Topological properties of hierarchical networks

Hierarchical networks are attracting a renewal interest for modeling the organization of a number of biological systems and for tackling the complexity of statistical mechanical models beyond mean-field limitations. Here we consider the Dyson hierarchical construction for ferromagnets, neural networks, and spin glasses, recently analyzed from a statistical-mechanics perspective, and we focus on the topological properties of the underlying structures. In particular, we find that such structures are weighted graphs that exhibit a high degree of clustering and of modularity, with a small spectral gap; the robustness of such features with respect to the presence of thermal noise is also studied. These outcomes are then discussed and related to the statistical-mechanics scenario in full consistency. Last, we look at these weighted graphs as Markov chains and we show that in the limit of infinite size, the emergence of ergodicity breakdown for the stochastic process mirrors the emergence of metastabilities in the corresponding statistical mechanical analysis.