The study of networks in such diverse areas as biology, technology, and social sciences has given rise to the interdisciplinary field of network science. Many real-world networks exhibit strongly connected communities and a degree distribution that follows a power law. This thesis explores these two topics - community structure and power law distributions - as they relate to network models and the diffusion of information on networks. We first consider the generation of heavy-tailed distributions in stochastic processes. We give a system of stochastic differential equations in which processes grow at an exponential rate, but are reset at exponentially distributed times. We show that this system has a stationary solution which is regularly varying. It is known that networks with a power law degree distribution are produced under the preferential attachment model, where edges are attached with preference to nodes of high degree. We analyze the effect of community structure on the degree distribution of a community-aware preferential attachment model. We also consider a generative network model where the metric for edge formation is not degree but the number of common neighbors. We further study the effect of community structure on information diffusion in networks. Under the Susceptible-Infected-Susceptible model, we show that the epidemic threshold of a network is closely related to the epidemic threshold of its strongest community. We consider the lifetime of an infection on a growing preferential attachment network and show that the lifetime distribution has heavier tails on the growing network than on static networks.