Sarika Jalan

Self-organized and driven phase synchronization in coupled maps.

We study the phase synchronization and cluster formation in coupled maps on different networks. We identify two different mechanisms of cluster formation: (a) self-organized phase synchronization which leads to clusters with dominant intracluster couplings and (b) driven phase synchronization which leads to clusters with dominant intercluster couplings. In the novel driven synchronization the nodes of one cluster are driven by those of the others. We also discuss the dynamical origin of these two mechanisms for small networks with two and three nodes.

Synchronized clusters in coupled map networks. I. Numerical studies

We study the synchronization of coupled maps on a variety of networks including regular one and two dimensional networks, scale free networks, small world networks, tree networks, and random networks. For small coupling strengths nodes show turbulent behavior but form phase synchronized clusters as coupling increases. We identify two different ways of cluster formation, self-organized clusters which have mostly intra-cluster couplings and driven clusters which have mostly inter-cluster couplings. The synchronized clusters may be of dominant self-organized type, dominant driven type or mixed type depending on the type of network and the parameters of the dynamics. There are some nodes of the floating type that show intermittent behaviour between getting attached to some clusters and evolving independently. The residence times of a floating node in a synchronized cluster show an exponential distribution. We define different states of the coupled dynamics by considering the number and type of synchronized clusters. For the local dynamics governed by the logistic map we study the phase diagram in the plane of the coupling constant (

Random matrix analysis of complex networks

\epsilon

Universality in complex networks: random matrix analysis.

) and the logistic map parameter (

Birth and death of chimera: Interplay of delay and multiplexing

\mu

Self-organized and driven phase synchronization in coupled map networks

). For large coupling strengths and nonlinear coupling we find that the scale free networks and the Caley tree networks lead to better cluster formation than the other types of networks with the same average connectivity. For most of our study we use the number of connections of the order of the number of nodes which allows us to distinguish between the two mechanisms of cluster formation. As the number of connections increases the number of nodes forming clusters and the size of the clusters in general increase.

Random matrix analysis of localization properties of gene coexpression network.

We study complex networks under random matrix theory (RMT) framework. Using nearest-neighbor and next-nearest-neighbor spacing distributions we analyze the eigenvalues of the adjacency matrix of various model networks, namely, random, scale-free, and small-world networks. These distributions follow the Gaussian orthogonal ensemble statistic of RMT. To probe long-range correlations in the eigenvalues we study spectral rigidity via the Delta_{3} statistic of RMT as well. It follows RMT prediction of linear behavior in semilogarithmic scale with the slope being approximately 1pi;{2} . Random and scale-free networks follow RMT prediction for very large scale. A small-world network follows it for sufficiently large scale, but much less than the random and scale-free networks.

Emergence of Chimera in Multiplex Network

We apply random matrix theory to complex networks. We show that nearest neighbor spacing distribution of the eigenvalues of the adjacency matrices of various model networks, namely scale-free, small-world, and random networks follow universal Gaussian orthogonal ensemble statistics of random matrix theory. Second, we show an analogy between the onset of small-world behavior, quantified by the structural properties of networks, and the transition from Poisson to Gaussian orthogonal ensemble statistics, quantified by Brody parameter characterizing a spectral property. We also present our analysis for a protein-protein interaction network in budding yeast.

Impact of a leader on cluster synchronization.

The chimera state with co-existing coherent and incoherent dynamics has recently attracted a lot of attention due to its wide applicability. We investigate non-locally coupled identical chaotic maps with delayed interactions in the multiplex network framework and find that an interplay of delay and multiplexing brings about an enhanced or suppressed appearance of the chimera state depending on the distribution as well as the parity of delay values in the layers. Additionally, we report a layer chimera state with the existence of one layer exhibiting coherent and another layer incoherent dynamical evolution. The rich variety of dynamical behavior demonstrated here can be used to gain further insight into the real-world networks which inherently possess such multi-layer architecture with delayed interactions.

Analytical results for stochastically growing networks : Connection to the zero-range process

We study synchronization of coupled logistic maps on networks. For small coupling strengths nodes show turbulent behaviour but form phase synchronized clusters as coupling increases. We identify two different ways of cluster formation, self-organized clusters which have mostly intra-cluster couplings and driven clusters which have mostly inter-cluster couplings. In the novel driven synchronization the nodes of one cluster are driven by those of the others.