Community Detection in Multilayer Networks Based on Matrix Factorization and Spectral Embedding Method
Abstract
:1. Introduction
2. Model Framework
2.1. Notations
2.2. Discussion of the Framework
2.2.1. Between-Layer Cluster
Methods
Updating Rules
2.2.2. Within-Layer Community Detection
- (Sa):
- Calculate the graph Laplacian
- (Sb):
- Find orthonormal eigenvectors corresponding to the eigenvalues that are largest in absolute value of and put them in a matrix .
- (Sc):
- Form the normalized version by normalizing each row of to have unit length. Then, is considered to be the spectral embedding derived from .
Algorithm 1 MFSE Algorithm |
Input: The adjacency matrices ; number of layer groups M; number of the node groups ; tuning parameter . Output: The network groups and the within-layer node communities . Algorithm:
|
3. Theoretical Guarantees
- (A1)
- for ;
- (A2)
- with representing the -th largest eigenvalue of ;
- (A3)
- cluster of layers and local communities balance condition: there exist positive constants such that
4. Experiments
4.1. Simulation Study
4.2. Real-World Network Data
4.2.1. FAO data
4.2.2. AU-CS Network
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
MFSE | Matrix factorization and spectral embedding method |
MMLSBM | Mixture multilayer stochastic block model |
TWIST | Tucker decomposition with integrated SVD transformation |
ALMA | Alternating minimization algorithm |
Appendix A
Appendix B
Appendix C
References
- Girvan, M.; Newman, M.E.J. Communitystructureinsocial and biological networks. Proc. Natl. Acad. Sci. USA 2002, 99, 7821–7826. [Google Scholar] [CrossRef] [Green Version]
- Javed, M.A.; Younis, M.S.; Latif, S.; Qadir, J.; Baig, A. Community detection in networks: A multidisciplinary review. J. Netw. Comput. Appl. 2018, 108, 87–111. [Google Scholar] [CrossRef]
- Kim, J.; Lee, J.G. Community detection in multi-layer graphs: A survey. ACM Sigmod Rec. 2015, 44, 37–48. [Google Scholar] [CrossRef] [Green Version]
- Kivelä, M.; Arenas, A.; Barthelemy, M.; Gleeson, J.P.; Moreno, Y.; Porter, M.A. Multilayer networks. J. Complex Netw. 2014, 2, 203–271. [Google Scholar] [CrossRef] [Green Version]
- Hric, D.; Darst, R.K.; Fortunato, S. Community detection in networks: Structural communities versus ground truth. Phys. Rev. E 2014, 90, 062805. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Traud, A.L.; Kelsic, E.D.; Mucha, P.J.; Porter, M.A. Comparing community structure to characteristics in online collegiate social networks. SIAM Rev. 2011, 53, 526–543. [Google Scholar]
- Mariadassou, M.; Robin, S.; Vacher, C. Uncovering latent structure in valued graphs: A variarion approach. Ann. Appl. Stat. 2010, 4, 715–742. [Google Scholar] [CrossRef]
- Amini, A.A.; Chen, A.; Bickel, P.J.; Levina, E. Pseudo-likelihood methods for community detection in large sparse networks. Ann. Stat. 2013, 41, 2097–2122. [Google Scholar] [CrossRef]
- Newman, M.E.J. Modularity and community structure in networks. Proc. Natl. Acad. Sci. USA 2006, 103, 8577–8582. [Google Scholar] [CrossRef] [Green Version]
- Karrer, B.; Newman, M.E. Stochastic blockmodels and community structure in networks. Phys. Rev. E 2011, 83, 016107. [Google Scholar] [CrossRef] [Green Version]
- Fortunato, S.; Darko, H. Community detection in networks: A user guide. Phys. Rep. 2016, 659, 1–44. [Google Scholar] [CrossRef] [Green Version]
