Overlapping Communities in Complex Networks

Communities are subsets of a network that are densely connected inside and share only few connections to the rest of the network. The aim of this research is the development and evaluation of an efficient algorithm for detection of overlapping, fuzzy communities. The algorithm gets as input some members of each community that we aim to discover. We call these members seed nodes. The algorithm then propagates this information by using random walks that start at non-seed nodes and end as they reach a seed node. The probability that a random walk starting at a non-seed node $v$ ends at a seed node $s$ is then equated with the probability that $v$ belongs to the communities of $s$. The algorithm runs in time $\tilde{O}(l \cdot m \cdot \log n)$, where $l$ is the number of communities to detect, $m$ is the number of edges, $n$ is the number of nodes. The $\tilde{O}$-notation hides a factor of at most $(\log \log n)^2$. The LFR benchmark proposed by Lancichinetti et al.\ is used to evaluate the performance of the algorithm. We found that, given a good set of seed nodes, it is able to reconstruct the communities of a network in a meaningful manner.