Topology and Dynamics of Complex Social Networks

The problem of modeling complex social networks is considered from three perspectives: The problem of describing network topology; the problem of modeling dynamic processes on networks; and the problem of network sampling. These perspectives are highly complementary, each providing results with applications to one other. With respect to network topology, two main results are presented: An algorithm is presented capable of combining two measures of network structure, the degree distribution and the clustering coefficient. It is found that just two mechanisms are required to achieve any desired combination of these metrics-- network growth, combined with preferential attachment. Secondly, a mathematical model of one class of complex network, semi-random networks, is presented which is capable of elucidating the structure of semi-random networks in greater detail then had been achieved with previous models. Among other results, this theory allows one to calculate the expected number of neighbors at a given distance from a randomly chosen node, and to compute the mean path length inside the giant component. Network dynamics are investigated with a simple epidemic model, the SIR (Susceptible Infected Removed) model. A mathematical theory is presented for predicting epidemic incidence for SIR dynamics in semi-random networks. Finally, the problem of network sampling is considered. A probability based estimation theory is presented for Respondent Driven Sampling (RDS). The theory enhances RDS by offering greater analytical tractability, analytical variance estimation, and the estimation of means of continuous variables