First-passage phenomena in hierarchical networks

In this paper we study Markov processes and related first passage problems on a class of weighted, modular graphs which generalize the Dyson hierarchical model. In these networks, the coupling strength between two nodes depends on their distance and is modulated by a parameter $\sigma$. We find that, in the thermodynamic limit, ergodicity is lost and the "distant" nodes can not be reached. Moreover, for finite-sized systems, there exists a threshold value for $\sigma$ such that, when $\sigma$ is relatively large, the inhomogeneity of the coupling pattern prevails and "distant" nodes are hardly reached. The same analysis is carried on also for generic hierarchical graphs, where interactions are meant to involve $p$-plets ($p>2$) of nodes, finding that ergodicity is still broken in the thermodynamic limit, but no threshold value for $\sigma$ is evidenced, ultimately due to a slow growth of the network diameter with the size.