Spectral properties of directed random networks with modular structure

Abstract

We study spectra of directed networks with inhibitory and excitatory couplings. We investigate in particular eigenvector localization properties of various model networks for different values of correlation among their entries. Spectra of random networks with completely uncorrelated entries show a circular distribution with delocalized eigenvectors, whereas networks with correlated entries have localized eigenvectors. In order to understand the origin of localization we track the spectra as a function of connection probability and directionality. As connections are made directed, eigenstates start occurring in complex-conjugate pairs and the eigenvalue distribution combined with the localization measure shows a rich pattern. Moreover, for a very well distinguished community structure, the whole spectrum is localized except few eigenstates at the boundary of the circular distribution. As the network deviates from the community structure there is a sudden change in the localization property for a very small value of deformation from the perfect community structure. We search for this effect for the whole range of correlation strengths and for different community configurations. Furthermore, we investigate spectral properties of a metabolic network of zebrafish and compare them with those of the model networks.

Randomness and preserved patterns in cancer network

“…Thus, the IPR quantifies the reciprocal of the number of eigenvector components that contribute significantly. We further calculate the average IPR in order to measure an overall localization of the network calculated as60, Note that IPR defined as above separates out the TCNs by keeping the threshold as 1/ IPR .…”

Extreme-value statistics of networks with inhibitory and excitatory couplings

“…Maximum eigenvalue for this network scales as R max ∼ pN [27], where quantity pN is referred to as average degree k of the network. Upon introduction of directionality, complex eigenvalues start appearing in conjugate pairs, and for p in = 0.5, bulk of the eigenvalues is distributed in a circular region of radius N p(1 − p) [28]. Note that for a random network with entries 1 and −1, the radius of circular bulk region scales with square root of the average degree of the network i.e.…”

Universality in the spectral and eigenfunction properties of random networks

“…In fact, both quantities have been already used to characterize the eigenfunctions of the adjacency matrices of random network models (see some examples in Refs. [31,36,39,42,48,[53][54][55][56][57][66][67][68][69]). …”