Core-Periphery Detection in Hypergraphs
Abstract.
Core-periphery detection is a key task in exploratory network analysis where one aims to find a core, a set of nodes well connected internally and with the periphery, and a periphery, a set of nodes connected only (or mostly) with the core. In this work we propose a model of core-periphery for higher-order networks modeled as hypergraphs, and we propose a method for computing a core-score vector that quantifies how close each node is to the core. In particular, we show that this method solves the corresponding nonconvex core-periphery optimization problem globally to an arbitrary precision. This method turns out to coincide with the computation of the Perron eigenvector of a nonlinear hypergraph operator, suitably defined in terms of the incidence matrix of the hypergraph, generalizing recently proposed centrality models for hypergraphs. We perform several experiments on synthetic and real-world hypergraphs showing that the proposed method outperforms alternative core-periphery detection algorithms, in particular those obtained by transferring established graph methods to the hypergraph setting via clique expansion.
Keywords
MSC codes
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
S. Agarwal, K. Branson, and S. Belongie, Higher order learning with graphs, in Proceedings of the 23rd International Conference on Machine Learning (ICML), Association for Computing Machinery, 2006, pp. 17–24.
2.
S. Agarwal, J. Lim, L. Zelnik-Manor, P. Perona, D. Kriegman, and S. Belongie, Beyond pairwise clustering, in Proceedings of the 2005 Conference on Computer Vision and Pattern Recognition (CVPR), Vol. 2, IEEE, 2005, pp. 838–845.
3.
I. Amburg, J. Kleinberg, and A. R. Benson, Planted hitting set recovery in hypergraphs, J. Phys. Complex., 2 (2021), 035004.
4.
F. Arrigo, D. J. Higham, and F. Tudisco, A framework for second-order eigenvector centralities and clustering coefficients, Proc. A, 476 (2020), 20190724.
5.
D. Arya, D. K. Gupta, S. Rudinac, and M. Worring, HyperSAGE: Generalizing inductive representation learning on hypergraphs, in Proceedings of the International Conference on Learning Representations (ICLR), 2021.
6.
A. Banerjee, On the spectrum of hypergraphs, Linear Algebra Appl., 614 (2021), pp. 82–110.
7.
R. Baños, J. Borge-Holthoefer, N. Wang, Y. Moreno, and S. González-Bailón, Diffusion dynamics with changing network composition, Entropy, 15 (2013), pp. 4553–4568.
8.
F. Battiston, E. Amico, A. Barrat, G. Bianconi, G. Ferraz de Arruda, B. Franceschiello, I. Iacopini, S. Kéfi, V. Latora, Y. Moreno, et al., The physics of higher-order interactions in complex systems, Nat. Phys., 17 (2021), pp. 1093–1098.
9.
F. Battiston, G. Cencetti, I. Iacopini, V. Latora, M. Lucas, A. Patania, J.-G. Young, and G. Petri, Networks beyond pairwise interactions: Structure and dynamics, Phys. Rep., 874 (2020), pp. 1–92.
10.
A. R. Benson, Three hypergraph eigenvector centralities, SIAM J. Math. Data Sci., 1 (2019), pp. 293–312.
11.
A. R. Benson, R. Abebe, M. T. Schaub, A. Jadbabaie, and J. Kleinberg, Simplicial closure and higher-order link prediction, Proc. Natl. Acad. Sci. USA, 115 (2018), pp. E11221–E11230.
12.
A. R. Benson, D. F. Gleich, and J. Leskovec, Higher-order organization of complex networks, Science, 353 (2016), pp. 163–166.
13.
G. Bianconi, Higher Order Networks: An Introduction to Simplicial Complexes, Cambridge University Press, Cambridge, UK, 2021.
14.
S. P. Borgatti and M. G. Everett, Models of core/periphery structures, Soc. Netw., 21 (2000), pp. 375–395.
15.
J. P. Boyd, W. J. Fitzgerald, M. C. Mahutga, and D. A. Smith, Computing continuous core/periphery structures for social relations data with MINRES/SVD, Soc. Netw., 32 (2010), pp. 125–137.
