Spectral analysis of complex networks

Abstract

This thesis investigates spectral properties of adjacency matrices of complex systems having underlying network structure. First part of the thesis considers the fluctuation behaviour of the largest real part of the eigenvalues (Rmax) of the adjacency matrices of networks having inhibitory and excitatory couplings. These networks are motivated from synaptic behaviour of brain. We find that as probability of inclusion of inhibitory nodes (pin) increases, Rmax exhibits a linear decrease until a certain pin value followed by a nonlinear behaviour. It indicates that an increase in the complexity, in terms of the inclusion of inhibitory nodes, increases the stability of the underlying system. For the range where Rmax mean exhibts the linear dependence on pin, the statistics mostly yields a normal distribution, and after this critical pin value there is a transition to the GEV statistics.Further, strictly balanced condition, which is defined in terms of equal strength of inhibitory and excitatory couplings, leads to convergence of extreme value statistics to a fixed Weibull statistics. This balanced condition is so strong that even changes in the interaction patterns do not affect the Rmax distribution. Origin of the Weibull distribution for the strictly balanced condition could be explained by the fact that this condition shifts the outliers into the bulk of the spectra. Additionally, the transition to the Weibull statistics and the small-world transition occur at the same rewiring probability, reflecting a more stable system. The second part of the thesis focuses on the evolutionary origin of various types of patterns observed in real-world systems. We show that inhibition leads to the clique structure in networks as they are evolved using Rmax as a fitness function. Moreover, the distribution of triads over nodes in the network, evolved from the mixed connections, reveals a negative correlation with its degree providing insight into the origin of this trend observed in many real-world. Furthermore, motivated by the mutation observed in real-world systems, we introduce the mutation in the interaction behaviour among the individuals during the evolution of networks. We find that maximization of the stability of the underlying system leads to the evolution of the disassortative structure. and the mutation probability governs the degree of saturation of the disassortativity coefficient, revealing the origin of a wide range of the disassortativity values found in real-world systems. Finally, we show, by imposing stability criterion through minimization of the Rmax, the evolutionary origin of (bi)multi-partite structure found in many real-world systems having inhibitory-excitatory couplings.