Modular networks with hierarchical organization:

The dynamical implications of complex structure

RAJ KUMAR PAN and SITABHRA SINHA^*

The Institute of Mathematical Sciences, C.I.T. Campus, Chennai 600 113, India

^*Corresponding author

E-mail: rajkp@imsc.res.in; sitabhra@imsc.res.in

Abstract. Several networks occurring in real life have modular

structures that are arranged in a 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.

Keywords. Modular network; hierarchical organization; stability;

robustness.

PACS Nos 89.75.Hc; 05.45.-a; 89.75.Fb