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  • Cited by 16
Publisher:
Cambridge University Press
Online publication date:
December 2021
Print publication year:
2022
Online ISBN:
9781108774116

Book description

Complex networks are typically not homogeneous, as they tend to display an array of structures at different scales. A feature that has attracted a lot of research is their modular organisation, i.e., networks may often be considered as being composed of certain building blocks, or modules. In this Element, the authors discuss a number of ways in which this idea of modularity can be conceptualised, focusing specifically on the interplay between modular network structure and dynamics taking place on a network. They discuss, in particular, how modular structure and symmetries may impact on network dynamics and, vice versa, how observations of such dynamics may be used to infer the modular structure. They also revisit several other notions of modularity that have been proposed for complex networks and show how these can be related to and interpreted from the point of view of dynamical processes on networks.

References

Abbe, E. (2017). Community detection and stochastic block models: recent developments. The Journal of Machine Learning Research, 18 (1), 64466531.
Abrams, D. M., Mirollo, R., Strogatz, S. H., & Wiley, D. A. (2008). Solvable model for chimera states of coupled oscillators. Physical Review Letters, 101 (8),084103.
Ahn, Y.-Y., Bagrow, J. P., & Lehmann, S. (2010). Link communities reveal multiscale complexity in networks. Nature, 466 (7307), 761764.
Almquist, Z. W. (2012, October). Random errors in egocentric networks. Social Networks, 34(4), 493505. doi: https://doi.org/10.1016/j.socnet.2012.03.002
Alpert, C. J., & Kahng, A. B. (1995). Recent directions in netlist partitioning: a survey. Integration, 19(1–2), 181.
Altafini, C. (2012). Dynamics of opinion forming in structurally balanced social networks. PloS One, 7(6), e38135.
Altafini, C. (2013, April). Consensus problems on networks with antagonistic interactions. IEEE Transactions on Automatic Control, 58(4), 935946. doi: https://doi.org/10.1109/TAC.2012.2224251
Arenas, A., Díaz-Guilera, A., Kurths, J., Moreno, Y., & Zhou, C. (2008). Synchronization in complex networks. Physics Reports, 469 (3), 93153.
Arenas, A., Díaz-Guilera, A., & Pérez-Vicente, C. J. (2006). Synchronization reveals topological scales in complex networks. Physical Review Letters, 96 (11),114102.
Asllani, M., Lambiotte, R., & Carletti, T. (2018). Structure and dynamical behavior of non-normal networks. Science Advances, 4(12),eaau9403.
Avella-Medina, M., Parise, F., Schaub, M. T., & Segarra, S. (2020). Centrality measures for graphons: accounting for uncertainty in networks. IEEE Transactions on Network Science and Engineering, 7 (1), 520537. doi: https://doi.org/10.1109/TNSE.2018.2884235
Aynaud, T., Blondel, V. D., Guillaume, J. L., & Lambiotte, R. (2013). Multilevel local optimization of modularity. In: Bichot, C.-E., Siarry, P. (eds.), Graph Partitioning, (pp. 315345). John Wiley and Sons, Hoboken, NJ.
Banisch, R., & Conrad, N. D. (2015). Cycle-flow–based module detection in directed recurrence networks. EPL (Europhysics Letters), 108 (6), 68008.
Barabási, A.-L., et al. (2016). Network science. Cambridge University Press, Cambridge, UK.
Barbarossa, S., & Sardellitti, S. (2020). Topological signal processing: making sense of data building on multiway relations. IEEE Signal Processing Magazine, 37 (6), 174183.
Battiston, F., Cencetti, G., Iacopini, I. et al. (2020). Networks beyond pairwise interactions: structure and dynamics. Physics Reports, 874, 192.
Belkin, M., & Niyogi, P. (2001). Laplacian eigenmaps and spectral techniques for embedding and clustering. Advances in Neural Information Processing Systems, 14, 585591.
