Clustering via hypergraph modularity

Despite the fact that many important problems (including clustering) can be described using hypergraphs, theoretical foundations as well as practical algorithms using hypergraphs are not well developed yet. In this paper, we propose a hypergraph modularity function that generalizes its well established and widely used graph counterpart measure of how clustered a network is. In order to define it properly, we generalize the Chung-Lu model for graphs to hypergraphs. We then provide the theoretical foundations to search for an optimal solution with respect to our hypergraph modularity function. A simple heuristic algorithm is described and applied to a few illustrative examples. We show that using a strict version of our proposed modularity function often leads to a solution where a smaller number of hyperedges get cut as compared to optimizing modularity of 2-section graph of a hypergraph.

[1]  Chuang Liu,et al.  A hypergraph model of social tagging networks , 2010, ArXiv.

[2]  Martine D. F. Schlag,et al.  Multi-level spectral hypergraph partitioning with arbitrary vertex sizes , 1996, Proceedings of International Conference on Computer Aided Design.

[3]  Santo Fortunato,et al.  Limits of modularity maximization in community detection , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[5]  S. Klamt,et al.  Modeling approaches for qualitative and semi-quantitative analysis of cellular signaling networks , 2013, Cell Communication and Signaling.

[6]  Giorgio Gallo,et al.  Directed Hypergraphs and Applications , 1993, Discret. Appl. Math..

[7]  M. Winlaw,et al.  An In-Depth Analysis of the Chung-Lu Model , 2015 .

[8]  Colin McDiarmid,et al.  Modularity of regular and treelike graphs , 2018, J. Complex Networks.

[9]  Kumar N. Sivarajan,et al.  Hypergraph models for cellular mobile communication systems , 1998 .

[10]  V. Voloshin.  Introduction to Graph and Hypergraph Theory , 2013 .

[11]  J. Rodríguez.  On the Laplacian Spectrum and Walk-regular Hypergraphs , 2003 .

[12]  Vipin Kumar,et al.  Multilevel k-way hypergraph partitioning , 1999, DAC '99.

[13]  I-Hsiang Wang,et al.  Community Detection in Hypergraphs: Optimal Statistical Limit and Efficient Algorithms , 2018, AISTATS.

[14]  M. Newman,et al.  Finding community structure in very large networks. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  Santo Fortunato,et al.  Community detection in graphs , 2009, ArXiv.

[16]  S. Fortunato,et al.  Resolution limit in community detection , 2006, Proceedings of the National Academy of Sciences.

[17]  Boleslaw K. Szymanski,et al.  Asymptotic resolution bounds of generalized modularity and statistically significant community detection , 2019, ArXiv.

[18]  Serge J. Belongie,et al.  Higher order learning with graphs , 2006, ICML.

[19]  F. Chung,et al.  Connected Components in Random Graphs with Given Expected Degree Sequences , 2002 .

[20]  Tamara G. Kolda,et al.  Community structure and scale-free collections of Erdös-Rényi graphs , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  François Théberge,et al.  Comparing Graph Clusterings: Set partition measures vs. Graph-aware measures , 2020, IEEE transactions on pattern analysis and machine intelligence.

[22]  F. Chung,et al.  Complex Graphs and Networks , 2006 .

[23]  Jean-Loup Guillaume,et al.  Fast unfolding of communities in large networks , 2008, 0803.0476.

[24]  M E J Newman,et al.  Fast algorithm for detecting community structure in networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  Cristian Sminchisescu,et al.  Efficient Hypergraph Clustering , 2012, AISTATS.

[26]  Mark E. J. Newman,et al.  Community detection in networks: Modularity optimization and maximum likelihood are equivalent , 2016, ArXiv.

[27]  Liudmila Ostroumova,et al.  Modularity of Complex Networks Models , 2016, WAW.

[28]  Michael A. Shepherd,et al.  Transient hypergraphs for citation networks , 1990, Inf. Process. Manag..

[29]  Pietro Perona,et al.  Beyond pairwise clustering , 2005, 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05).

[30]  M E J Newman,et al.  Community structure in social and biological networks , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[31]  Jitendra Malik,et al.  Normalized cuts and image segmentation , 1997, Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[32]  M. Newman.  Community detection in networks: Modularity optimization and maximum likelihood are equivalent , 2016, Physical review. E.

[33]  Colin McDiarmid,et al.  Modularity of Erdős‐Rényi random graphs , 2018, AofA.

[34]  Tamara G. Kolda,et al.  A Scalable Generative Graph Model with Community Structure , 2013, SIAM J. Sci. Comput..

[35]  M E J Newman,et al.  Finding and evaluating community structure in networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[36]  Colin McDiarmid,et al.  Modularity in random regular graphs and lattices , 2013, Electron. Notes Discret. Math..

[37]  Colin McDiarmid,et al.  Modularity of tree-like and random regular graphs , 2016, ArXiv.

[38]  Bernhard Schölkopf,et al.  Learning with Hypergraphs: Clustering, Classification, and Embedding , 2006, NIPS.

[39]  Mark Newman,et al.  Networks: An Introduction , 2010 .