Random talk: Random walk and synchronizability in a moving neighborhood network (original) (raw)

Stochastic synchronization over a moving neighborhood network

2007 American Control Conference, 2007

We examine the synchronization problem for a group of dynamic agents that communicate via a moving neighborhood network. Each agent is modeled as a random walker in a finite lattice and is equipped with an oscillator. The communication network topology changes randomly and is dictated by the agents' locations in the lattice. Information sharing (talking) is possible only for geographically neighboring agents. The complex system is a time-varying jump nonlinear system. We introduce the concept of long-time expected communication network defined as the ergodic limit of the stochastic time-varying network. We show that if the long-time expected network supports synchronization, then so does the stochastic network when the agents diffuse sufficiently fast in the lattice.

Synchronization in time-varying random networks with vanishing connectivity

Scientific Reports

A sufficiently connected topology linking the constituent units of a complex system is usually seen as a prerequisite for the emergence of collective phenomena such as synchronization. We present a random network of heterogeneous phase oscillators in which the links mediating the interactions are constantly rearranged with a characteristic timescale and, possibly, an extremely low instantaneous connectivity. We show that with strong coupling and sufficiently fast rewiring the network reaches partial synchronization even in the vanishing connectivity limit. In particular, we provide an approximate analytical argument, based on the comparison between the different characteristic timescales of our system in the low connectivity regime, which is able to predict the transition to synchronization threshold with satisfactory precision beyond the formal fast rewiring limit. We interpret our results as a qualitative mechanism for emergence of consensus in social communities. In particular, our result suggest that groups of individuals are capable of aligning their opinions under extremely sparse exchanges of views, which is reminiscent of fast communications that take place in the modern social media. Our results may also be relevant to characterize the onset of collective behavior in engineered systems of mobile units with limited wireless capabilities.

Network synchronization, diffusion, and the paradox of heterogeneity

Physical Review E, 2005

Many complex networks display strong heterogeneity in the degree (connectivity) distribution. Heterogeneity in the degree distribution often reduces the average distance between nodes but, paradoxically, may suppress synchronization in networks of oscillators coupled symmetrically with uniform coupling strength. Here we offer a solution to this apparent paradox. Our analysis is partially based on the identification of a diffusive process underlying the communication between oscillators and reveals a striking relation between this process and the condition for the linear stability of the synchronized states. We show that, for a given degree distribution, the maximum synchronizability is achieved when the network of couplings is weighted and directed, and the overall cost involved in the couplings is minimum. This enhanced synchronizability is solely determined by the mean degree and does not depend on the degree distribution and system size. Numerical verification of the main results is provided for representative classes of small-world and scale-free networks.

Synchronization in complex networks

Physics Reports, 2008

Synchronization processes in populations of locally interacting elements are in the focus of intense research in physical, biological, chemical, technological and social systems. The many efforts devoted to understand synchronization phenomena in natural systems take now advantage of the recent theory of complex networks. In this review, we report the advances in the comprehension of synchronization phenomena when oscillating elements are constrained to interact in a complex network topology. We also overview the new emergent features coming out from the interplay between the structure and the function of the underlying pattern of connections. Extensive numerical work as well as analytical approaches to the problem are presented. Finally, we review several applications of synchronization in complex networks to different disciplines: biological systems and neuroscience, engineering and computer science, and economy and social sciences.

Paths to synchronization on complex networks

Physical review letters, 2007

The understanding of emergent collective phenomena in natural and social systems has driven the interest of scientists from different disciplines during decades. Among these phenomena, the synchronization of a set of interacting individuals or units has been intensively studied because of its ubiquity in the natural world. In this paper, we show how for fixed coupling strengths local patterns of synchronization emerge differently in homogeneous and heterogeneous complex networks, driving the process towards a certain global synchronization degree following different paths. The dependence of the dynamics on the coupling strength and on the topology is unveiled. This study provides a new perspective and tools to understand this emerging phenomena. PACS numbers: 05.45.Xt, 89.75.Fb In 1998 Watts and Strogatz in an effort to understand the synchronization of cricket chirps, which show a high degree of coordination over long distances as though the insects where "invisibly" connected, end up with a seminal paper about the small-world effect [1] that was the seed of the modern theory of complex networks . Many natural and man-made networks have been, since then, successfully described within this framework. Nevertheless, the understanding of the synchronization dynamics in complex networks remains a challenge.

Synchronization on small-world networks

2001

We investigate collective synchronization in a system of coupled oscillators on small-world networks. The order parameters which measure synchronization of phases and frequencies are introduced and analyzed by means of dynamic simulations and finite-size scaling. Phase synchronization is observed to emerge in the presence of even a tiny fraction PPP of shortcuts and to display saturated behavior for Pgtrsim0.5P \gtrsim 0.5Pgtrsim0.5. This indicates that the same synchronizability as the random network (P=1) can be achieved with relatively small number of shortcuts. The transient behavior of the synchronization, obtained from the measurement of the relaxation time, is also discussed.

Synchronization in time-varying networks

Physical Review E, 2014

We study the stability of the synchronized state in time varying complex networks using the concept of basin stability which is a nonlocal and nonlinear measure of stability that can be easily applied to high−dimensional systems [P. J. Menck, J. Heitzig, N. Marwan, and J. Kurths, Nature Physics 9, 89 (2013)]. The time varying character is included by stochastically rewiring each link with the average frequency f . We find that the time taken to reach synchronization is lowered and the stability range of the synchronized state increases considerably in dynamic networks. Further we uncover that small−world networks are much more sensitive to link changes than random ones, with the time varying character of the network having a significant effect at much lower rewiring frequencies. At very high rewiring frequencies, random networks perform better than small−world networks and the synchronized state is stable over a much wider window of coupling strengths. Lastly we show that the stability range of the synchronized state may be quite different for small and large perturbations, and so the linear stability analysis and the basin stability criterion provide complementary indicators of stability.

Synchronizability of Highly Clustered Scale-Free Networks

Chinese Physics Letters, 2006

In this letter, we consider the effect of clustering coefficient on the synchronizability of coupled oscillators located on scale-free networks. The analytic result for the value of clustering coefficient aiming at a highly clustered scale-free network model, the Holme-Kim is obtained, and the relationship between network synchronizability and clustering coefficient is reported. The simulation results strongly suggest that the more clustered the network, the poorer its synchronizability. PACS numbers: 89.75,-k, 05.45.Xt

Average distance as a predictor of synchronisability in networks of coupled oscillators

2010

Abstract The importance of networks of coupled oscillators is widely recognized. Such networks occur in biological systems like the heart, in chemical systems, in computational problems, and in engineering systems. Systems of coupled oscillators can also be used as an abstract model for synchronisation in organisations. Here we show that synchronisability in a specific coupledoscillator model, the Kuramoto model, is best predicted using the average distance (or characteristic path length) between nodes in the network.