Motif based hierarchical random graphs: structural properties and critical points of an Ising model

TL;DR

This paper introduces a new class of random graphs built from specific motifs, analyzes their structural properties, and investigates the critical point of the Ising model on one such graph, revealing insights into their complex behavior.

Contribution

It presents a novel motif-based hierarchical graph construction and studies its structural properties and phase transition behavior of the Ising model on these graphs.

Findings

Graphs exhibit small-world and clustering properties.

Degree distributions follow specific patterns.

Critical point of the Ising model identified for one motif.

Abstract

A class of random graphs is introduced and studied. The graphs are constructed in an algorithmic way from five motifs which were found in [Milo R., Shen-Orr S., Itzkovitz S., Kashtan N., Chklovskii D., Alon U., Science, 2002, 298, 824-827]. The construction scheme resembles that used in [Hinczewski M., A. Nihat Berker, Phys. Rev. E, 2006, 73, 066126], according to which the short-range bonds are non-random, whereas the long-range bonds appear independently with the same probability. A number of structural properties of the graphs have been described, among which there are degree distributions, clustering, amenability, small-world property. For one of the motifs, the critical point of the Ising model defined on the corresponding graph has been studied.