Bhatia, R. (2013). Matrix analysis (vol. 169). Springer Science & Business Media, London.
Blondel, V. D., Gajardo, A., Heymans, M., Senellart, P., & Van Dooren, P. (2004). A measure of similarity between graph vertices: applications to synonym extraction and web searching. SIAM Review, 46 (4), 647666.
Blondel, V. D., Guillaume, J.-L., Lambiotte, R., & Lefebvre, E. (2008). Fast unfolding of communities in large networks. Journal of Statistical Mechanics: Theory and Experiment, 2008 (10),P10008.
Bohlin, L., Edler, D., Lancichinetti, A., & Rosvall, M. (2014). Community detection and visualization of networks with the map equation framework. In: Ding, Y., Rousseau, R., & Wolfram, D. (eds.), Measuring scholarly impact, (pp. 334). Springer, New York.
Borgatti, S. P., Carley, K. M., & Krackhardt, D. (2006). On the robustness of centrality measures under conditions of imperfect data. Social Networks, 28(2), 124136. doi: https://doi.org/10.1016/j.socnet.2005.05.001
Brandes, U. (2005). Network analysis: methodological foundations, (vol. 3418). Springer Science & Business Media, London.
Brandes, U., Delling, D., Gaertler, M. et al. (2007). On modularity clustering. IEEE Transactions on Knowledge and Data Engineering, 20(2), 172188.
Brin, S., & Page, L. (1998). The anatomy of a large-scale hypertextual web search engine. Computer Networks and ISDN Systems, 30(1–7), 107117.
Broido, A. D., & Clauset, A. (2019). Scale-free networks are rare. Nature Communications, 10 (1), 110.
Bui-Xuan, B.-M., & Jones, N. S. (2014). How modular structure can simplify tasks on networks: parameterizing graph optimization by fast local community detection. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 470 (2170),20140224.
Bullo, F. (2019). Lectures on network systems. Kindle Direct Publishing. ISBN 978-1-986425-64-3.
Burt, R. S. (2004). Structural holes and good ideas. American Journal of Sociology, 110 (2), 349399.
Cardoso, D. M., Delorme, C., & Rama, P. (2007). Laplacian eigenvectors and eigenvalues and almost equitable partitions. European Journal of Combinatorics, 28 (3), 665673.
Carletti, T., Fanelli, D., & Lambiotte, R. (2020). Random walks and community detection in hypergraphs. arXiv preprint arXiv:2010.14355.
Cason, T. P. (2014). Role extraction in networks. (unpublished doctoral dissertation). Catholic University of Louvain.
Cavallari, S., Zheng, V. W., Cai, H., Chang, K. C.-C., & Cambria, E. (2017). Learning community embedding with community detection and node embedding on graphs. In Proceedings of the 2017 ACM on Conference on Information and Knowledge Management, (pp. 377–386).
Chan, A., & Godsil, C. D. (1997). Symmetry and eigenvectors. In: Hahn, G., & Sabidussi, G. (eds.), Graph Symmetry, (pp. 75106). Springer, New York.
Chandra, A. K., Raghavan, P., Ruzzo, W. L., Smolensky, R., & Tiwari, P. (1996). The electrical resistance of a graph captures its commute and cover times. Computational Complexity, 6 (4), 312340.
Chodrow, P. S., Veldt, N., & Benson, A. R. (2021). Hypergraph clustering: from blockmodels to modularity. arXiv preprint arXiv:2101.09611.
Chung, F., & Lu, L. (2002). Connected components in random graphs with given expected degree sequences. Annals of Combinatorics, 6 (2), 125145.
Chung, F. R. (1997). Spectral graph theory, (vol. 92). American Mathematical Society, Providence, RI.
Coifman, R. R., Lafon, S., Lee, A. B. et al. (2005). Geometric diffusions as a tool for harmonic analysis and structure definition of data: diffusion maps. Proceedings of the National Academy of Sciences of the United States of America, 102 (21), 74267431.
