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.

Hierarchical graphs were invented to formalize heuristic Migdal-Kadanoff renormalization arguments. In such graphs, certain characteristic patterns (motifs) appear as construction elements. Real-world complex networks may also contain such patterns. S. Itzkovitz and U. Alon in Phys. Rev. E, 2005, 71, selected five most typical motifs, which include the triangle. In Cond. Matt. Phys., 2011, 14, M. Kotorowicz and Yu. Kozitsky introduced and described hierarchical random graphs in which these five motifs appear at each hierarchy level. In the present work, we study the equilibrium states of the Ising spin model living on the graph of this kind based on the triangle. The main result is the description of annealed phase transitions in this model. In particular, we show that -- depending on the parameters -- the model may be in an unordered or ordered states at all temperatures, as well as to have a critical point. The key aspect of our theory is detecting the appearance of an ordered state by the non-ergodicity of a certain nonhomogeneous Markov chain.

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 graph models are proposed in which the motifs appear at each hierarchical level. A rigorous construction of such graphs is performed 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.