Social networks, as well as many other real-world networks, exhibit overlapping community structure. In this paper, we present formulas which facilitate the computation for characterizing the overlapping community structure of networks. A hypergraph representation of networks with overlapping community structure is introduced. Using the hypergraph representation we setup a social network model which exhibits innate tunable overlapping community structure. By comparing simulation results of our model with results of the Hyves network and the SourceForge network, we show that our model captures most of the common features of large social networks. We analytically give the relation between the maximum membership number of a network and the adjacency eigenvalues of the simple graph representation of the network, which is the line graph of the hypergraph representation.