Phase Transitions in Random Graphs- Outbreak of Epidemics to Network Robustness and fragility

Inspired by empirical studies researchers have tried to model various systems like human populations, the World Wide Web or electric power grids by random graphs. Here we first examine different properties of random graphs(both undirected and directed) having arbitrary degree distributions using the generating functon formalism. We present some empirical data about the structure of random graphs in real life especially the WWW. Then we modify our results to include site and bond percolation to address questions about the onset of large scale connectivity and the formation of a giant component in random graphs. This point corresponds to a phase transition. We use this information to examine questions about the onset of epidemic or the robustness or fragility of a network to random or targeted deletion of nodes.