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We study the synchronization transition (ST) of a modified Kuramoto model on two different types of modular complex networks. It is found that the ST depends on the type of inter-modular connections. For the network with decentralized (centralized) inter-modular connections, the ST occurs at finite coupling constant (behaves abnormally). Such distinct features are found in the yeast protein interaction network and the Internet, respectively. Moreover, by applying the finite-size scaling analysis to an artificial network with decentralized inter-modular connections, we obtain the exponent associated with the order parameter of the ST to be $\beta \approx 1$ different from $\beta_{\rm MF} \approx 1/2$ obtained from the scale-free network with the same degree distribution but the absence of modular structure, corresponding to the mean field value., Comment: 9pages, 6figures. PRE (in press)

We study collective behavior of locally-coupled limit-cycle oscillators with scattered intrinsic frequencies on $d$-dimensional lattices. A linear analysis shows that the system should be always desynchronized up to $d=4$. On the other hand, numerical investigation for $d= 5$ and 6 reveals the emergence of the synchronized (ordered) phase via a continuous transition from the fully random desynchronized phase. This demonstrates that the lower critical dimension for the phase synchronization in this system is $d_{l}=4$

While shorter characteristic path length has in general been believed to enhance synchronizability of a coupled oscillator system on a complex network,the suppressing tendency of the heterogeneity of the degree distribution, even for shorter characteristic path length, has also been reported. To see this, we investigate the effects of various factors such as the degree, characteristic path length, heterogeneity, and betweenness centrality on synchronization, and find a consistent trend between the synchronization and the betweenness centrality. The betweenness centrality is thus proposed as a good indicator for synchronizability., to appear in Phys. Rev. E

We study numerically quantum diffusion of a particle on small-world networks by integrating the time-dependent Schr\"odinger equation with a localized initial state. The participation ratio, which corresponds to the number of visited sites in the case of classical diffusion, as a function of time is measured and the corresponding diffusion time $\tau$ is computed. In a local regular network, i.e., in the network with the rewiring probability $p=0$, the diffusion time depends on the network size $N$ as $\tau \sim N$, while the behavior $\tau \sim \log N$ is observed as $p$ becomes finite. Such fast diffusion of a particle on a complex network suggests that the small-world transition is also the fast-world transition from a dynamic point of view. The classical diffusion behavior is also studied and compared with the quantum behavior., Comment: 5 pages, to appear in PRB

We study the collective behavior of an Ising system on a small-world network with the interaction $J(r) \propto r^{-\alpha}$, where $r$ represents the Euclidean distance between two nodes. In the case of $\alpha = 0$ corresponding to the uniform interaction, the system is known to possess a phase transition of the mean-field nature, while the system with the short-range interaction $(\alpha\to\infty)$ does not exhibit long-range order at any finite temperature. Monte Carlo simulations are performed at various values of $\alpha$, and the critical value $\alpha_c$ beyond which the long-range order does not emerge is estimated to be zero. Thus concluded is the absence of a phase transition in the system with the algebraically decaying interaction $r^{-\alpha}$ for any nonzero positive value of $\alpha$.

We investigate the response behavior of an Ising system, driven by an oscillating field, on a small-world network, with particular attention to the effects of the system size. The responses of the magnetization to the driving field are probed by means of Monte Carlo dynamic simulations with the rewiring probability varied. It is found that at low and high temperatures the occupancy ratio, measuring how many spins follow the driving field, behaves monotonically with the system size. At intermediate temperatures, on the other hand, the occupancy ratio first grows and then reduces as the size is increased, displaying a resonance-like peak at a finite value of the system size. In all cases, further increase of the size eventually leads to saturation to finite values; the size at which saturation emerges is observed to depend on the temperature, similarly to the correlation length of the system., Comment: 5 pages, 6 figures. To appear in PRE

In the recent study of the Ising model on a small-world network by A. P\c{e}kalski [Phys. Rev. E {\bf 64}, 057104 (2001)], a surprisingly small value of the critical exponent $\beta \approx 0.0001$ has been obtained for the temperature dependence of the magnetization. We perform extensive Monte Carlo simulations of the same model and conclude, via the standard finite-size scaling of various quantities,that the phase transition in the model is of the mean-field nature, in contrast to the work by A. P\c{e}kalski but in accord with other existing studies., Comment: to be published in PRE

We investigate global phase coherence in a system of coupled oscillators on a small-world networks constructed from a ring with nearest-neighbor edges. The effects of both thermal noise and quenched randomness on phase ordering are examined and compared with the global coherence in the corresponding \xy model without quenched randomness. It is found that in the appropriate regime phase ordering emerges at finite temperatures, even for a tiny fraction of shortcuts. Nature of the phase transition is also discussed., 5 pages, 4 figures, Phys. Rev. E (in press)

We investigate collective synchronization in a system of coupled oscillators on small-world networks. The order parameters that 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 P of shortcuts and to display saturated behavior for $P\ensuremath{\gtrsim}0.5.$ 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.

We consider a system of coupled oscillators with finite inertia and time-delayed interaction, and investigate the interplay between inertia and delay both analytically and numerically. The phase velocity of the system is examined; revealed in numerical simulations is emergence of spontaneous phase oscillation without external driving, which turns out to be in good agreement with analytical results derived in the strong-coupling limit. Such self-oscillation is found to suppress synchronization and its frequency is observed to decrease with inertia and delay. We obtain the phase diagram, which displays oscillatory and stationary phases in the appropriate regions of the parameters., Comment: 5 pages, 6 figures, to pe published in PRE

The phase transition in the XY model on one-dimensional small-world networks is investigated by means of Monte-Carlo simulations. It is found that long-range order is present at finite temperatures, even for very small values of the rewiring probability, suggesting a finite-temperature transition for any nonzero rewiring probability. Nature of the phase transition is discussed in comparison with the globally-coupled XY model., Comment: 5 pages, accepted in PRE

We study synchronization and noise-induced resonance phenomena in systems of globally coupled oscillators, each possessing finite inertia. The behavior of the order parameter, which measures collective synchronization of the system, is investigated as the noise level and the coupling strength are varied, and hysteretic behavior is manifested. The power spectrum of the phase velocity is also examined and the quality factor as well as the response function is obtained to reveal noise-induced resonance behavior., to be published in Phys. Rev. E

We study the noise effects in a driven system of globally coupled oscillators, with particular attention to the interplay between driving and noise. The self-consistency equation for the order parameter, which measures the collective synchronization of the system, is derived; it is found that the total order parameter decreases monotonically with noise, indicating overall suppression of synchronization. Still, for large coupling strengths, there exists an optimal noise level at which the periodic (ac) component of the order parameter reaches its maximum. The response of the phase velocity is also examined and found to display resonance behavior., Comment: 17 pages, 3 figures

We study the synchronization phenomena in a system of globally coupled oscillators with time delay in the coupling. The self-consistency equations for the order parameter are derived, which depend explicitly on the amount of delay. Analysis of these equations reveals that the system in general exhibits discontinuous transitions in addition to the usual continuous transition, between the incoherent state and a multitude of coherent states with different synchronization frequencies. In particular, the phase diagram is obtained on the plane of the coupling strength and the delay time, and ubiquity of multistability as well as suppression of the synchronization frequency is manifested. Numerical simulations are also performed to give consistent results.