| The goal of this thesis is to study the behaviors of complex networks in several aspects using methods from statistical physics. Networks are structures that consist of nodes and links. By changing the way links connect to nodes, different complex network structures can be constructed such as Erd&huml;s-Renyi (ER) networks and scale-free (SF) networks. Complex networks have wide relevance to many real world problems, including the spread of disease in human society, message routing in the Internet, etc. In this thesis analytical and simulation results are obtained regarding optimal paths in disordered networks, fragmentation of social networks, and improved strategies for immunization against diseases.;In the study of disordered systems, of particular current interest is the scaling behavior of the optimal path length ℓopt from strong disorder to weak disorder state for different weight distributions P(w). Here we derive analytically a new criterion. Using this criterion we find that for all P(w) that possess a strong-weak disorder crossover, the distributions p(ℓ) of the optimal path lengths display the same universal behavior.;Fragmentation in social networks is also studied using methods from percolation theory. Recently, a new measure of fragmentation F has been developed in social network studies. For each removal of a subset of links or nodes, F is defined as the ratio between the number of pairs of nodes that are not connected in the fragmented network after removal, and the total number of pairs in the original fully connected network. We study the statistical behavior of F using both analytical and numerical methods and relate it to the traditional measure of fragmentation, the relative size of the largest cluster, Pinfinity, used in percolation theory. Finally, we tried to find a better immunization strategy. It is widely accepted that the most efficient immunization strategies are based on "targeted" strategies. Here we propose a novel "equal graph partitioning" immunization strategy which we find to be significantly better than targeted methods, with 5% to 50% fewer immunization doses required. We confirm the improved efficiency of our approach on ER and SF networks, random regular graphs, and on several real networks. |
