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This review presents an account of the major works done on spectra of adjacency matrices drawn on networks and the basic understanding attained so far. We have divided the review under three sections: (a) extremal eigenvalues, (b) bulk part of the spectrum, and (c) degenerate eigenvalues, based on the intrinsic properties of eigenvalues and the phenomena they capture. We have reviewed the works done for spectra of various popular model networks, such as the Erdős-Rényi random networks, scale-free networks, 1-d lattice, small-world networks, and various different real-world networks. Additionally, potential applications of spectral properties for natural processes have been reviewed. [ABSTRACT FROM AUTHOR]

The spectra of many real world networks exhibit properties which are different from those of random networks generated using various models. One such property is the existence of a very high degeneracy at the zero eigenvalue. In this work, we provide all the possible reasons behind the occurrence of the zero degeneracy in the network spectra, namely, the complete and partial duplications, as well as their implications. The power-law degree sequence and the preferential attachment are the properties which enhances the occurrence of such duplications and hence leading to the zero degeneracy. A comparison of the zero degeneracy in protein-protein interaction networks of six different species and in their corresponding model networks indicates importance of the degree sequences and the power-law exponent for the occurrence of zero degeneracy. [ABSTRACT FROM AUTHOR]

We investigate the impact of degree-degree correlations on the spectra of networks. Even though density distributions exhibit drastic changes depending on the (dis)assortative mixing and the network architecture, the short-range correlations in eigenvalues exhibit universal random matrix theory predictions. The long-range correlations turn out to be a measure of randomness in (dis)assortative networks. The analysis further provides insight into the origin of high degeneracy at the zero eigenvalue displayed by a majority of biological networks. [ABSTRACT FROM AUTHOR]

Though biological and artificial complex systems having inhibitory connections exhibit a high degree of clustering in their interaction pattern, the evolutionary origin of clustering in such systems remains a challenging problem. Using genetic algorithm we demonstrate that inhibition is required in the evolution of clique structure from primary random architecture, in which the fitness function is assigned based on the largest eigenvalue. Further, the distribution of triads over nodes of the network evolved from mixed connections reveals a negative correlation with its degree providing insight into origin of this trend observed in real networks. [ABSTRACT FROM AUTHOR]

Despite the key role played by inhibitory-excitatory couplings in the functioning of brain networks, the impact of a balanced condition on the stability properties of underlying networks remains largely unknown. We investigate properties of the largest eigenvalues of networks having such couplings, and find that they follow completely different statistics when in the balanced situation. Based on numerical simulations, we demonstrate that the transition from Weibull to Fréchet via the Gumbel distribution can be controlled by the variance of the column sum of the adjacency matrix, which depends monotonically on the denseness of the underlying network. As a balanced condition is imposed, the largest real part of the eigenvalue emulates a transition to the generalized extreme-value statistics, independent of the inhibitory connection probability. Furthermore, the transition to the Weibull statistics and the small-world transition occur at the same rewiring probability, reflecting a more stable system. [ABSTRACT FROM AUTHOR]

Inspired by the importance of inhibitory and excitatory couplings in the brain, we analyze the largest eigenvalue statistics of random networks incorporating such features. We find that the largest real part of eigenvalues of a network, which accounts for the stability of an underlying system, decreases linearly as a function of inhibitory connection probability up to a particular threshold value, after which it exhibits rich behaviors with the distribution manifesting generalized extreme value statistics. Fluctuations in the largest eigenvalue remain somewhat robust against an increase in system size but reflect a strong dependence on the number of connections, indicating that systems having more interactions among its constituents are likely to be more unstable. [ABSTRACT FROM AUTHOR]

We investigate the spectra of adjacency matrices of multiplex networks under random matrix theory (RMT) framework. Through extensive numerical experiments, we demonstrate that upon multiplexing two random networks, the spectra of the combined multiplex network exhibit superposition of two Gaussian orthogonal ensemble (GOE)s for very small multiplexing strength followed by a smooth transition to the GOE statistics with an increase in the multiplexing strength. Interestingly, randomness in the connection architecture, introduced by random rewiring to 1D lattice, of at least one layer may govern nearest neighbor spacing distribution (NNSD) of the entire multiplex network, and in fact, can drive to a transition from the Poisson to the GOE statistics or vice versa. Notably, this transition transpires for a very small number of the random rewiring corresponding to the small-world transition. Ergo, only one layer being represented by the small-world network is enough to yield GOE statistics for the entire multiplex network. Spectra of adjacency matrices of underlying interaction networks have been contemplated to be related with dynamical behavior of the corresponding complex systems, the investigations presented here have implications in achieving better structural and dynamical control to the systems represented by multiplex networks against structural perturbation in only one of the layers. • Impact of structural randomness on the spectra of an entire multiplex network. • The interplay of spectra, interlayer coupling, and structural randomness in multiplex networks. • Poisson-GOE transition is observed in the course of SW transition in one of the layers. • Results are consistent with different kind of randomness and 3-layer multiplex networks. [ABSTRACT FROM AUTHOR]

We study localization properties of principal eigenvectors (PEVs) of multilayer networks (MNs). Starting with a multilayer network corresponding to a delocalized PEV, we rewire the network edges using an optimization technique such that the PEV of the rewired multilayer network becomes more localized. The framework allows us to scrutinize structural and spectral properties of the networks at various localization points during the rewiring process. We show that rewiring only one layer is enough to attain a MN having a highly localized PEV. Our investigation reveals that a single edge rewiring of the optimized MN can lead to the complete delocalization of a highly localized PEV. This sensitivity in the localization behavior of PEVs is accompanied with the second largest eigenvalue lying very close to the largest one. This observation opens an avenue to gain a deeper insight into the origin of PEV localization of networks. Furthermore, analysis of multilayer networks constructed using real-world social and biological data shows that the localization properties of these real-world multilayer networks are in good agreement with the simulation results for the model multilayer network. This paper is relevant to applications that require understanding propagation of perturbation in multilayer networks. [ABSTRACT FROM AUTHOR]

Abstract: We analyze protein–protein interaction networks for six different species under the framework of random matrix theory. Nearest neighbor spacing distribution of the eigenvalues of adjacency matrices of the largest connected part of these networks emulate universal Gaussian orthogonal statistics of random matrix theory. We demonstrate that spectral rigidity, which quantifies long range correlations in eigenvalues, for all protein–protein interaction networks follow random matrix prediction up to certain ranges indicating randomness in interactions. After this range, deviation from the universality evinces underlying structural features in network. [Copyright &y& Elsevier]