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

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Publication date: 25 March 2023

Abstract: We study a new connection between a technical measure called

m u

-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

m u

-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

m u

-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.

Has companion code repository: https://github.com/luotuoqingshan/mu-conductance-low-rank-sdp

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