Statistics And Dynamics Of Complex Network

Complex network theory has become one of the most important approaches to study complexsystem, in which the nodes represent the individuals and the links represent the interactions betweenthem. It is empirically found that the real-world complex networks in different fields exhibit somecommon structural features. Moreover, the macroscopic properties and dynamical behavior have closerelation to the structure of the system. Therefore, the main issue in complex network theory is toestablish network model to precisely mimic the real-world system, and to clarify the relationshipbetween the macroscopic properties and the microscopic structures, which is also the main goal of thisthesis. Our obtained results are summarized as follow:(1) Universal scaling behavior of clustering coefficient induced by deactivation mechanism: Wepropose a model of network growth that generalizes the deactivation model previously suggested forcomplex networks. Several topological features of this generalized model, such as the degreedistribution and clustering coefficient, have been investigated analytically and by simulations. A scalingbehavior of clustering coefficient C:1/Mis theoretically obtained, where M refers to the numberof active nodes in the network. We discuss the relationship between the recently observed numericalbehavior of clustering coefficient in the coauthor and paper citation networks and our theoretical result.It shows that both of them are induced by deactivation mechanism. By introducing a perturbation, thegenerated network undergoes a transition from large to small world, meanwhile the scaling behavior ofC is conserved. It indicates that C:1/Mis a universal scaling behavior induced by deactivationmechanism. On the other hand, we also investigate the effect of generalized deactivation mechanism inweighted networks. By introducing the special aging mechanism, the model can produce power-lawdistributions of degree, strength, and weight, as confirmed in many real networks. We also characterizethe clustering and correlation properties of this class of networks. It is found that the generated networksimultaneously exhibits hierarchical organization and disassortative degree correlation(2) Scaling of disordered recursive scale-free networks: we present a solvable model of disorderedrecursive scale-free networks. The structure, fractality, and dimensionality are studied theoretically andnumerically, which are shown to be totally different from those in ordered recursive networks. Thetransfinite fractal (transfractal) dimension, which is recently introduced to distinguish the structuraldifferences between the networks with infinite dimension, exhibits an interesting scaling behavior. Wealso investigate the diffusion process on this family of networks, and it is found that the transfractaldimension can identify detailed scaling behavior of diffusion dynamics on transfractal networks. (3) Spatial evolutionary game with bond dilution: we study numerically the prisoner’s dilemmagame (PDG) and snowdrift game (SG) on a two-dimensional square lattice with both quenched andannealed bond dilution. For quenched bond dilution, the system undergoes a dynamical transition at thecritical occupation probabilityq*, which is higher than the bond percolation transition point for asquare lattice. In the critical region, the defined order parameter has a scaling form asP:(q q*e)forq q*with the critical exponents1.42for PDG and1.52for SG, which differ fromthose with quenched site dilution. For annealed bond dilution, the system exhibits a distinct cooperativebehavior. We find that the cooperation is much enhanced in the range of small payoff parameters on alattice with slightly annealed bond dilution.(4) Temporal Behavior of Evolutionary Dynamics on Graph: we systematically study the temporalbehavior of evolutionary dynamics on graph. With evolutionary graph theory, two time scales, mutantfixation time and mutant spreading time, and their relationship are theoretically and numericallyinvestigated on three characteristic population structures, such as fully-connected graph, Bethe lattice,and hyper-cubic lattice. The ingredients and scaling behaviors of the interface between mutants andwild-type individuals are analyzed in detail, which have significant impact on temporal behaviors inevolutionary dynamics. This research provides further understanding of temporal behavior inevolutionary dynamics at theoretical level.