16.
T. Bühler and M. Hein, Spectral clustering based on the graph p-Laplacian, in Proceedings of the 26th Annual International Conference on Machine Learning, Association for Computing Machinery, 2009, pp. 81–88.
17.
T. Carletti, F. Battiston, G. Cencetti, and D. Fanelli, Random walks on hypergraphs, Phys. Rev. E, 101 (2020), 022308.
18.
P. Csermely, A. London, L.-Y. Wu, and B. Uzzi, Structure and dynamics of core/periphery networks, J. Complex Netw., 1 (2013), pp. 93–123.
19.
M. Cucuringu, P. Rombach, S. H. Lee, and M. A. Porter, Detection of core-periphery structure in networks using spectral methods and geodesic paths, Eur. J. Appl. Math., 27 (2016), pp. 846–887.
20.
M. R. Da Silva, H. Ma, and A.-P. Zeng, Centrality, network capacity, and modularity as parameters to analyze the core-periphery structure in metabolic networks, Proc. IEEE, 96 (2008), pp. 1411–1420.
21.
G. F. de Arruda, M. Tizzani, and Y. Moreno, Phase transitions and stability of dynamical processes on hypergraphs, Commun. Phys., 4 (2021), pp. 1–9.
22.
F. Della Rossa, F. Dercole, and C. Piccardi, Profiling core-periphery network structure by random walkers, Sci. Rep., 3 (2013), 1467.
23.
A. Elmoataz, O. Lezoray, and S. Bougleux, Nonlocal discrete regularization on weighted graphs: A framework for image and manifold processing, IEEE Trans. Image Process., 17 (2008), pp. 1047–1060.
24.
R. J. Gallagher, J.-G. Young, and B. F. Welles, A clarified typology of core-periphery structure in networks, Sci. Adv., 7 (2021), eabc9800.
25.
C. L. Giles, K. D. Bollacker, and S. Lawrence, Citeseer: An automatic citation indexing system, in Proceedings of the Third ACM Conference on Digital Libraries, 1998, pp. 89–98.
26.
X. Gong, D. J. Higham, and K. Zygalakis, Directed network Laplacians and random graph models, R. Soc. Open Sci., 8 (2021), 211144.
27.
P. Grindrod, Range-dependent random graphs and their application to modeling large small-world proteome datasets, Phys. Rev. E, 66 (2002), 066702.
28.
M. Hein, S. Setzer, L. Jost, and S. S. Rangapuram, The total variation on hypergraphs-learning on hypergraphs revisited, in Advances in Neural Information Processing Systems 26 (NIPS 2013), Curran Associates, Red Hook, NY, 2013, pp. 2427–2435.
29.
D. J. Higham and H.-L. de Kergorlay, Epidemics on hypergraphs: Spectral thresholds for extinction, Proc. A, 477 (2021), 20210232.
30.
R. Ibrahim and D. Gleich, Nonlinear diffusion for community detection and semi-supervised learning, in Proceedings of The World Wide Web Conference, Association for Computing Machinery, 2019, pp. 739–750.
31.
P. M. Kim, J. O. Korbel, and M. B. Gerstein, Positive selection at the protein network periphery: Evaluation in terms of structural constraints and cellular context, Proc. Natl. Acad. Sci. USA, 104 (2007), pp. 20274–20279.
32.
B. Lemmens and R. D. Nussbaum, Nonlinear Perron–Frobenius Theory, Cambridge University Press, Cambridge, UK, 2012.
33.
P. Li and O. Milenkovic, Inhomogeneous hypergraph clustering with applications, in Advances in Neural Information Processing Systems 30 (NIPS 2017), Curran Associates, Red Hook, NY, 2017, pp. 2305–2315.
34.
P. Li and O. Milenkovic, Submodular hypergraphs: p-Laplacians, Cheeger inequalities and spectral clustering, in Proceedings of the 35th International Conference on Machine Learning, Proc. Mach. Learn. Res. (PMLR) 80, JMLR, Cambridge, MA, 2018, pp. 3014–3023.
