## The Annals of Applied Probability

### Consistency of modularity clustering on random geometric graphs

#### Abstract

Given a graph, the popular “modularity” clustering method specifies a partition of the vertex set as the solution of a certain optimization problem. In this paper, we discuss scaling limits of this method with respect to random geometric graphs constructed from i.i.d. points $\mathcal{X}_{n}=\{X_{1},X_{2},\ldots,X_{n}\}$, distributed according to a probability measure $\nu$ supported on a bounded domain $D\subset\mathbb{R}^{d}$. Among other results, we show, via a Gamma convergence framework, a geometric form of consistency: When the number of clusters, or partitioning sets of $\mathcal{X}_{n}$ is a priori bounded above, the discrete optimal modularity clusterings converge in a specific sense to a continuum partition of the underlying domain $D$, characterized as the solution to a “soap bubble” or “Kelvin”-type shape optimization problem.

#### Article information

Source
Ann. Appl. Probab., Volume 28, Number 4 (2018), 2003-2062.

Dates
Revised: February 2017
First available in Project Euclid: 9 August 2018

https://projecteuclid.org/euclid.aoap/1533780266

Digital Object Identifier
doi:10.1214/17-AAP1313

Mathematical Reviews number (MathSciNet)
MR3843822

Zentralblatt MATH identifier
06974744

#### Citation

Davis, Erik; Sethuraman, Sunder. Consistency of modularity clustering on random geometric graphs. Ann. Appl. Probab. 28 (2018), no. 4, 2003--2062. doi:10.1214/17-AAP1313. https://projecteuclid.org/euclid.aoap/1533780266