Conrad, N. D., Weber, M., & Schütte, C. (2016). Finding dominant structures of nonreversible Markov processes. Multiscale Modeling & Simulation, 14 (4), 13191340.
Cooper, K., & Barahona, M. (2010). Role-based similarity in directed networks. arXiv preprint arXiv:1012.2726.
Dasgupta, A., Hopcroft, J. E., & McSherry, F. (2004). Spectral analysis of random graphs with skewed degree distributions. In 45th Annual IEEE Symposium on Foundations of Computer Science, (pp. 602–610). doi: https://doi.org/10.1109/FOCS.2004.61.
De Domenico, M., Solé-Ribalta, A., Cozzo, E. et al. (2013). Mathematical formulation of multilayer networks. Physical Review X, 3 (4),041022.
Delvenne, J.-C., & Libert, A.-S. (2011). Centrality measures and thermodynamic formalism for complex networks. Physical Review E, 83(4), 046117.
Delvenne, J.-C., Schaub, M. T., Yaliraki, S. N., & Barahona, M. (2013). The stability of a graph partition: a dynamics-based framework for community detection. In A. Mukherjee, M. Choudhury, F. Peruani, N. Ganguly, & B. Mitra (eds.), Dynamics On and Of Complex Networks, Volume 2 (pp. 221–242). Springer, New York. doi: https://doi.org/10.1007/978-1-4614-6729-8_11
Delvenne, J.-C., Yaliraki, S. N., & Barahona, M. (2010). Stability of graph communities across time scales. Proceedings of the National Academy of Sciences, 107 (29),12755–12760. doi: https://doi.org/10.1073/pnas.0903215107
Derrida, B., & Flyvbjerg, H. (1986). Multivalley structure in Kauffman’s model: analogy with spin glasses. Journal of Physics A: Mathematical and General, 19 (16),L1003.
Devriendt, K. (2020). Effective resistance is more than distance: Laplacians, simplices and the Schur complement. arXiv preprint arXiv:2010.04521.
Devriendt, K. (2020). Effective resistance is more than distance: Laplacians, simplices and the Schur complement. arXiv preprint arXiv:2010.04521.
Doreian, P., Batagelj, V., & Ferligoj, A. (2020). Advances in Network Clustering and Blockmodeling. John Wiley & Hoboken, NJ.
Egerstedt, M., Martini, S., Cao, M., Camlibel, K., & Bicchi, A. (2012). Interacting with networks: how does structure relate to controllability in single-leader, consensus networks? Control Systems, IEEE, 32 (4),6673. doi: https://doi.org/10.1109/MCS.2012.2195411
Eriksson, A., Edler, D., Rojas, A., & Rosvall, M. (2020). Mapping flows on hypergraphs. arXiv preprint arXiv:2101.00656.
Eriksson, A., Edler, D., Rojas, A., & Rosvall, M. (2021). Mapping flows on hypergraphs. arXiv preprint arXiv:2101.00656.
Estrada, E., & Hatano, N. (2008). Communicability in complex networks. Physical Review E, 77 (3),036111.
Everett, M. G., & Borgatti, S. P. (1994). Regular equivalence: general theory. Journal of Mathematical Sociology, 19 (1), 2952.
Expert, P., Evans, T. S., Blondel, V. D., & Lambiotte, R. (2011). Uncovering space-independent communities in spatial networks. Proceedings of the National Academy of Sciences, 108 (19), 76637668.
Faccin, M., Schaub, M. T., & Delvenne, J.-C. (2018). Entrograms and coarse graining of dynamics on complex networks. Journal of Complex Networks, 6 (5), 661678.
Faccin, M., Schaub, M. T., & Delvenne, J.-C. (2020). State aggregations in Markov chains and block models of networks. arXiv preprint arXiv:2005.00337.
