title: Topological properties of hierarchical networks.
abstract: 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.
authors.affiliation: Dipartimento di Matematica, Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185 Roma, Italy.
authors.forename: Elena
authors.initials: E
authors.lastname: Agliari
authors.affiliation: Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy.
authors.forename: Adriano
authors.initials: A
authors.lastname: Barra
authors.affiliation: Dipartimento di Matematica, Sapienza Università di Roma, Piazzale Aldo Moro 5, 00185 Roma, Italy.
authors.forename: Andrea
authors.initials: A
authors.lastname: Galluzzi
authors.affiliation: Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy.|Istituto Nazionale di Fisica Nucleare, Sezione di Roma.
authors.forename: Francesco
authors.initials: F
authors.lastname: Guerra
authors.affiliation: Centro di Ricerca Matematica "Ennio De Giorgi", Scuola Normale Superiore, Piazza dei Cavalieri 3, 56100 Pisa, Italy.
authors.forename: Daniele
authors.initials: D
authors.lastname: Tantari
authors.affiliation: Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, Via Antonio Scarpa 16, 00185 Roma, Italy.
authors.forename: Flavia
authors.initials: F
authors.lastname: Tavani