- Taylor, D.; Caceres, R.S.; Mucha, P.J. Super-resolution community detection for layer-aggregated multilayer networks. Phys. Rev. X 2017, 7, 031056. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Taylor, D.; Shai, S.; Stanley, N.; Mucha, P.J. Enhanced detectability of community structure in multilayer networks through layer aggregation. Phys. Rev. Lett. 2016, 116, 228301. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Dong, X.; Frossard, P.; Vandergheynst, P.; Nefedov, N. Clustering on multi-layer graphs via subspace analysis on grassmann manifolds. IEEE Trans. Signal. Proces. 2013, 62, 905–918. [Google Scholar] [CrossRef] [Green Version]
- Tang, L.; Wang, X.; Liu, H. Community detection via hetero- geneous interaction analysis. Data Min. Knowl. Discov. 2012, 25, 1–33. [Google Scholar] [CrossRef]
- Xu, K.S.; Hero, A.O. Dynamic stochastic blockmodels for time-evolving social networks. IEEE J. Sel. Top. Signal Process. 2014, 8, 552–562. [Google Scholar] [CrossRef] [Green Version]
- Han, Q.; Kevin, X.; Edoardo, A. Consistent estimation of dynamic and multi-layer block models. In Proceedings of the International Conference on Machine Learning, Miami, FL, USA, 9–11 December 2015; pp. 1511–1520. [Google Scholar]
- Tang, R.; Tang, M.; Vogelstein, J.T.; Priebe, C.E. Robust estimation from multiple graphs under gross error contamination. arXiv 2017, arXiv:1707.03487. [Google Scholar]
- De Bacco, C.; Power, E.A.; Larremore, D.B.; Moore, C. Community detection, link prediction, and layer interdependence in multilayer networks. Phys. Rev. E 2017, 95, 042317. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Dong, X.; Frossard, P.V.; Ergheynst, P.; Nefedov, N. Clustering with multi-layer graphs: A spectral perspective. IEEE. Trans. Signal. Proces. 2012, 60, 5820–5831. [Google Scholar] [CrossRef] [Green Version]
- Bhattacharyya, S.; Chatterjee, S. Spectral clustering for multiple sparse networks: I. arXiv 2018, arXiv:1805.10594. [Google Scholar]
- Tang, W.; Lu, Z.; Dhillon, I.S. Clustering with multiple graphs. In Proceedings of the Ninth IEEE International Conference on Data Mining, Miami, FL, USA, 6–9 December 2009; pp. 1016–1021. [Google Scholar]
- Lei, J.; Chen, K.; Lynch, B. Consistent community detection in multi-layer network data. Biometrika 2020, 107, 61–73. [Google Scholar] [CrossRef]
- Mucha, P.J.; Richardson, T.; Macon, K.; Porter, M.A.; Onnela, J.P. Community structure in time-dependent, multiscale, and multiplex networks. Science 2010, 328, 876–878. [Google Scholar] [CrossRef]
- Bazzi, M.; Porter, M.A.; Williams, S.; McDonald, M.; Fenn, D.J.; Howison, S.D. Community detection in temporal multilayer networks, with an application to correlation networks. Multiscale Model. Simul. 2016, 14, 1–41. [Google Scholar] [CrossRef]
- Stanley, N.; Shai, S.; Taylor, D.; Mucha, P.J. Clusteringnetwork layers with the strata multilayer stochastic block model. IEEE. Trans. Netw. Sci. Eng. 2016, 3, 95–105. [Google Scholar] [CrossRef] [Green Version]
- Arroyo, J.; Athreya, A.; Cape, J.; Chen, G.; Priebe, C.E.; Vogelstein, J.T. Inference for multiple heterogeneous networks with a common invariant subspace. J. Mach. Learn. Res. 2021, 22, 6303–6351. [Google Scholar]
