Query log analysis has received substantial attention in recent years, in which the click graph is an important technique for describing the relationship between queries and URLs. State-of-the-art approaches based on the raw click frequencies for modeling the click graph, however, are not noise-eliminated. Nor do they handle heterogeneous query-URL pairs well. In this paper, we investigate and develop a novel entropy-biased framework for modeling click graphs. The intuition behind this model is that various query-URL pairs should be treated differently, i.e., common clicks on less frequent but more specific URLs are of greater value than common clicks on frequent and general URLs. Based on this intuition, we utilize the entropy information of the URLs and introduce a new concept, namely the inverse query frequency (IQF), to weigh the importance (discriminative ability) of a click on a certain URL. The IQF weighting scheme is never explicitly explored or statistically examined for any bipartite graphs in the information retrieval literature. We not only formally define and quantify this scheme, but also incorporate it with the click frequency and user frequency information on the click graph for an effective query representation. To illustrate our methodology, we conduct experiments with the AOL query log data for query similarity analysis and query suggestion tasks. Experimental results demonstrate that considerable improvements in performance are obtained with our entropy-biased models. Moreover, our method can also be applied to other bipartite graphs.