The structure and dynamics of complex networks: a statistical physics and dynamical systems approach

In this thesis, we attempt to fill many important gaps in understanding the structure and dynamics of complex networks, using techniques from statistical and computational physics as well as dynamical systems. First, we study the property of community structure or modularity. We propose and study a new method for detecting community structure in networks and show that it is highly effective at discovering known structure. In addition, we introduce a measure for the strength of the community structure of a graph, which gives us a notion of the significance of the structure detected by our method. Next, we explore how optimization processes can influence network structure. Expanding ideas from the theory of highly-optimized tolerance, we provide an analytical solution for the existence of power law distributions in simple optimized network model. We hypothesize on how these ideas can extend to more complex topologies and provide a partial explanation for the power-law degree distributions seen in many engineered and evolved networks. Turning to the interplay between topology and dynamics, we study a simple model of epidemics that incorporates pathogen mutation. Using both anlytical and numerical techniques we investigate the effects of two network properties—connectivity and community structure. We find that pathogens in highly connected populations must mutate rapidly in order to remain viable, and we locate the phase transition for pathogen extinction as a function of the mutation rate and connectivity. Investigating the effects of modularity, we find that the addition of community structure tends to foster pathogen persistence. Finally, we explore the effect of network topology on synchronization dynamics and basin size distributions in a system of phase oscillators. We investigate the effects of topology in two ways—through the addition of structured links (more connections to nearest neighbors) and through the addition of randomness. We find that, while random rewiring leads to synchrony more quickly than the addition of structured links, the difference between these two approaches is not nearly as significant as one might expect from intuition about ‘small-world’ networks.