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We quantify the dynamical implications of the small-world phenomenon. We consider the generic synchronization of oscillator networks of arbitrary topology, and link the linear stability of the synchronous state to an algebraic condition of the Laplacian of the graph. We show numerically that the addition of random shortcuts produces improved network synchronizability. Further, we use a perturbation analysis to place the synchronization threshold in relation to the boundaries of the small-world region. Our results also show that small-worlds synchronize as efficiently as random graphs and hypercubes, and more so than standard constructive graphs. PACS numbers:Recently, Watts and Strogatz [1] showed that the addition of a few long-range shortcuts to an otherwise locally connected lattice (the "pristine world") produces a sharp reduction of the average distance between arbitrary nodes. The ensuing semi-random lattice was denoted a small-world (SW) because the sudden appearance of short paths occurs early on, while the system is still relatively localized. This concept has wide appeal: the SW property seems to be a quantifiable characteristic of many real-world structures [1, 2, 3, 4], both human generated (social networks, WWW, power grid), or of biological origin (neural and biochemical networks).A spur of ongoing research [2] has concentrated on static and combinatoric properties [5,6,7,8,9, 10] of a tractable SW model [1,11]. Monasson [11] considered the SW effect on the distribution of eigenvalues of the connectivity matrix (the graph Laplacian) which specifies the coupling between nodes-a relevant topic for polymer networks [12]. However, despite their central role in real-world networks, there are fewer studies of dynamical processes taking place on SW lattices. Among those, automata epidemics simulations [13] In this paper, we explicitly link the SW addition of random shortcuts to the synchronization of networks of coupled dynamical systems. This is an example of dynamics on networks-leaving aside the distinct problem of evolution of networks here. By using a generic synchronization formulation [17,18] to factor out the connectivity of the network, we identify the synchronization threshold with an algebraic condition of the graph Laplacian. Through numerics and analysis, we quantify how the SW scheme improves the synchronizability of the pristine world, mainly as a result of the steep increase of the first-non-zero eigenvalue (FNZE). The synchronization threshold is found to lie in the SW region [3, 13], but does not coincide with its onset-it can in fact be linked to the effective randomization that ends SW. Within this framework, we show that the synchronization efficiency of semi-random SW networks is higher than standard deterministic graphs, and comparable to both fully random and ideal constructive graphs.Consider n identical dynamical systems (placed at the nodes of a graph) that are linearly and symmetrically coupled (as represented by the edges of the undirected graph) with global coupling strengt...

Synchronization is of central importance in power distribution, telecommunication, neuronal and biological networks. Many networks are observed to produce patterns of synchronized clusters, but it has been difficult to predict these clusters or understand the conditions under which they form. Here we present a new framework and develop techniques for the analysis of network dynamics that shows the connection between network symmetries and cluster formation. The connection between symmetries and cluster synchronization is experimentally confirmed in the context of real networks with heterogeneities and noise using an electrooptic network. We experimentally observe and theoretically predict a surprising phenomenon in which some clusters lose synchrony without disturbing the others. Our analysis shows that such behaviour will occur in a wide variety of networks and node dynamics. The results could guide the design of new power grid systems or lead to new understanding of the dynamical behaviour of networks ranging from neural to social.

The field of chaotic synchronization has grown considerably since its advent in 1990. Several subdisciplines and "cottage industries" have emerged that have taken on bona fide lives of their own. Our purpose in this paper is to collect results from these various areas in a review article format with a tutorial emphasis. Fundamentals of chaotic synchronization are reviewed first with emphases on the geometry of synchronization and stability criteria. Several widely used coupling configurations are examined and, when available, experimental demonstrations of their success (generally with chaotic circuit systems) are described. Particular focus is given to the recent notion of synchronous substitution-a method to synchronize chaotic systems using a larger class of scalar chaotic coupling signals than previously thought possible. Connections between this technique and well-known control theory results are also outlined. Extensions of the technique are presented that allow so-called hyperchaotic systems (systems with more than one positive Lyapunov exponent) to be synchronized. Several proposals for "secure" communication schemes have been advanced; major ones are reviewed and their strengths and weaknesses are touched upon. Arrays of coupled chaotic systems have received a great deal of attention lately and have spawned a host of interesting and, in some cases, counterintuitive phenomena including bursting above synchronization thresholds, destabilizing transitions as coupling increases (short-wavelength bifurcations), and riddled basins. In addition, a general mathematical framework for analyzing the stability of arrays with arbitrary coupling configurations is outlined. Finally, the topic of generalized synchronization is discussed, along with data analysis techniques that can be used to decide whether two systems satisfy the mathematical requirements of generalized synchronization. (c) 1997 American Institute of Physics.