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Algorithms for detecting clusters (including overlapping clusters) in graphshave received significant attention in the research community. A closelyrelated important aspect of the problem -- quantification of statisticalsignificance of overlap of clusters, remains relatively unexplored. This paperpresents the first theoretical and practical results on quantifyingstatistically significant interactions between clusters in networks. Suchproblems commonly arise in diverse applications, ranging from social networkanalysis to systems biology. The paper addresses the problem of quantifying thestatistical significance of the observed overlap of the two clusters in anErd\H{o}s-R\'enyi graph model. The analytical framework presented in the paperassigns a $p$-value to overlapping subgraphs by combining information aboutboth the sizes of the subgraphs and their edge densities in comparison to thecorresponding values for their overlapping component. This $p$-value isdemonstrated to have excellent discrimination properties in real applicationsand is shown to be robust across broad parameter ranges.  Our results are comprehensively validated on synthetic, social, andbiological networks. We show that our framework: (i) derives insight from boththe density and the size of overlap among communities (circles/pathways), (ii)consistently outperforms state-of-the-art methods over all tested datasets, and(iii) when compared to other measures, has much broader application scope. Inthe context of social networks, we identify highly interdependent (social)circles and show that our predictions are highly co-enriched with known userfeatures. In networks of biomolecular interactions, we show that our methodidentifies novel cross-talk between pathways, sheds light on their mechanismsof interaction, and provides new opportunities for investigations ofbiomolecular interactions.

Combining Density and Overlap (CoDO): A New Method for Assessing the Significance of Overlap Among Subgraphs