Modular networks with hierarchical organization: The dynamical implications of complex structure

TL;DR

This paper introduces a stochastic model for hierarchical modular networks, showing that certain scaling relations are not necessary and analyzing how hierarchy affects network stability.

Contribution

It presents a new stochastic model for hierarchical modular networks and investigates the impact of hierarchy and modularity on network stability.

Findings

Scaling relation between clustering and degree is not necessary.

Increasing hierarchy or modularity raises instability probability.

Hierarchical structure requires additional constraints for robustness.

Abstract

Several networks occurring in real life have modular structures that are arranged in an hierarchical fashion. In this paper, we have proposed a model for such networks, using a stochastic generation method. Using this model we show that, the scaling relation between the clustering and degree of the nodes is not a necessary property of hierarchical modular networks, as had previously been suggested on the basis of a deterministically constructed model. We also look at dynamics on such networks, in particular, the stability of equilibria of network dynamics and of synchronized activity in the network. For both of these, we find that, increasing modularity or the number of hierarchical levels tends to increase the probability of instability. As both hierarchy and modularity are seen in natural systems, which necessarily have to be robust against environmental fluctuations, we conclude that additional constraints are necessary for the emergence of hierarchical structure, similar to the occurrence of modularity through multi-constraint optimization as shown by us previously.