Title: Modelling complex networks by random hierarchical graphs
Author(s):

M.Wróbel (Institute of Mathematics, Maria Curie-Skłodowska University, Lublin, Poland)

Numerous complex networks contain special patterns, called network motifs. These are specific subgraphs, which occur oftener than in randomized networks of Erdős-Rényi type. We choose one of them, the triangle, and build a family of random hierarchical graphs, being Sierpiński gasket-based graphs with random "decorations". We calculate the important characteristics of these graphs - average degree, average shortest path length, small-world graph family characteristics. They depend on probability of decorations. We analyze the Ising model on our graphs and describe its critical properties using a renormalization-group technique.

Key words: random graphs, Ising model, complex networks, network motifs
PACS: 05.50.+q, 05.70.Fh, 75.10.Nr, 89.75.-k