35.
A. K. McCallum, K. Nigam, J. Rennie, and K. Seymore, Automating the construction of internet portals with machine learning, Inf. Retr., 3 (2000), pp. 127–163.
36.
A. P. Millán, J. J. Torres, and G. Bianconi, Explosive higher-order Kuramoto dynamics on simplicial complexes, Phys. Rev. Lett., 124 (2020), 218301.
37.
R. J. Mondragón, Network partition via a bound of the spectral radius, J. Complex Netw., 5 (2017), pp. 513–526.
38.
L. Neuhäuser, R. Lambiotte, and M. T. Schaub, Consensus Dynamics and Opinion Formation on Hypergraphs, in Higher-Order Systems, Understanding Complex Systems, F. Battiston, G. Petri, eds., Springer, Cham, 2022, https://doi.org/10.1007/978-3-030-91374-8_14.
39.
K. Prokopchik, A. R. Benson, and F. Tudisco, Nonlinear feature diffusion on hypergraphs, in Proceedings of the 39th International Conference on Machine Learning, Proc. Mach. Learn. Res. (PMLR) 162, JMLR, Cambridge, MA, 2022, pp. 17945–17958.
40.
S. Rao, A. van der Schaft, and B. Jayawardhana, A graph-theoretical approach for the analysis and model reduction of complex-balanced chemical reaction networks, J. Math. Chem., 51 (2013), pp. 2401–2422.
41.
J. A. Rodriguez, On the Laplacian eigenvalues and metric parameters of hypergraphs, Linear Multilinear Algebra, 50 (2002), pp. 1–14.
42.
J. A. Rodriguez, On the Laplacian spectrum and walk-regular hypergraphs, Linear Multilinear Algebra, 51 (2003), pp. 285–297.
43.
J. A. Rodriguez, Laplacian eigenvalues and partition problems in hypergraphs, Appl. Math. Lett., 22 (2009), pp. 916–921.
44.
M. P. Rombach, M. A. Porter, J. H. Fowler, and P. J. Mucha, Core-periphery structure in networks (revisited), SIAM Rev., 59 (2017), pp. 619–646.
45.
K. S. Sandhu, G. Li, H. M. Poh, Y. L. K. Quek, Y. Y. Sia, S. Q. Peh, F. H. Mulawadi, J. Lim, M. Sikic, F. Menghi, et al., Large-scale functional organization of long-range chromatin interaction networks, Cell Rep., 2 (2012), pp. 1207–1219.
46.
M. T. Schaub, N. O’Clery, Y. N. Billeh, J.-C. Delvenne, R. Lambiotte, and M. Barahona, Graph partitions and cluster synchronization in networks of oscillators, Chaos, 26 (2016), 094821.
47.
P. Sen, G. Namata, M. Bilgic, L. Getoor, B. Galligher, and T. Eliassi-Rad, Collective classification in network data, AI Mag., 29 (2008), pp. 93–93.
48.
M. V. Tomasello, M. Napoletano, A. Garas, and F. Schweitzer, The rise and fall of R & D networks, Ind. Corporate Change, 26 (2017), pp. 617–646.
49.
L. Torres, A. S. Blevins, D. S. Bassett, and T. Eliassi-Rad, The why, how, and when of representations for complex systems, SIAM Rev., 63 (2021), pp. 435–485.
50.
F. Tudisco, A. R. Benson, and K. Prokopchik, Nonlinear higher-order label spreading, in Proceedings of the Web Conference, Association for Computing Machinery, 2021.
51.
F. Tudisco and D. J. Higham, A fast and robust kernel optimization method for core-periphery detection in directed and weighted graphs, Appl. Netw. Sci., 4 (2019), pp. 1–13.
52.
F. Tudisco and D. J. Higham, A nonlinear spectral method for core-periphery detection in networks, SIAM J. Math. Data Sci., 1 (2019), pp. 269–292.
53.
