Hierarchical Random Graphs Based on Motifs

TL;DR

This paper introduces hierarchical random graphs constructed from network motifs, analyzing their structural properties and phase transitions, providing a new framework for understanding complex network organization.

Contribution

It presents a rigorous method for building hierarchical graphs based on motifs and analyzes their structural and phase transition properties.

Findings

Graphs exhibit characteristic degree distributions and clustering.

Hierarchical motifs display small world properties.

Phase transitions in Ising models are observed on these graphs.

Abstract

Network motifs are characteristic patterns which occur in the networks essentially more frequently than the other patterns. For five motifs found in S. Itzkovitz, U. Alon, Phys. Rev.~E, 2005, 71, 026117-1, hierarchical random graphs are proposed in which the motifs appear at each hierarchical level. A rigorous construction of such graphs is given and a number of their structural properties are analyzed. This includes degree distribution, amenability, clustering, and the small world property. For one of the motifs, annealed phase transitions in the Ising model based on the corresponding graph are also studied.