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

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.

6 pages, 6 figures, Published version

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• Phase of Ising spins on modular networks analogous to social polarization
• Modular organization enhances the robustness of attractor network dynamics
• Modular hierarchical and power-law small-world networks bear structural optima for minimal first passage times and cover time
• Multiple dynamical time-scales in networks with hierarchically nested modular organization