The Size of the Giant Component of a Random Graph with a Given Degree Sequence
Combinatorics, Probability and Computing
Assessing significance of connectivity and conservation in protein interaction networks
RECOMB'06 Proceedings of the 10th annual international conference on Research in Computational Molecular Biology
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Knowledge about protein function is often encoded in the form of large and sparse undirected graphs where vertices are proteins and edges represent their functional relationships. One elementary task in the computational utilization of these networks is that of quantifying the density of edges, referred to as connectedness, inside a prescribed protein set. For instance, many functional modules can be identified because of their high connectedness. Since individual proteins can have very different numbers of interactions, a connectedness measure should be well-normalized for vertex degree. Namely, its distribution across random sets of vertices should not be affected when these sets are biased for hubs. We show that such degree-robustness can be achieved via an analytical framework based on a model of random graph with given expected degrees. We also introduce the concept of connectedness profile, which characterizes the relation between adjacency in a graph and a prescribed order of its vertices. A straightforward application to gene expression data and protein networks is the identification of tissue-specific functional modules or cellular processes perturbed in an experiment. The strength of the mapping between gene-expression score and interaction in the network is measured by the area of the connectedness profile. Deriving the distribution of this area under the random graph enables us to define degree-robust statistics that can be computed in O(M), M being the network size. These statistics can identify groups of microarray experiments that are pathway-coherent, and more generally, vertex attributes that relate to adjacency in a graph.