- Jing, B.Y.; Li, T.; Lyu, Z.; Xia, D. Community detection on mixture multilayer networks via regularized tensor decomposition. Ann. Stat. 2021, 49, 3181–3205. [Google Scholar] [CrossRef]
- Fan, X.; Pensky, M.; Yu, F.; Zhang, T. ALMA: Alternating Minimization Algorithm for Clustering Mixture Multilayer Network. J. Mach. Learn. Res. 2022, 23, 6303–6351. [Google Scholar]
- Le, C.M.; Levin, K.; Levina, E. Estimating a network from multiple noisy realizations. Electron. J. Stat 2018, 12, 4697–4740. [Google Scholar] [CrossRef]
- Chen, S.; Liu, S.; Ma, Z. Global and individualized community detection in inhomogeneous multilayer networks. Ann. Stat. 2022, 50, 2664–2693. [Google Scholar] [CrossRef]
- Lei, J.; Lin, K.Z. Bias-adjusted spectral clustering in multi-layer stochastic block models. J. Am. Stat. Assoc. 2022, 1–13. [Google Scholar] [CrossRef]
- Gupta, A.K.; Nagar, D.K. Matrix Variate Distributions; Chapman Hall/CRC Press: Boca Raton, FL, USA, 2000. [Google Scholar]
- De Domenico, M.; Granell, C.; Porter, M.A.; Arenas, A. The physics of spreading processes in multilayer networks. Nat. Phys. 2016, 12, 901–906. [Google Scholar] [CrossRef] [Green Version]
- De Domenico, M.; Biamonte, J. Spectral entropies as information-theoretic tools for complex network comparison. Phys. Rev. X 2016, 6, 041062. [Google Scholar] [CrossRef] [Green Version]
- Nicosia, V.; Latora, V. Measuring and modeling correlations in multiplex networks. Phys. Rev. X 2015, 92, 032805. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Yang, L.; Cao, X.; Jin, D.; Wang, X.; Meng, D. A unified semi-supervised community detection framework using latent space graph regularization. IEEE Trans. Cybern. 2015, 45, 2585–2598. [Google Scholar] [CrossRef] [PubMed]
- Ma, X.; Gao, L.; Yong, X.; Fu, L. Semi-supervised clustering algorithm for community structure detection in complex networks. Phys. A 2010, 389, 187–197. [Google Scholar] [CrossRef]
- Luxburg, V. A tutorial on spectral clustering. Stat. Comput. 2007, 17, 395–416. [Google Scholar] [CrossRef]
- Lu, H.; Sang, X.; Zhao, Q.; Lu, J. Community detection algorithm based on nonnegative matrix factorization and pairwise constraints. Phys. A 2020, 545, 123491. [Google Scholar] [CrossRef]
- Amini, A.A.; Levina, E. On semidefinite relaxations for the block model. Ann. Stat. 2018, 46, 149–179. [Google Scholar] [CrossRef] [Green Version]
- Tang, F.; Wang, C.; Su, J.; Wang, Y. Spectral clustering-based community detection using graph distance and node attributes. Computation. Stat. 2020, 35, 69–94. [Google Scholar] [CrossRef]
- Rossi, L.; Magnani, M. Towards effective visual analytics on multiplex and multilayer networks. Chaos. Soliton. Farct. 2015, 72, 68–76. [Google Scholar] [CrossRef] [Green Version]
- Attouch, H.; Bolte, J.; Svaiter, B.F. Convergence of descent methods for semi-algebraic and tame problems: Proximal algorithms, forward-backward splitting, and regularized gauss-seidel methods. Math. Programm. 2013, 137, 91–129. [Google Scholar] [CrossRef] [Green Version]
- Liu, J.; Wang, J.; Liu, B. Community detection of multi-Layer attributed networks via penalized alternating factorization. Mathematics 2020, 8, 239. [Google Scholar] [CrossRef] [Green Version]
- Lei, J.; Rinaldo, A. Consistency of spectral clustering in stochastic block models. Ann. Stat. 2015, 43, 215–237. [Google Scholar] [CrossRef]