F. Tudisco and D. J. Higham, Node and edge nonlinear eigenvector centrality for hypergraphs, Commun. Phys., 4 (2021), 201.
54.
P. Upadhyaya, E. Jarlbering, and F. Tudisco, The Self-Consistent Field Iteration for p-Spectral Clustering, preprint, arXiv:2111.09750, 2021.
55.
A. Van der Schaft, S. Rao, and B. Jayawardhana, A network dynamics approach to chemical reaction networks, Internat. J. Control, 89 (2016), pp. 731–745.
56.
N. Veldt, A. R. Benson, and J. Kleinberg, Hypergraph cuts with general splitting functions, SIAM Rev., 64 (2022), pp. 650–685.
57.
Y. Yoshida, Cheeger inequalities for submodular transformations, in Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, 2019, pp. 2582–2601.
58.
D. Zhou, J. Huang, and B. Schölkopf, Learning with hypergraphs: Clustering, classification, and embedding, in Advances in Neural Information Processing Systems 19 (NIPS 2006), MIT Press, Cambridge, MA, 2007, pp. 1601–1608.
59.
J. Y. Zien, M. D. Schlag, and P. K. Chan, Multilevel spectral hypergraph partitioning with arbitrary vertex sizes, IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., 18 (1999), pp. 1389–1399.
Information & Authors
Information
Published In
SIAM Journal on Mathematics of Data Science
Pages: 1 - 21
DOI: 10.1137/22M1480926
ISSN (online): 2577-0187
Copyright
© 2023 Society for Industrial and Applied Mathematics.
History
Submitted: 28 February 2022
Accepted: 11 October 2022
Published online: 25 January 2023
Keywords
MSC codes
Authors
Funding Information
Engineering and Physical Sciences Research Council (EPSRC): EP/P020720/1, EP/V015605/1
Funding: The work of the second author was supported the Engineering and Physical Sciences Research Council under grants EP/P020720/1 and EP/V015605/1.
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- A Survey on Hypergraph Mining: Patterns, Tools, and GeneratorsACM Computing Surveys, Vol. 57, No. 8 | 24 March 2025
- Finding influential cores via normalized Ricci flows in directed and undirected hypergraphs with applicationsPhysical Review E, Vol. 111, No. 4 | 23 April 2025
- Multiplex measures for higher-order networksApplied Network Science, Vol. 9, No. 1 | 3 September 2024
- An application of node and edge nonlinear hypergraph centrality to a protein complex hypernetworkPLOS ONE, Vol. 19, No. 10 | 3 October 2024
- Learning the effective order of a hypergraph dynamical systemScience Advances, Vol. 10, No. 19 | 10 May 2024
- Patterns in Temporal Networks with Higher-Order Egocentric StructuresEntropy, Vol. 26, No. 3 | 13 March 2024
- Influence Maximization in Hypergraphs Using Multi-Objective Evolutionary AlgorithmsParallel Problem Solving from Nature – PPSN XVIII | 7 September 2024
- Hyper-cores promote localization and efficient seeding in higher-order processesNature Communications, Vol. 14, No. 1 | 6 October 2023
- Generative hypergraph models and spectral embeddingScientific Reports, Vol. 13, No. 1 | 11 January 2023
- Connectivity of Random Geometric HypergraphsEntropy, Vol. 25, No. 11 | 17 November 2023
- How Transitive Are Real-World Group Interactions? - Measurement and ReproductionProceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining | 4 August 2023
- Community detection in large hypergraphsScience Advances, Vol. 9, No. 28 | 14 Jul 2023
- Hypergraph Artificial Benchmark for Community Detection (h–ABCD)Journal of Complex Networks, Vol. 11, No. 4 | 9 August 2023
- Nonlinear Perron--Frobenius Theorems for Nonnegative TensorsSIAM Review, Vol. 65, No. 2 | 9 May 2023
- Hypergraphx: a library for higher-order network analysisJournal of Complex Networks, Vol. 11, No. 3 | 26 May 2023
View Options
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].