Theoretical Study Of Important Phase Transition Dynamics On Complex Networks

Recently, complex networks research has been one of the most active topics in statistical physics and closely related disciplines. The central issue in this field is to study phase transition and critical phenomena. Usually, computer simulations have been widely used to study such dynamics. It is known that brute-force simulations are quite expensive and sometimes even become impossible. While phenomenological models, such as mean-field description, may capture certain properties of the system, but often ignore microscopic details and fluctuation effects that may be important near some critical points. Therefore a promising way to bridge the gap between the microscopic details and system level behaviors is to develop coarse-grained (CG) approaches, aiming at significantly reducing the degree of freedom while properly preserving the microscopic information of interest. In the paper, we propose a CG Monte Carlo (CGMC) method and study the phase transition dynamics in complex networks. We develop forward flux sampling method proposed recently, and apply to study nucleaion pathways of Ising model in homogeneous and heterogeneous networks, and propose an optimal strategy to suppress epidemic explosion in heterogeneous metapopulation networks. In the following, we will give the brief summary.·CGMC simulations of the phase transition of the Potts model on complex networksWe propose a strength-based CG (s-CG) method to study critical phenomena of the Potts model on weighted complex networks. By merging nodes with close strengths together, the original network is reduced to a CG network with much smaller size, on which the CG Hamiltonian can be well defined. In particular, we propose the consistent condition for the statistical probability of the CG model and its corresponding microscopic model. Further more, we make an error analysis and show that our s-CG approach exactly or approximately holds. Extensive numerical simulations are performed on scale-free networks and random networks, without or with strength correlation, showing that this s-CG approach works very well in reproducing the phase diagrams, fluctuations, and finite-size effects of the microscopic model, while the d-CG approach proposed in our recent work [Phys. Rev. E82,011107(2010)] does not. Our study may open more perspectives and potential applications in developing other efficient coarse-grained simulation methods.·Nucleation pathways on complex networksNucleation is a fluctuation-driven process that initiates the decay of a metastable state into a more stable one. Many important first order phase transition in nature are associated with nucleation. For many decades, our understanding of nucleation has been dominated by the classical nucleation theory, and is limited to regular lattices in Euclidean space. Since many real systems can be properly modeled by network-organized structure, it is thus an interesting topic to explore nucleation process in complex networks. In the present work, we study nucleation pathways of the Ising model in homogeneous and heterogeneous networks using the forward flux sampling method, and fund that the nucleation processes represent distinct features along pathways for different network topologies. For homogeneous networks, there always exists a dominant nucleating cluster to which relatively small clusters are attached gradually to form the critical nucleus. For heterogeneous ones, many small isolated nucleating clusters emerge at the early stage of the nucleation process, until suddenly they form the critical nucleus through a sharp merging process. Further more, we also study nucleation pathway of the Ising model in correlated complex networks and find that with the increment of the correlation coefficient the critical nucleus size increase monotonically. At the early nucleation stage, the dominant nucleating cluster for assortative networks grows faster than the others. Strikingly, a giant cluster emerges suddenly and forms the critical nucleus through a sharp merging process in assortative networks and uncorrelated networks, especially this jump in assortative networks is highest, but it disappears in disassortative networks.·An optimal strategy to suppress epidemic explosion on complex networksStudy of epidemic spreading in complex networks is of great importance in practical infectious disease and computer viruses control. Very recently, the metapopulation dynamics on heteregeneous networks have gained great research attention. However, all the studies so far have treated the curing rate as a homogenous parameter, Note, however, in reality the curing rate of individuals should certainly be associated with the available medical resources in the local subpopulation, that is the local property of the network node, such as the degree k. In the present work, we propose an optimal strategy to suppress epidemic explosion in heterogeneous metapopulation networks, wherein each node represents a subpopulation with any number of individuals and is assigned a curing rate that is proportional to kα with k the node degree and a an adjustable parameter. We have performed stochastic simulations of the dynamical reaction-diffusion processes associated with the susceptible-infected-susceptible model in scale-free networks. We found that the epidemic threshold reaches a maximum when the exponent α is tuned to be αopt=1.3. This nontrivial phenomenon is robust to the change of the network size and the average degree. In addition, we have carried out a mean field analysis to further validate our scheme, which also demonstrates that epidemic explosion follows different routes for α larger or less than αopt. Our work suggests that in order to effectively suppress epidemic spreading on heterogeneous complex networks, subpopulations with higher degrees should be allocated more resources than just being linearly dependent on the degree k.