Topological properties of hierarchical networks

TL;DR

This paper investigates the topological features of hierarchical networks used in biological and statistical models, revealing high clustering, modularity, and ergodicity breakdown linked to meta-stability.

Contribution

It provides a detailed analysis of the topological properties of Dyson hierarchical networks and connects these features to their statistical mechanics behavior.

Findings

Hierarchical networks are highly clustered and modular.

Small spectral gap indicates robustness to link removal.

Ergodicity breakdown correlates with meta-stability in the models.

Abstract

Hierarchical networks are attracting a renewal interest for modelling 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 high degree of clustering and of modularity, with small spectral gap; the robustness of such features with respect to link removal is also studied. These outcomes are then discussed and related to the statistical mechanics scenario in full consistency. Lastly, 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 meta-stabilities in the corresponding statistical mechanical analysis.