Faqeeh, A., Osat, S., & Radicchi, F. (2018). Characterizing the analogy between hyperbolic embedding and community structure of complex networks. Physical Review Letters, 121(9), 098301.
Fiedler, M. (1973). Algebraic connectivity of graphs. Czechoslovak Mathematical Journal, 23(2), 298305.
Fiedler, M. (2011). Matrices and Graphs in Geometry. Cambridge University Press, Cambridge, UK. doi: https://doi.org/10.1017/cbo9780511973611
Fortunato, S. (2010). Community detection in graphs. Physics Reports, 486(3–5), 75174. doi: https://doi.org/10.1016/j.physrep.2009.11.002
Fortunato, S., & Barthelemy, M. (2007). Resolution limit in community detection. Proceedings of the National Academy of Sciences, 104(1),3641.
Fortunato, S., & Hric, D. (2016). Community detection in networks: a user guide. Physics Reports, 659, 144.
Fosdick, B. K., Larremore, D. B., Nishimura, J., & Ugander, J. (2018). Configuring random graph models with fixed degree sequences. SIAM Review, 60 (2), 315355.
Fouss, F., Saerens, M., & Shimbo, M. (2016). Algorithms and models for network data and link analysis. Cambridge University Press, Cambridge, UK.
Ghasemian, A., Hosseinmardi, H., & Clauset, A. (2019). Evaluating overfit and underfit in models of network community structure. IEEE Transactions on Knowledge and Data Engineering, 32(9), 17221735. doi: https://doi.org/10.1109/TKDE.2019.2911585.
Gleich, D. F. (2015). Pagerank beyond the web. SIAM Review, 57 (3), 321363.
Godsil, C., & Royle, G. F. (2013). Algebraic graph theory (vol. 207). Springer Science & Business Media, London.
Golub, G., & Van Loan, C. (2013). Matrix computations. 4th ed. Johns Hopkins. University Press, Baltimore, MD.
Golubitsky, M., & Stewart, I. (2006). Nonlinear dynamics of networks: the groupoid formalism. Bulletin of the American Mathematical Society, 43 (3), 305364.
Golubitsky, M., & Stewart, I. (2015). Recent advances in symmetric and network dynamics. Chaos: An Interdisciplinary Journal of Nonlinear Science, 25(9),097612.
Good, B. H., De Montjoye, Y.-A., & Clauset, A. (2010). Performance of modularity maximization in practical contexts. Physical Review E, 81 (4),046106.
Grohe, M., Kersting, K., Mladenov, M., & Selman, E. (2014). Dimension reduction via colour refinement. In European Symposium on Algorithms, (pp. 505–516). Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44777-2_42
Grohe, M., & Schweitzer, P. (2020). The graph isomorphism problem. Communications of the ACM, 63 (11), 128134.
Grover, A., & Leskovec, J. (2016). node2vec: Scalable feature learning for networks. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, (pp. 855–864).
Gvishiani, A. D., & Gurvich, V. A. (1987). Metric and ultrametric spaces of resistances. Uspekhi Matematicheskikh Nauk, 42 (6), 187188.
Hage, P., Harary, F., & Harary, F. (1983). Structural models in anthropology. Cambridge Studies in Anthropology, Social. No. 46. Cambridge University Press, Cambridge, UK.
Higham, N. J. (2008). Functions of matrices: theory and computation. SIAM.
Holland, P. W., Laskey, K. B., & Leinhardt, S. (1983). Stochastic blockmodels: first steps. Social Networks, 5 (2), 109137.
Holland, P. W., & Leinhardt, S. (1973). The structural implications of measurement error in sociometry. Journal of Mathematical Sociology, 3 (1), 85111.
Holme, P., & Saramäki, J. (2019). Temporal Network Theory. Springer.
