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Hierarchical Random Graphs Based on Motifs

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.