We study the clique diameter problem. Given an undirected, unweighted graph containing a set of cliques, we are interested in finding the maximum distance among all pairs of cliques. In the context of social network analysis, a clique represents an ideal community, a clique distance represents the sparsity between the two communities, and a clique diameter indicates the farthest distance among the social network. Let n denote the number of nodes, m denote the number of edges and r denote the number of given cliques. We provide an O(r(n + m)) time algorithm to compute approximate clique distances with additive error of one. Another contribution is a reduction which transforms any instance of the clique distance problem into an instance of the all-pairs shortest paths problem.