It has been suggested in [77], and recently formally proved in [39], that the property, observed in real-world networks, in which traffic between nodes tends to go through a relatively small core of the network, as if the shortest paths between them are curved inwards, is due to the hyperbolicity of the network. Bending property of the eccentricity function in hyperbolic graphs were used in [16,15] to identify core-periphery structures in biological networks. Small hyperbolicity in real-world graphs provides also many algorithmic advantages.…”