The Study Of Critical Phenomena On Complex Networks

Recently complex networks have been of much interest in many scientifc disci-plines. In this paper, we use both rigorous approaches and mean-feld methods to studythe structural properties of dynamic random networks and stochastic processes takingplaceonthesenetworks. Weareespeciallyinterestedinthecriticalphenomenainthesemodels. Here we mainly focus on three kinds of phase change problems: the thresh-old of an epidemic model on scale-free networks, the phase transition time of a clusteraggregation network and the emergence of a giant component in growing networks.First, we consider a two-stage contact process on scale-free networks as a modelfor the spread of epidemics. By studying the behavior of the process around the hubsandusingtheneighborhoodexpansionmethod, weshowthatifthepowerlawexponentα>2then any virus starting from a randomly chosen vertex with arbitrarily smallinfection rates can last for a super-polynomial time with positive probability and hencethe critical value of the model is zero. This is in sharp contrast with the mean-feldanalysis. Wealsoprovidetheestimationoftheupperandlowerboundofthemetastabledensity.Second, a dynamical graph process is presented to study a cluster aggregationnetwork with epidemic on it. Each connected-component of the graph can be healthyor infected and the components can change their state or merge together according tospecifed rates. The existence and uniqueness of the limit density of connected com-ponents under certain hypothesis on the reaction rates are proved rigorously using themartingale approach. The gelation time of the process is also discussed in the mean-feld approximation. We show that both spontaneous gelation and induced gelation canbe observed in the model.Finally, we investigate the emergence of the giant connected component in anevolving scale-free network. At each step a new node is added with a random number of edges linking to existing nodes according to the preferential attachment rule. Weobtain the critical value of the phase transition and the size of the giant componentclose to the critical region. At the transition the average cluster size is discontinuouswhile the size of the giant component is infnitely diferentiable. The distribution ofcomponentsizeisofpowerlawformbelowor at the thresholdandisexponentialabovethe transition.The thesis is organized in fve chapters. In chapter1we introduce the backgroundand give an overview of our work. Chapter2to4are the core of the paper and reportall the results of the three models we mentioned above. In the last chapter, we oferconclusion and some open problems for further research.