Existing query suggestion techniques mainly revolve around mining existing queries that are most similar to a given query. If the query fails to precisely capture a user's real intent, for example, in most exploratory search tasks, suggested queries are likely to fail as well. If suggested queries are not only relevant to the query but also diverse in nature, it is likely that some of them are close to the user's real intent. In this paper, we propose a novel social-knowledge-directed query suggestion approach for exploratory search, which integrates the social knowledge into the probabilistic model based on query-URL bipartite graphs. Social knowledge is discovered by conducting kernel principle component analysis on the related queries, and incorporating the social knowledge with random walk on the bipartite graph can obtain diverse queries that are relevant to a given one. We have conducted a set of experiments to validate this approach and the results show that this approach outperforms other query suggestion methods in terms of supporting exploratory search.