We propose an algorithm for finding overlapping community structure in very large networks. The algorithm is based on the label propagation technique of Raghavan, Albert, and Kumara, but is able to detect communities that overlap. Like the original algorithm, vertices have labels that propagate between neighbouring vertices so that members of a community reach a consensus on their community membership. Our main contribution is to extend the label and propagation step to include information about more than one community: each vertex can now belong to up to v communities, where v is the parameter of the algorithm. Our algorithm can also handle weighted and bipartite networks. Tests on an independentlydesigned set of benchmarks, and on real networks, show the algorithm to be highly effective in recovering overlapping communities. It is also very fast and can process very large and dense networks in a short time.

Abstract. Recent years have seen the development of many graph clustering algorithms, which can identify community structure in networks. The vast majority of these only find disjoint communities, but in many real-world networks communities overlap to some extent. We present a new algorithm for discovering overlapping communities in networks, by extending Girvan and Newman's well-known algorithm based on the betweenness centrality measure. Like the original algorithm, ours performs hierarchical clustering -partitioning a network into any desired number of clusters -but allows them to overlap. Experiments confirm good performance on randomly generated networks based on a known overlapping community structure, and interesting results have also been obtained on a range of real-world networks.

Abstract. Networks commonly exhibit a community structure, whereby groups of vertices are more densely connected to each other than to other vertices. Often these communities overlap, such that each vertex may occur in more than one community. However, two distinct types of overlapping are possible: crisp (where each vertex belongs fully to each community of which it is a member) and fuzzy (where each vertex belongs to each community to a different extent). We investigate the effects of the fuzziness of community overlap. We find that it has a strong effect on the performance of community detection methods: some algorithms perform better with fuzzy overlapping while others favour crisp overlapping. We also evaluate the performance of some algorithms that recover the belonging coefficients when the overlap is fuzzy. Finally, we investigate whether real networks contain fuzzy or crisp overlapping.

Postpartum depression (PPD) affects up to 19% of women, negatively impacting maternal and infant health. Reductions in plasma oxytocin levels have been associated with PPD and heritability studies have established a genetic contribution. Epigenetic regulation of the oxytocin receptor gene (OXTR) has been demonstrated and we hypothesized that individual epigenetic variability at OXTR may impact the development of PPD and that such variability may be central to predicting risk. This case-control study is nested within the Avon Longitudinal Study of Parents and Children and included 269 cases with PPD and 276 controls matched on age group, parity, and presence or absence of depressive symptoms in pregnancy as assessed by the Edinburgh Postnatal Depression Scale. OXTR DNA methylation (CpG site -934) and genotype (rs53576 and rs2254298) were assayed from DNA extracted from blood collected during pregnancy. Conditional logistic regression was used to estimate odds ratios (ORs) and 95% confidence intervals (CIs) for the association of elevated symptoms of PPD with genotype, methylation, and their interaction adjusted for psychosocial factors (n = 500). There was evidence of an interaction between rs53576 and methylation in the OXTR gene amongst women who did not have depression prenatally but developed PPD (p interaction = 0.026, adjusted for covariates, n = 257). Those women with GG genotype showed 2.63 greater odds of PPD for every 10% increase in methylation level (95% CI: 1.37, 5.03), whereas methylation was unrelated to PPD amongst “A” carriers (OR = 1.00, 95% CI: 0.58, 1.73). There was no such interaction among women with PPD and prenatal depression. These data indicate that epigenetic variation that decreases expression of OXTR in a susceptible genotype may play a contributory role in the etiology of PPD.

Many networks possess a community structure, such that vertices form densely connected groups which are more sparsely linked to other groups. In some cases these groups overlap, with some vertices shared between two or more communities. Discovering communities in networks is a computationally challenging task, especially if they overlap. In previous work we proposed an algorithm, CONGA, that could detect overlapping communities using the new concept of split betweenness. Here we present an improved algorithm based on a local form of betweenness, which yields good results but is much faster. It is especially effective in discovering small-diameter communities in large networks, and has a time complexity of only O(n log n) for sparse networks.