Motivated by the analysis of social networks, we study a model of network that has both a tunable degree distribution and a tunable clustering coefficient. We compute the asymptotic (as the size of the population tends to infinity) for the number of acquaintances and the clustering for this model. We analyze a contagion model with threshold effects and obtain conditions for the existence of a large cascade. We also analyze a diffusion process with a given probability of contagion. In both cases, we characterize conditions under which a global cascade is possible.