Theoretical bounds on the network community profile from low-rank semi-definite programming

Yufan Huang; C. Seshadhri; David F. Gleich

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arXiv · 2303.14550

Theoretical bounds on the network community profile from low-rank semi-definite programming

Yufan Huang·C. Seshadhri·David F. Gleich

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Abstract

We study a new connection between a technical measure called $\mu$-conductance that arises in the study of Markov chains for sampling convex bodies and the network community profile that characterizes size-resolved properties of clusters and communities in social and information networks. The idea of $\mu$-conductance is similar to the traditional graph conductance, but disregards sets with small volume. We derive a sequence of optimization problems including a low-rank semi-definite program from which we can derive a lower bound on the optimal $\mu$-conductance value. These ideas give the first theoretically sound bound on the behavior of the network community profile for a wide range of cluster sizes. The algorithm scales up to graphs with hundreds of thousands of nodes and we demonstrate how our framework validates the predicted structures of real-world graphs.

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Yufan Huang, C. Seshadhri, David F. Gleich. 2023-03-25. Theoretical bounds on the network community profile from low-rank semi-definite programming. https://arxiv.org/abs/2303.14550

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