Hermitian matrices for clustering directed graphs: insights and applications

Mihai Cucuringu, Huan Li, He Sun, Luca Zanetti
Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR 108:983-992, 2020.

Abstract

Graph clustering is a basic technique in machine learning, and has widespread applications in different domains. While spectral techniques have been successfully applied for clustering undirected graphs, the performance of spectral clustering algorithms for directed graphs (digraphs) is not in general satisfactory: these algorithms usually require symmetrising the matrix representing a digraph, and typical objective functions for undirected graph clustering do not capture cluster-structures in which the information given by the direction of the edges is crucial. To overcome these downsides, we propose a spectral clustering algorithm based on a complex-valued matrix representation of digraphs. We analyse its theoretical performance on a Stochastic Block Model for digraphs in which the cluster-structure is given not only by variations in edge densities, but also by the direction of the edges. The significance of our work is highlighted on a data set pertaining to internal migration in the United States: while previous spectral clustering algorithms for digraphs can only reveal that people are more likely to move between counties that are geographically close, our approach is able to cluster together counties with a similar socio-economical profile even when they are geographically distant, and illustrates how people tend to move from rural to more urbanised areas.

Cite this Paper


BibTeX
@InProceedings{pmlr-v108-cucuringu20a, title = {Hermitian matrices for clustering directed graphs: insights and applications}, author = {Cucuringu, Mihai and Li, Huan and Sun, He and Zanetti, Luca}, booktitle = {Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics}, pages = {983--992}, year = {2020}, editor = {Silvia Chiappa and Roberto Calandra}, volume = {108}, series = {Proceedings of Machine Learning Research}, month = {26--28 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v108/cucuringu20a/cucuringu20a.pdf}, url = { http://proceedings.mlr.press/v108/cucuringu20a.html }, abstract = {Graph clustering is a basic technique in machine learning, and has widespread applications in different domains. While spectral techniques have been successfully applied for clustering undirected graphs, the performance of spectral clustering algorithms for directed graphs (digraphs) is not in general satisfactory: these algorithms usually require symmetrising the matrix representing a digraph, and typical objective functions for undirected graph clustering do not capture cluster-structures in which the information given by the direction of the edges is crucial. To overcome these downsides, we propose a spectral clustering algorithm based on a complex-valued matrix representation of digraphs. We analyse its theoretical performance on a Stochastic Block Model for digraphs in which the cluster-structure is given not only by variations in edge densities, but also by the direction of the edges. The significance of our work is highlighted on a data set pertaining to internal migration in the United States: while previous spectral clustering algorithms for digraphs can only reveal that people are more likely to move between counties that are geographically close, our approach is able to cluster together counties with a similar socio-economical profile even when they are geographically distant, and illustrates how people tend to move from rural to more urbanised areas.} }
Endnote
%0 Conference Paper %T Hermitian matrices for clustering directed graphs: insights and applications %A Mihai Cucuringu %A Huan Li %A He Sun %A Luca Zanetti %B Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2020 %E Silvia Chiappa %E Roberto Calandra %F pmlr-v108-cucuringu20a %I PMLR %P 983--992 %U http://proceedings.mlr.press/v108/cucuringu20a.html %V 108 %X Graph clustering is a basic technique in machine learning, and has widespread applications in different domains. While spectral techniques have been successfully applied for clustering undirected graphs, the performance of spectral clustering algorithms for directed graphs (digraphs) is not in general satisfactory: these algorithms usually require symmetrising the matrix representing a digraph, and typical objective functions for undirected graph clustering do not capture cluster-structures in which the information given by the direction of the edges is crucial. To overcome these downsides, we propose a spectral clustering algorithm based on a complex-valued matrix representation of digraphs. We analyse its theoretical performance on a Stochastic Block Model for digraphs in which the cluster-structure is given not only by variations in edge densities, but also by the direction of the edges. The significance of our work is highlighted on a data set pertaining to internal migration in the United States: while previous spectral clustering algorithms for digraphs can only reveal that people are more likely to move between counties that are geographically close, our approach is able to cluster together counties with a similar socio-economical profile even when they are geographically distant, and illustrates how people tend to move from rural to more urbanised areas.
APA
Cucuringu, M., Li, H., Sun, H. & Zanetti, L.. (2020). Hermitian matrices for clustering directed graphs: insights and applications. Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 108:983-992 Available from http://proceedings.mlr.press/v108/cucuringu20a.html .

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