Statistical analyis of complex network

Abstract

Spectra of networks adjacency matrices are known to be fingerprint of the underlying complex systems. Most of the works on the network spectra have revolved around analysis of largest eigenvalue. In this work, we analyse the second largest eigen value (2) of the networks adjacency matrices. We find that may follow an entirely different behaviour than that of the largest eigen value. While, the largest eigenvalue contains information of the largest degree of the underlying network, thereby, exhibiting an increasing behvaiour throughout with an increase in the average connectivity of a network emulates this behaviour until a critical connectivity, after which it exhibits a continuous decrease. The behaviour of provides an insight to degree of freedom available to the nodes to form connected pairs. More interesting is the behvaiour of fluctuations, in the presence of inhibitory couplings. Inspired by the presence as well as importance of inhibitory couplings in many complex systems, including brain and ecological systems, we introduce such couplings in networks and investigate its impact on For this case, while average behaviour of the largest and the second largest eigenvalues are exactly same for various value of inhibitory probability, the fluctuation statistics for the exhibit a very different behavior indicating importance of second largest eigenvalue in getting information of underlying system.