- Binkiewicz, N.; Vogelstein, J.T.; Rohe, K. Covariate-assisted spectral clustering. Biometrika 2017, 104, 361–377. [Google Scholar] [CrossRef] [PubMed] [Green Version]
Notations | Explanations |
---|---|
number of nodes, number of network layers | |
the l-th layer; the corresponding adjacency matrix | |
the adjacency tensor | |
number of layer groups; number of layers in the m-th layer group | |
number of node communities in the m-th layer group | |
the network groups such that | |
the k-th node community in the m-th layer group | |
the membership (clustering) matrix of the m-th layer group | |
link probability matrix of the m-th layer group | |
the expected matrix of the m-th layer | |
the membership matrix of layers | |
the vectoried version of matrix by sequentially stacking its columns | |
the vectoried versions of matrices , respectively | |
matrices with rows and , respectively | |
, | the i-th row and j-th column of a matrix A, respectively |
the i-to-jth entities in the r-th row of matrix A | |
the Laplacian matrix of Q | |
a diagonal-matrix with -th element | |
eigenvector matrix corresponding to ; normalized version of | |
an estimator of matrix A; the population version of A | |
the k-th absolute large eigenvalue of matrix W | |
an orthonormal rotation whose rank varies from one line to another |
Layer Groups | Products |
---|---|
Group 1 | Cheese_whole_cow_milk, Bread, Chocolate_products_nes, Fatty_acids, Anise_badian_fennel_coriander, Beer_of_barley, Chillies_and_peppers_dry, Beverages_distilled_alcoholic, Coffee_extracts, Coffee_roasted, Fruit_dried_nes, Juice_fruit |
_nes, Pepper_(piper_spp.), Sugar_nes, Tea_mate_extracts, Vegetables_frozen | |
Group 2 | Beverages_non_alcoholic, Macaroni, Food_prep_nes, Cigarettes, Beans_dry, Flour_wheat, Crude_materials, Food_preparations_flour_malt_extract, Food_wastes, |
Fruit_prepared_nes, Cereals_breakfast, Coffee_green, Cocoa_powder_and_cake, Pastry, Pet_food, Tea, Nuts_prepared_(exc._groundnuts), Mixes_and_doughs, Oil_vegetable_origin_nes, Oil_essential_nes, Rice_milled, Sugar_refined, Sugar_confectionery, Wine, Spices_nes, Vegetables_dehydrated, Vegetables_preserved_nes, Waters ice_etc, Vegetables_in_vinegar, Tobacco_unmanufactured |
Layer Groups | Layers |
---|---|
Group 1 | Coauthor, Leisure, Facebook |
Group 2 | Lunch, Work |
Layers | MFSE | SPC-Mean |
---|---|---|
Coauthor | 0.95 | 0.76 |
0.76 | 0.70 | |
Leisure | 0.83 | 0.76 |
Lunch | 0.72 | 0.62 |
Work | 0.77 | 0.71 |
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Tang, F.; Zhao, X.; Li, C. Community Detection in Multilayer Networks Based on Matrix Factorization and Spectral Embedding Method. Mathematics 2023, 11, 1573. https://doi.org/10.3390/math11071573
Tang F, Zhao X, Li C. Community Detection in Multilayer Networks Based on Matrix Factorization and Spectral Embedding Method. Mathematics. 2023; 11(7):1573. https://doi.org/10.3390/math11071573
Chicago/Turabian StyleTang, Fengqin, Xuejing Zhao, and Cuixia Li. 2023. "Community Detection in Multilayer Networks Based on Matrix Factorization and Spectral Embedding Method" Mathematics 11, no. 7: 1573. https://doi.org/10.3390/math11071573
APA StyleTang, F., Zhao, X., & Li, C. (2023). Community Detection in Multilayer Networks Based on Matrix Factorization and Spectral Embedding Method. Mathematics, 11(7), 1573. https://doi.org/10.3390/math11071573