#### References

• [1] Adams, R. A. and Fournier, J. J. F. (2003). Sobolev Spaces, 2nd ed. Pure and Applied Mathematics (Amsterdam) 140. Elsevier/Academic Press, Amsterdam.
• [2] Alberti, G. and Bellettini, G. (1998). A non-local anisotropic model for phase transitions: Asymptotic behaviour of rescaled energies. European J. Appl. Math. 9 261–284.
• [3] Ambrosio, L., Fusco, N. and Pallara, D. (2000). Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford Univ. Press, New York.
• [4] Ambrosio, L., Gigli, N. and Savaré, G. (2005). Gradient Flows in Metric Spaces and in the Space of Probability Measures. Birkhäuser, Basel.
• [5] Antonioni, A., Eglof, M. and Tomassini, M. (2013). An energy-based model for spatial social networks. In Advances in Artificial Life ECAL 2013 226–231. MIT Press, Cambridge, MA.
• [6] Arias-Castro, E. and Pelletier, B. (2013). On the convergence of maximum variance unfolding. J. Mach. Learn. Res. 14 1747–1770.
• [7] Arias-Castro, E., Pelletier, B. and Pudlo, P. (2012). The normalized graph cut and Cheeger constant: From discrete to continuous. Adv. in Appl. Probab. 44 907–937.
• [8] Belkin, M. and Niyogi, P. (2003). Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput. 15 1373–1396.
• [9] Belkin, M. and Niyogi, P. (2008). Towards a theoretical foundation for Laplacian-based manifold methods. J. Comput. System Sci. 74 1289–1308.
• [10] Bettstetter, C. (2002). On the minimum node degree and connectivity of a wireless multihop network. In Proceedings of the 3rd ACM International Symposium on Mobile Ad Hoc Networking & Computing 80–91. ACM, New York.
• [11] Bickel, P. and Chen, A. (2009). A nonparametric view of network models and Newman–Girvan and other modularities. Proc. Natl. Acad. Sci. USA 106 21068–21073.
• [12] Blondel, V., Guillaume, J., Lambiotte, R. and Lefebvre, E. (2008). Fast unfolding of communities in large networks. J. Stat. Mech. Theory Exp. 2008 10008–10020.
• [13] Braides, A. (2002). $\Gamma$-Convergence for Beginners. Oxford Lecture Series in Mathematics and Its Applications 22. Oxford Univ. Press, Oxford.
• [14] Braides, A. and Gelli, M. S. (2006). From discrete systems to continuous variational problems: An introduction. In Topics on Concentration Phenomena and Problems with Multiple Scales. Lect. Notes Unione Mat. Ital. 2 3–77. Springer, Berlin.
• [15] Braides, A. and Truskinovsky, L. (2008). Asymptotic expansions by $\Gamma$-convergence. Contin. Mech. Thermodyn. 20 21–62.
• [16] Brakke, K. A. (1992). The surface evolver. Exp. Math. 1 141–165.
• [17] Brandes, U., Delling, D., Gaertler, M., Görke, R., Hoefer, M., Nikoloski, Z. and Wagner, D. (2008). On modularity clustering. IEEE Trans. Knowl. Data Eng. 20 172–188.
• [18] Cañete, A. and Ritoré, M. (2004). Least-perimeter partitions of the disk into three regions of given areas. Indiana Univ. Math. J. 53 883–904.
• [19] Clauset, A., Newman, M. and Moore, C. (2004). Finding community structure in very large networks. Phys. Rev. E 70 066111.
• [20] Coifman, R. R. and Lafon, S. (2006). Diffusion maps. Appl. Comput. Harmon. Anal. 21 5–30.
• [21] Cox, S. J. and Flikkema, E. (2010). The minimal perimeter for $N$ confined deformable bubbles of equal area. Electron. J. Combin. 17 Research Paper 45.
• [22] Davis, E. and Sethuraman, S. (2017). Consistency of modularity clustering on random geometric graphs. Available at arXiv:1604.03993v1.
• [23] de la Peña, V. H. and Montgomery-Smith, S. J. (1995). Decoupling inequalities for the tail probabilities of multivariate $U$-statistics. Ann. Probab. 23 806–816.
• [24] Dhara, M. and Shukla, K. K. (2012). Advanced cost based graph clustering algorithm for random geometric graphs. Int. J. Comput. Appl. 60 20–34.
• [25] Díaz, J., Petit, J. and Serna, M. (2002). A survey of graph layout problems. ACM Comput. Surv. 34 313–356.
• [26] Dí az, J., Penrose, M. D., Petit, J. and Serna, M. (2001). Approximating layout problems on random geometric graphs. J. Algorithms 39 78–116.
• [27] Dudley, R. M. (2002). Real Analysis and Probability. Cambridge Studies in Advanced Mathematics 74. Cambridge Univ. Press, Cambridge. Revised reprint of the 1989 original.
