Theoretical Bounds on the Network Community Profile from Low-rank Semi-definite Programming

Yufan Huang, C. Seshadhri, David F. Gleich
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:13976-13992, 2023.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-huang23l, title = {Theoretical Bounds on the Network Community Profile from Low-rank Semi-definite Programming}, author = {Huang, Yufan and Seshadhri, C. and Gleich, David F.}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {13976--13992}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/huang23l/huang23l.pdf}, url = {https://proceedings.mlr.press/v202/huang23l.html}, 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.} }
Endnote
%0 Conference Paper %T Theoretical Bounds on the Network Community Profile from Low-rank Semi-definite Programming %A Yufan Huang %A C. Seshadhri %A David F. Gleich %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-huang23l %I PMLR %P 13976--13992 %U https://proceedings.mlr.press/v202/huang23l.html %V 202 %X 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.
APA
Huang, Y., Seshadhri, C. & Gleich, D.F.. (2023). Theoretical Bounds on the Network Community Profile from Low-rank Semi-definite Programming. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:13976-13992 Available from https://proceedings.mlr.press/v202/huang23l.html.

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