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We present an analytical framework to analyze lists of proteins with large undirected graphs representing their known functional relationships. We consider edge-count variables such as the number of interactions between a protein and a list, the size of a subgraph induced by a list, and the number of interactions bridging two lists. We derive approximate analytical expressions for the probability distributions of these variables in a model of a random graph with given expected degrees. Probabilities obtained with the analytical expressions are used to mine a protein interaction network for functional modules, characterize the connectedness of protein functional categories, and measure the strength of relations between modules.