We study spectra of directed networks with inhibitory and excitatory couplings. Particularly, we investigate eigenvector localization properties of various model networks with varying correlations \tau among their entries. Spectra of random directed networks, where entries are completely un-correlated (\tau=0) show circular distribution with delocalized eigenvectors, where non-random 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 both. The kind of inhibitory and excitatory connections we are considering, low connection probability leads to localized eigenstates near boundary of the circular region, whereas large connection probabilities give rise to isolated delocalized eigenstates. As connections are made directed by making some nodes inhibitory, some of eigenstates start occurring in complex conjugate pairs . The eigenvalue distribution along with localization measure show rich pattern. Spectra of networks having modular structure show distinguishable different features than the random networks. For a very well distinguished community structure (rewiring probability p_r \sim 0) , the whole spectra is localized except some of eigenstates at boundary . As p_r is increased and network deviates from  community structure there is a sudden change in the localization property for  very small value of deformation from the perfect community structure.  Furthermore, we investigate spectral properties of a metabolic networks of Zebra-fish and compare with spectral properties of various model networks.
We study spectra of directed networks with inhibitory and excitatory couplings. We investigate in particular eigenvector localization properties of various model networks for different value of correlation among their entries. Spectra of random networks, with completely uncorrelated entries show a circular distribution with delocalized eigenvectors, where as 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 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.
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We search for this effect for the whole range of correlation strengths and for different community configurations.
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