By manipulating the clustering coefficient of a network without changing its degree distribution, we examine the effect of clustering on the synchronization of phase oscillators on networks withPoisson and scale-free degree distributions. For both types of network, increased clustering hinders global synchronization as the network splits into dynamical clusters that oscillate at different frequencies. Surprisingly, in scale-free networks, clustering promotes the synchronization of the most connected nodes (hubs) even though it inhibits global synchronization. As a result, they show an additional, advanced transition instead of a single synchronization threshold. This clusterenhanced synchronization of hubs may be relevant to the brain which is scale-free and highly clustered.

We examine numerically the three-way relationships among structure, Laplacian spectra and frequency synchronization dynamics on complex networks. We study the effects of clustering, degree distribution and a particular type of coupling asymmetry (input normalization), all of which are known to have effects on the synchronizability of oscillator networks. We find that these topological factors produce marked signatures in the Laplacian eigenvalue distribution and in the localization properties of individual eigenvectors. Using a set of coordinates based on the Laplacian eigenvectors as a diagnostic tool for synchronization dynamics, we find that the process of frequency synchronization can be visualized as a series of quasiindependent transitions involving different normal modes. Particular features of the partially synchronized state can be understood in terms of the behavior of particular modes or groups of modes.For example, there are important partially synchronized states in which a set of low-lying modes remain unlocked while those in the main spectral peak are locked. We find therefore that spectra influence dynamics in ways that go beyond results relating a single threshold to a single extremal eigenvalue.

We describe a numerical simulation of the evolution of an S 3 cosmic string network which takes fully into account the noncommutative nature of the cosmic string fluxes and the topological obstructions which hinder strings from moving past each other or intercommuting. The influence of initial conditions, string tensions, and other parameters on the network's evolution is explored. Contrary to some previous suggestions, we find no strong evidence of the ''freezing'' required for a string-dominated cosmological scenario. Instead, the results in a broad range of regimes are consistent with the familiar scaling law, i.e., a constant number of strings per horizon volume. The size of this number, however, can vary quite a bit, as can other overall features. There is a surprisingly strong dependence on the statistical properties of the initial conditions. We also observe a rich variety of interesting new structures, such as light string webs stretched between heavier strings, which are not seen in Abelian networks. ͓S0556-2821͑98͒00304-X͔ PACS number͑s͒: 98.80.Cq

We perform numerical studies of a reaction-diffusion system that is both Turing and Hopf unstable, and that grows by addition at a moving boundary (which is equivalent by a Galilean transformation to a reaction-diffusion-advection system with a fixed boundary and a uniform flow). We model the conditions of a recent set of experiments which used a temporally varying illumination in the boundary region to control the formation of patterns in the bulk of the photosensitive medium. The frequency of the illumination variations can select patterns from among the competing instabilities of the medium. In the usual case, the waves that are excited have frequencies (as measured at a constant distance from the upstream boundary) matching the driving frequency. In contrast to the usual case, we find that both Turing patterns and flow-distributed oscillation waves can be excited by forcing at subharmonic multiples of the wave frequencies. The final waves (with frequencies at integer multiples of the driving frequency) are formed by a process in which transient wave fronts break up and reconnect. We find ratios of response to driving frequency as high as 10.

To explore the relation between network structure and function, we studied the computational performance of Hopfield-type attractor neural nets with regular lattice, random, small-world, and scale-free topologies. The random configuration is the most efficient for storage and retrieval of patterns by the network as a whole. However, in the scale-free case retrieval errors are not distributed uniformly among the nodes. The portion of a pattern encoded by the subset of highly connected nodes is more robust and efficiently recognized than the rest of the pattern. The scale-free network thus achieves a very strong partial recognition. The implications of these findings for brain function and social dynamics are suggestive.

We analyze the synchronization dynamics of phase oscillators far from the synchronization manifold, including the onset of synchronization on scale-free networks with low and high clustering coefficients. We use normal coordinates and corresponding time-averaged velocities derived from the Laplacian matrix, which reflects the network's topology. In terms of these coordinates, synchronization manifests itself as a contraction of the dynamics onto progressively lower-dimensional submanifolds of phase space spanned by Laplacian eigenvectors with lower eigenvalues. Differences between high and low clustering networks can be correlated with features of the Laplacian spectrum. For example, the inhibition of full synchoronization at high clustering is associated with a group of low-lying modes that fail to lock even at strong coupling, while the advanced partial synchronization at low coupling noted elsewhere is associated with high-eigenvalue modes.

Random networks of symmetrically coupled, excitable elements can self-organize into coherently oscillating states if the networks contain loops (indeed loops are abundant in random networks) and if the initial conditions are sufficiently random. In the oscillating state, signals propagate in a single direction and one or a few network loops are selected as driving loops in which the excitation circulates periodically. We analyze the mechanism, describe the oscillating states, identify the pacemaker loops and explain key features of their distribution. This mechanism may play a role in epileptic seizures.PACS numbers: 89.75. Hc, 05.65.+b, 87.19.lj, 87.19.xm The coherent oscillation (CO) of a collection of units that are non-oscillatory on their own is relevant to biological and physical sciences: CO has been identified and analyzed in populations of excitable biological cells ( [7]. In contrast with well-studied synchronization phenomena of self-oscillating units [8], in these cases the ability to oscillate derives from the interactions of the elements. CO can occur also on complex networks if the nodes are excitable [9][10] or even monostable [9], provided that the network contains loops, and that the directional symmetry of couplings is somehow broken to allow a signal to propagate in a single direction around a loop [10].Networks of these types include some neural [11][12] and genetic regulatory networks.[9] Some studies of excitable networks have been inspired by target and spiral waves in continuous media. [13][7] [10] In complex networks, loops are both generic and abundant. [15][14] While short loops of length L ≪ N (where N is the network size) are rare in large random networks, ones with L ln N occur generically in numbers growing exponentially with N . Their number also grows roughly exponentially with L up to a maximum at a length L m ∼ N.[15] [14] In this letter we show that random excitable networks readily self-organize into a CO state following a transient phase during which one or a few loops are dynamically selected as driving (or pacemaker) loops. We describe the mechanism of signal propagation and gain an understanding of the resulting distributions of the oscillating states and associated driving loops.We consider networks of diffusively coupled, excitable elements with dynamics described by the Bär model [16] N is the number of nodes, u i and v i are dynamical variables, D is the coupling strength, and A ij is the adjacency matrix. a, b, and ε are parameters, for which we adopt the values a = 0.84, b = 0.07, and ε = 0.04. In the absence of coupling, the individual nodes display excitable dynamics, with a stable equilibrium at (u, v) = (0, 0) and an excitation threshold u th ≈ 0.1. The dynamical equations were integrated numerically using a fourth-order Runge-Kutta algorithm with time step ∆t = 0.1. For the topology, we chose the undirected random regular network of degree 3 (RRN3), where nodes are randomly and symmetrically connected with the constraint that all have the same degr...

It is shown that a model with a spontaneously broken global symmetry can support defects analogous to Alice strings, and a process analogous to Cheshire charge exchange can take place. A possible realization in superfluid He-3 is pointed out.