Though biological and artificial complex systems having inhibitory connections exhibit high degree of clustering in their interaction pattern, the evolutionary origin of clustering in such systems remains a challenging problem. Using genetic algorith
m 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 exhibits a negative correlation with its degree providing insight into origin of this trend observed in real networks.
We analyze the gene expression data of Zebrafish under the combined framework of complex networks and random matrix theory. The nearest neighbor spacing distribution of the corresponding matrix spectra follows random matrix predictions of Gaussian or
thogonal statistics. Based on the eigenvector analysis we can divide the spectra into two parts, first part for which the eigenvector localization properties match with the random matrix theory predictions, and the second part for which they show deviation from the theory and hence are useful to understand the system dependent properties. Spectra with the localized eigenvectors can be characterized into three groups based on the eigenvalues. We explore the position of localized nodes from these different categories. Using an overlap measure, we find that the top contributing nodes in the different groups carry distinguished structural features. Furthermore, the top contributing nodes of the different localized eigenvectors corresponding to the lower eigenvalue regime form different densely connected structure well separated from each other. Preliminary biological interpretation of the genes, associated with the top contributing nodes in the localized eigenvectors, suggests that the genes corresponding to same vector share common features.
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 netwo
rks, 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.
We use symbolic dynamics to study discrete-time dynamical systems with multiple time delays. We exploit the concept of avoiding sets, which arise from specific non-generating partitions of the phase space and restrict the occurrence of certain symbol
sequences related to the characteristics of the dynamics. In particular, we show that the resulting forbidden sequences are closely related to the time delays in the system. We present two applications to coupled map lattices, namely (1) detecting synchronization and (2) determining unknown values of the transmission delays in networks with possibly directed and weighted connections and measurement noise. The method is applicable to multi-dimensional as well as set-valued maps, and to networks with time-varying delays and connection structure.
We analyze gene co-expression network under the random matrix theory framework. The nearest neighbor spacing distribution of the adjacency matrix of this network follows Gaussian orthogonal statistics of random matrix theory (RMT). Spectral rigidity
test follows random matrix prediction for a certain range, and deviates after wards. Eigenvector analysis of the network using inverse participation ratio (IPR) suggests that the statistics of bulk of the eigenvalues of network is consistent with those of the real symmetric random matrix, whereas few eigenvalues are localized. Based on these IPR calculations, we can divide eigenvalues in three sets; (A) The non-degenerate part that follows RMT. (B) The non-degenerate part, at both ends and at intermediate eigenvalues, which deviate from RMT and expected to contain information about {it important nodes} in the network. (C) The degenerate part with $zero$ eigenvalue, which fluctuates around RMT predicted value. We identify nodes corresponding to the dominant modes of the corresponding eigenvectors and analyze their structural properties.
We study spectral behavior of sparsely connected random networks under the random matrix framework. Sub-networks without any connection among them form a network having perfect community structure. As connections among the sub-networks are introduced
, the spacing distribution shows a transition from the Poisson statistics to the Gaussian orthogonal ensemble statistics of random matrix theory. The eigenvalue density distribution shows a transition to the Wigners semicircular behavior for a completely deformed network. The range for which spectral rigidity, measured by the Dyson-Mehta $Delta_3$ statistics, follows the Gaussian orthogonal ensemble statistics depends upon the deformation of the network from the perfect community structure. The spacing distribution is particularly useful to track very slight deformations of the network from a perfect community structure, whereas the density distribution and the $Delta_3$ statistics remain identical to the undeformed network. On the other hand the $Delta_3$ statistics is useful for the larger deformation strengths. Finally, we analyze the spectrum of a protein-protein interaction network for Helicobacter, and compare the spectral behavior with those of the model networks.
We analyze complex networks under random matrix theory framework. Particularly, we show that $Delta_3$ statistic, which gives information about the long range correlations among eigenvalues, provides a qualitative measure of randomness in networks. A
s networks deviate from the regular structure, $Delta_3$ follows random matrix prediction of linear behavior, in semi-logarithmic scale with the slope of $1/pi^2$, for the longer scale.
We show how a simple scheme of symbolic dynamics distinguishes a chaotic from a random time series and how it can be used to detect structural relationships in coupled dynamics. This is relevant for the question at which scale in complex dynamics regularities and patterns emerge.
