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Simplified Energy Landscape for Modularity Using Total Variation by Boyd, Zachary; Bae, Egil; Tai, Xue-Cheng; Bertozzi, Andrea L. is a scholarly article available to read on EtoBox.

What is Simplified Energy Landscape for Modularity Using Total Variation about?

Networks capture pairwise interactions between entities and are frequently used in applications such as social networks, food networks, and protein interaction networks, to name a few. Communities, cohesive groups of nodes, often form in these applications, and identifying them gives insight into the overall organization of the network. One common quality function used to identify community structure is modularity. In Hu et al. [SIAM J. App. Math., 73(6), 2013], it was shown that modularity optimization is equivalent to minimizing a particular nonconvex total variation (TV) based functional over a discrete domain. They solve this problem, assuming the number of communities is known, using a Merriman, Bence, Osher (MBO) scheme. We show that modularity optimization is equivalent to minimizing a convex TV-based functional over a discrete domain, again, assuming the number of communities is known. Furthermore, we show that modularity has no convex relaxation satisfying certain natural conditions. We therefore, find a manageable non-convex approximation using a Ginzburg Landau functional, which provably converges to the correct energy in the limit of a certain parameter. We then derive

Author

Boyd, Zachary; Bae, Egil; Tai, Xue-Cheng; Bertozzi, Andrea L.

Published

2017

Language

EN