We study the impact of network heterogeneity on relaxation dynamics of the Kuramoto model on uncorrelated complex networks with scale-free degree distributions. Using the Ott-Antonsen method and the annealed-network approach, we find that the critical behavior of the relaxation rate near the synchronization phase transition does not depend on network heterogeneity and critical slowing down takes place at the critical point when the second moment of the degree distribution is finite. In the case of a complete graph we obtain an explicit result for the relaxation rate when the distribution of natural frequencies is Lorentzian. We also find a response of the Kuramoto model to an external field and show that the susceptibility of the model is inversely proportional to the relaxation rate. We reveal that network heterogeneity strongly impacts a field dependence of the relaxation rate and the susceptibility when the network has a divergent fourth moment of degree distribution. We introduce a pair correlation function of phase oscillators and show that it has a sharp peak at the critical point, signaling emergence of long-range correlations. Our numerical simulations of the Kuramoto model support our analytical results.

Active anticancer molecules have been searched from natural products; many drugs were developed from either natural products or their derivatives following the conventional pharmaceutical paradigm of drug discovery. However, the advances in the knowledge of cancer biology have led to personalized medicine using molecular-targeted agents which create new paradigm. Clinical benefit is dependent on individual biomarker and overall survival is prolonged through cytostatic rather than cytotoxic effects to cancer cell. Therefore, a different approach is needed from the single lead compound screening model based on cytotoxicity. In our experience, the Rhus verniciflua stoke (RVS) extract traditionally used for cancer treatment is beneficial to some advanced cancer patients though it is herbal extract not single compound, and low cytotoxic in vitro. The standardized RVS extract's action mechanisms as well as clinical outcomes are reviewed here. We hope that these preliminary results would stimulate different investigation in natural products from conventional chemicals.

Reduction of frustration was the driving force in an approach to social balance as it was recently considered by Antal [T. Antal, P. L. Krapivsky, and S. Redner, Phys. Rev. E 72, 036121 (2005)]. We generalize their triad dynamics to k-cycle dynamics for arbitrary integer k. We derive the phase structure, determine the stationary solutions, and calculate the time it takes to reach a frozen state. The main difference in the phase structure as a function of k is related to k being even or odd. As a second generalization we dilute the all-to-all coupling as considered by Antal to a random network with connection probability w<1. Interestingly, this model can be mapped to a satisfiability problem of computer science. The phase of social balance in our original interpretation then becomes the phase of satisfaction of all logical clauses in the satisfiability problem. In common to the cases we study, the ideal solution without any frustration always exists, but the question actually is as to whether this solution can be found by means of a local stochastic algorithm within a finite time. The answer depends on the choice of parameters. After establishing the mapping between the two classes of models, we generalize the social-balance problem to a diluted network topology for which the satisfiability problem is usually studied. On the other hand, in connection with the satisfiability problem we generalize the random local algorithm to a p-random local algorithm, including a parameter p that corresponds to the propensity parameter in the social balance problem. The qualitative effect of the inclusion of this parameter is a bias towards the optimal solution and a reduction of the needed simulation time.

We study the statistical properties of the sampled networks by a random walker. We compare topological properties of the sampled networks such as degree distribution, degree-degree correlation, and clustering coefficient with those of the original networks. From the numerical results, we find that most of topological properties of the sampled networks are almost the same as those of the original networks for γ 3. In contrast, we find that the degree distribution exponent of the sampled networks for γ > 3 somewhat deviates from that of the original networks when the ratio of the sampled network size to the original network size becomes smaller. We also apply the sampling method to various real networks such as collaboration of movie actor, world wide web, and peer-topeer networks. All topological properties of the sampled networks show the essentially same as the original real networks.