• [28] Durrett, R. (2010). Probability: Theory and Examples, 4th ed. Cambridge Series in Statistical and Probabilistic Mathematics 31. Cambridge Univ. Press, Cambridge.
• [29] El Gamal, A., Mammen, J., Prabhakar, B. and Shah, D. (2004). Throughput-delay trade-off in wireless networks. In Twenty-Third Annual Joint Conference Proceedings of the IEEE Computer and Communications Societies.
• [30] Folland, G. B. (2013). Real Analysis: Modern Techniques and Their Applications, 2nd ed. Wiley, New York.
• [31] Fortuna, M., Stouffer, D., Olesen, J., Jordano, P., Mouillot, D., Krasnov, B., Poulin, R. and Bascompte, J. (2010). Nestedness versus modularity in ecological networks: Two sides of the same coin? J. Anim. Ecol. 79 811–817.
• [32] Fortunato, S. (2010). Community detection in graphs. Phys. Rep. 486 75–174.
• [33] Fortunato, S. and Barthélemy, M. (2006). Resolution limit in community detection. Proc. Natl. Acad. Sci. USA 104 36–41.
• [34] Franceschetti, M. and Meester, R. (2007). Random Networks for Communication: From Statistical Physics to Information Systems. Cambridge Series in Statistical and Probabilistic Mathematics 24. Cambridge Univ. Press, Cambridge.
• [35] García Trillos, N. and Slepčev, D. (2016). A variational approach to the consistency of spectral clustering. Appl. Comput. Harmon. Anal.
• [36] García Trillos, N., Slepčev, D. and von Brecht, J. (2016). Estimating perimeter using graph cuts. Available at arXiv:1602.04102.
• [37] García Trillos, N. and Slepčev, D. (2015). On the rate of convergence of empirical measures in $\infty$-transportation distance. Canad. J. Math. 67 1358–1383.
• [38] García Trillos, N. and Slepčev, D. (2016). Continuum limit of total variation on point clouds. Arch. Ration. Mech. Anal. 220 193–241.
• [39] García Trillos, N., Slepčev, D., von Brecht, J., Laurent, T. and Bresson, X. (2016). Consistency of Cheeger and ratio graph cuts. J. Mach. Learn. Res. 17 Paper No. 181.
• [40] Giné, E. and Koltchinskii, V. (2006). Empirical graph Laplacian approximation of Laplace–Beltrami operators: Large sample results. In High Dimensional Probability. Institute of Mathematical Statistics Lecture Notes—Monograph Series 51 238–259. IMS, Beachwood, OH.
• [41] Giné, E., Latała, R. and Zinn, J. (2000). Exponential and moment inequalities for $U$-statistics. In High Dimensional Probability, II (Seattle, WA, 1999). Progress in Probability 47 13–38. Birkhäuser, Boston, MA.
• [42] Good, B. H., de Montjoye, Y.-A. and Clauset, A. (2010). Performance of modularity maximization in practical contexts. Phys. Rev. E (3) 81 046106.
• [43] Guimera, R. and Amaral, L. (2005). Functional cartography of complex metabolic networks. Nature 433 895–900.
• [44] Guimera, R., Sales-Pardo, M. and Amaral, L. (2004). Modularity from fluctuations in random graphs and complex networks. Phys. Rev. E 70 025101.
• [45] Gupta, P. and Kumar, P. R. (2000). The capacity of wireless networks. IEEE Trans. Inform. Theory 46 388–404.
• [46] Hagmann, P., Cammoun, L., Gigandet, X., Meuli, R., Honey, C. J., Wedeen, V. J. and Sporns, O. (2008). Mapping the structural core of human cerebral cortex. PLoS Biol. 6 e159.
• [47] Hartigan, J. A. (1981). Consistency of single linkage for high-density clusters. J. Amer. Statist. Assoc. 76 388–394.
• [48] Hein, M., Audibert, J.-Y. and von Luxburg, U. (2005). From graphs to manifolds—Weak and strong pointwise consistency of graph Laplacians. In Learning Theory. Lecture Notes in Computer Science 3559 470–485. Springer, Berlin.
• [49] Hu, H., Laurent, T., Porter, M. A. and Bertozzi, A. L. (2013). A method based on total variation for network modularity optimization using the MBO scheme. SIAM J. Appl. Math. 73 2224–2246.
• [50] Lancichinetti, A. and Fortunato, S. (2011). Limits of modularity maximization in community detection. Phys. Rev. E 84 066122.
• [51] Le, C. M., Levina, E. and Vershynin, R. (2016). Optimization via low-rank approximation for community detection in networks. Ann. Statist. 44 373–400.
