Kwangho Park

Synchronization in complex networks with a modular structure

Networks with a community (or modular) structure arise in social and biological sciences. In such a network individuals tend to form local communities, each having dense internal connections. The linkage among the communities is, however, much more sparse. The dynamics on modular networks, for instance synchronization, may be of great social or biological interest. (Here by synchronization we mean some synchronous behavior among the nodes in the network, not, for example, partially synchronous behavior in the network or the synchronizability of the network with some external dynamics.) By using a recent theoretical framework, the master-stability approach originally introduced by Pecora and Carroll in the context of synchronization in coupled nonlinear oscillators, we address synchronization in complex modular networks. We use a prototype model and develop scaling relations for the network synchronizability with respect to variations of some key network structural parameters. Our results indicate that random, long-range links among distant modules is the key to synchronization. As an application we suggest a viable strategy to achieve synchronous behavior in social networks.

Complex networks: Dynamics and security

This paper presents a perspective in the study of complex networks by focusing on how dynamics may affect network security under attacks. In particular, we review two related problems: attack-induced cascading breakdown and range-based attacks on links. A cascade in a network means the failure of a substantial fraction of the entire network in a cascading manner, which can be induced by the failure of or attacks on only a few nodes. These have been reported for the internet and for the power grid (e.g., the August 10, 1996 failure of the western United States power grid). We study a mechanism for cascades in complex networks by constructing a model incorporating the flows of information and physical quantities in the network. Using this model we can also show that the cascading phenomenon can be understood as a phase transition in terms of the key parameter characterizing the node capacity. For a parameter value below the phase-transition point, cascading failures can cause the network to disintegrate almost entirely. We will show how to obtain a theoretical estimate for the phase-transition point. The second problem is motivated by the fact that most existing works on the security of complex networks consider attacks on nodes rather than on links. We address attacks on links. Our investigation leads to the finding that many scale-free networks are more sensitive to attacks on short-range than on long-range links. Considering that the small-world phenomenon in complex networks has been identified as being due to the presence of long-range links, i.e., links connecting nodes that would otherwise be separated by a long node-to-node distance, our result, besides its importance concerning network efficiency and security, has the striking implication that the small-world property of scale-free networks is mainly due to short-range links.

Optimal structure of complex networks for minimizing traffic congestion

To design complex networks to minimize traffic congestion, it is necessary to understand how traffic flow depends on network structure. We study data packet flow on complex networks, where the packet delivery capacity of each node is not fixed. The optimal configuration of capacities to minimize traffic congestion is derived and the critical packet generating rate is determined, below which the network is at a free flow state but above which congestion occurs. Our analysis reveals a direct relation between network topology and traffic flow. Optimal network structure, free of traffic congestion, should have two features: uniform distribution of load over all nodes and small network diameter. This finding is confirmed by numerical simulations. Our analysis also makes it possible to theoretically compare the congestion conditions for different types of complex networks. In particular, we find that network with low critical generating rate is more susceptible to congestion. The comparison has been made on the following complex-network topologies: random, scale-free, and regular.

Binary spreading process with parity conservation.

Recently there has been a debate concerning the universal properties of the phase transition in the pair contact process with diffusion (PCPD) 2A-->3A, 2A-->0. Although some of the critical exponents seem to coincide with those of the so-called parity-conserving universality class, it was suggested that the PCPD might represent an independent class of phase transitions. This point of view is motivated by the argument that the PCPD does not conserve parity of the particle number. In the present work we question what happens if the parity conservation law is restored. To this end, we consider the reaction-diffusion process 2A-->4A, 2A-->0. Surprisingly, this process displays the same type of critical behavior, leading to the conclusion that the most important characteristics of the PCPD is the use of binary reactions for spreading, regardless of whether parity is conserved or not.

Attack induced cascading breakdown in complex networks

The possibility that a complex network can be brought down by attack on a single or very few nodes through the process of cascading failures is of significant concern. In this paper, we investigate cascading failures in complex networks and uncover a phase-transition phenomenon in terms of the key parameter characterizing the node capacity. For parameter value below the phase-transition point, cascading failures can cause the network to disintegrate almost entirely. Then we show how to design networks of finite capacity that are safe against cascading breakdown. Our theory yields estimates for the maximally achievable network integrity via controlled removal of a small set of low-degree nodes.

Effect of common noise on phase synchronization in coupled chaotic oscillators

We report a general phenomenon concerning the effect of noise on phase synchronization in coupled chaotic oscillators: the average phase-synchronization time exhibits a nonmonotonic behavior with the noise amplitude. In particular, we find that the time exhibits a local minimum for relatively small noise amplitude but a local maximum for stronger noise. We provide numerical results, experimental evidence from coupled chaotic circuits, and a heuristic argument to establish the generality of this phenomenon.

Frequency dependence of phase-synchronization time in nonlinear dynamical systems

It has been found recently that the averaged phase-synchronization time between the input and the output signals of a nonlinear dynamical system can exhibit an extremely high sensitivity to variations in the noise level. In real-world signal-processing applications, sensitivity to frequency variations may be of considerable interest. Here we investigate the dependence of the averaged phase-synchronization time on frequency of the input signal. Our finding is that, for typical nonlinear oscillator systems, there can be a frequency regime where the time exhibits significant sensitivity to frequency variations. We obtain an analytic formula to quantify the frequency dependence, provide numerical support, and present experimental evidence from a simple nonlinear circuit system.

Dynamics of an interface driven through random media: The effect of a spatially correlated noise

We studied the quenched Edwards–Wilkinson (QEW) equation with power-law type of a correlated noise near the depinning threshold. We solved analytically the QEW equation by using a functional renorm...