Hierarchical community decomposition via oblivious routing techniques

Abstract

The detection of communities in real-world large-scale complex networks is a fundamental step in many applications, such as describing community structure and predicting the dissemination of information. Unfortunately, community detection is a computationally expensive task. Indeed, many approaches with strong theoretic guarantees are infeasible when applied to networks of large scale. Numerous approaches have been designed to scale community detection algorithms, many of which leverage local optimizations or local greedy decisions to iteratively find the communities. Solely relying on local techniques to detect communities, rather than a global objective function, can fail to detect global structure of the network. In this work, we instead formulate a notion of a hierarchical community decomposition (HCD), which takes a more global view of hierarchical community structure. Our main contributions are as follows. We formally define a (λ, delta)-HCD where λ parametrizes the connectivity within each sub-community at the same hierarchical level and δ parametrizes the relationship between communities across two consecutive levels. Based on a method of Racke originally designed for oblivious routing, we provide an algorithm to construct a HCD and prove that an (O(log n);O(1))-HCD can always be found for any n-node input graph. Further, our algorithm does not rely on a pre-specified number of communities or depth of decomposition. Since the algorithm is of exponential complexity, we also describe a practical efficient, yet heuristic, implementation and perform an experimental validation on synthetic and real-world networks. We experiment first with synthetic networks with well-defined "intended" decompositions, on which we verify the quality of the decompositions produced by our method. Armed with the confidence these positive results provide, we use our implementation to compute the hierarchical community structure of more complex, real-world networks.