On function homophily of microbial Protein-Protein Interaction Networks.

We present a new method for assessing homophily in networks whose vertices have categorical attributes, namely when the vertices of networks come partitioned into classes. We apply this method to Protein- Protein Interaction networks, where vertices correspond to proteins, partitioned according to they func- tional role, and edges represent potential interactions between proteins. Similarly to other classical and well consolidated approaches, our method compares the relative edge density of the subgraphs induced by each class with the corresponding expected relative edge density under a null model. The novelty of our approach consists in prescribing an endogenous null model, namely, the sample space of the null model is built on the input network itself. This allows us to give exact explicit expression for the z-score of the relative edge density of each class as well as other related statistics. The z-scores directly quantify the statistical significance of the observed homophily via ?Ceby?s ?ev inequality. The expression of each z-score is entered by the network structure through basic combinatorial invariant such as the number of subgraphs with two spanning edges. Each z-score is computed in O(n3) worst-case time for a network with n vertices. This leads to an overall effective computational method for assesing homophily. Theoretical results are then exploited to prove that Protein-Protein Interaction networks networks are significantly homophillous.