• [52] Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge Tracts in Mathematics 119. Cambridge Univ. Press, Cambridge.
• [53] Mill, J., Tang, T., Kaminsky, Z., Khare, T., Yazdanpanah, S., Bouchard, L., Jia, P., Assadzadeh, A., Flanagan, J., Schumacher, A., Wang, S.-C. and Petronis, A. (2008). Epigenomic profiling reveals DNA-methylation changes associated with major psychosis. Am. J. Hum. Genet. 82 696–711.
• [54] Morgan, F. (2009). Geometric Measure Theory: A Beginner’s Guide, 4th ed. Elsevier/Academic Press, Amsterdam.
• [55] Newman, M. (2006). Modularity and community structure in networks. Proc. Natl. Acad. Sci. USA 103 8577–8582.
• [56] Newman, M. (2013). Spectral methods for community detection and graph partitioning. Phys. Rev. E 88 042822.
• [57] Newman, M. and Girvan, M. (2004). Finding and evaluating community structure in networks. Phys. Rev. E (3) 69 026113.
• [58] Newman, M. E. J. (2006). Finding community structure in networks using the eigenvectors of matrices. Phys. Rev. E (3) 74 036104.
• [59] Oudet, É. (2011). Approximation of partitions of least perimeter by $\Gamma$-convergence: Around Kelvin’s conjecture. Exp. Math. 20 260–270.
• [60] Penrose, M. (2003). Random Geometric Graphs. Oxford Studies in Probability 5. Oxford Univ. Press, Oxford.
• [61] Pollard, D. (1981). Strong consistency of $k$-means clustering. Ann. Statist. 9 135–140.
• [62] Porter, M., Mucha, P., Newman, M. and Warmbrand, C. (2005). A network analysis of committees in the US House of Representatives. Proc. Natl. Acad. Sci. USA 102 7057–7062.
• [63] Porter, M. A., Onnela, J.-P. and Mucha, P. J. (2009). Communities in networks. Notices Amer. Math. Soc. 56 1082–1097.
• [64] Przulj, N., Corneil, D. G. and Jurisica, I. (2004). Modeling interactome: Scale-free or geometric? Bioinformatics 20 3508–3515.
• [65] Reichardt, J. and Bornholdt, S. (2006). Statistical mechanics of community detection. Phys. Rev. E (3) 74 016110.
• [66] Rohe, K., Chatterjee, S. and Yu, B. (2011). Spectral clustering and the high-dimensional stochastic blockmodel. Ann. Statist. 39 1878–1915.
• [67] Sabin, M. (1987). Convergence and consistency of fuzzy c-means/ISODATA algorithms. IEEE Trans. Pattern Anal. Mach. Intell. 9 661–668.
• [68] Shorack, G. R. and Wellner, J. A. (2009). Empirical Processes with Applications to Statistics. Classics in Applied Mathematics 59. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. Reprint of the 1986 original.
• [69] Singer, A. (2006). From graph to manifold Laplacian: The convergence rate. Appl. Comput. Harmon. Anal. 21 128–134.
• [70] Singer, A. and Wu, H.-T. (2017). Spectral convergence of the connection Laplacian from random samples. Inf. Inference 6 58–123.
• [71] Thorpe, M., Theil, F., Johansen, A. M. and Cade, N. (2015). Convergence of the $k$-means minimization problem using $\Gamma$-convergence. SIAM J. Appl. Math. 75 2444–2474.
• [72] Ting, D., Huang, L. and Jordan, M. I. (2010). An analysis of the convergence of graph Laplacians. In Proceedings of the 27th International Conference on Machine Learning.
• [73] van Gennip, Y. and Bertozzi, A. L. (2012). $\Gamma$-convergence of graph Ginzburg–Landau functionals. Adv. Differential Equations 17 1115–1180.
• [74] van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics 3. Cambridge Univ. Press, Cambridge.
• [75] Villani, C. (2004). Topics in Optimal Transportation. American Mathematical Society, Providence, RI.
• [76] Villani, C. (2009). Optimal Transport: Old and New. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer, Berlin.
• [77] von Luxburg, U., Belkin, M. and Bousquet, O. (2008). Consistency of spectral clustering. Ann. Statist. 36 555–586.
• [78] Wets, R. J.-B. (1999). Statistical estimation from an optimization viewpoint. Ann. Oper. Res. 85 79–101.
• [79] Zhang, X. and Newman, M. (2015). Multiway spectral community detection in networks. Phys. Rev. E 92 052808.
• [80] Zhao, Y., Levina, E. and Zhu, J. (2012). Consistency of community detection in networks under degree-corrected stochastic block models. Ann. Statist. 40 2266–2292.