Title: Motif based hierarchical random graphs: structural properties and critical points of an Ising model
Author(s):

	Monika Kotorowicz (Institute of Mathematics, Maria Curie-Skłodowska University, Lublin, Poland) ,
	Yuri Kozitsky (Institute of Mathematics, Maria Curie-Skłodowska University, Lublin, Poland)

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.

Key words: amenability, degree distribution, clustering, small-world graph, Ising model, critical point
PACS: 89.75.Fb, 89.75.Kd, 05.10.Cc, 05.70.Jk
Comments: Figs. 5, Refs. 30, Tab. 1

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