Karrer, B., & Newman, M. E. J. (2011). Stochastic blockmodels and community structure in networks. Physical Review E, 83(1), 016107. doi: https://doi.org/10.1103/PhysRevE.83.016107
Kivela, M., Arenas, A., Barthelemy, M. et al. (2014). Multilayer networks. Journal of Complex Networks, 2 (3),203–271. doi: https://doi.org/10.1093/comnet/cnu016
Klein, D. J., & Randić, M. (1993). Resistance distance. Journal of Mathematical Chemistry, 12 (1), 8195. doi: https://doi.org/10.1007%2Fbf01164627
Klimm, F., Jones, N. S., & Schaub, M. T. (2021). Modularity maximisation for graphons. arXiv preprint arXiv:2101.00503.
Kloumann, I. M., Ugander, J., & Kleinberg, J. (2017). Block models and personalized pagerank. Proceedings of the National Academy of Sciences, 114 (1), 3338.
Komarek, A., Pavlik, J., & Sobeslav, V. (2015). Network visualization survey. In Computational Collective Intelligence, (pp. 275284). Springer.
Kondor, R., & Lafferty, J. (2002). Diffusion kernels on graphs and other discrete input spaces. In Proceedings of the ICML’02: Nineteenth International Joint Conference on Machine Learning, (pp. 315–322).
Kossinets, G. (2006). Effects of missing data in social networks. Social Networks, 28(3),247268. Accessed 1 October 2020 from https://linkinghub.elsevier.com/retrieve/pii/S0378873305000511 doi: https://doi.org/10.1016/j.socnet.2005.07.002
Krzakala, F., Moore, C., Mossel, E. et al. (2013). Spectral redemption in clustering sparse networks. Proceedings of the National Academy of Sciences, 110 (52), 2093520940.
Kunegis, J., Schmidt, S., Lommatzsch, A. et al. (2010). Spectral analysis of signed graphs for clustering, prediction and visualization. In Proceedings of the 2010 SIAM International Conference on Data Mining (SDM) (vol. 10, pp. 559–570). Society for Industrial and Applied Mathematics.
Lafon, S., & Lee, A. (2006). Diffusion maps and coarse-graining: a unified framework for dimensionality reduction, graph partitioning, and data set parameterization. Pattern Analysis and Machine Intelligence, IEEE Transactions on, 28(9), 13931403. doi: https://doi.org/10.1109/TPAMI.2006.184
Lambiotte, R., Ausloos, M., & Holyst, J. (2007). Majority model on a network with communities. Physical Review E, 75 (3),030101.
Lambiotte, R., Delvenne, J.-C., & Barahona, M. (2014). Random walks, Markov processes and the multiscale modular organization of complex networks. IEEE Transactions on Network Science and Engineering, 1(2), 7690. doi: https://doi.org/10.1109/TNSE.2015.2391998
Lambiotte, R., & Rosvall, M. (2012). Ranking and clustering of nodes in networks with smart teleportation. Physical Review E, 85 (5),056107.
Lambiotte, R., Rosvall, M., & Scholtes, I. (2019). From networks to optimal higher-order models of complex systems. Nature Physics, 15(4), 313320.
Langville, A. N., & Meyer, C. D. (2011). Google’s PageRank and beyond: the science of search engine rankings. Princeton University Press.
Le, C. M., Levina, E., & Vershynin, R. (2017). Concentration and regularization of random graphs. Random Structures & Algorithms, 51 (3), 538561.
Lei, J., & Rinaldo, A. (2015). Consistency of spectral clustering in stochastic block models. The Annals of Statistics, 43 (1), 215237.
Leskovec, J., Lang, K. J., Dasgupta, A., & Mahoney, M. W. (2008). Statistical properties of community structure in large social and information networks. In Proceedings of the 17th International Conference on World Wide Web (pp. 695–704).
Lorrain, F., & White, H. C. (1971). Structural equivalence of individuals in social networks. The Journal of Mathematical Sociology, 1 (1), 4980.
Malliaros, F. D., & Vazirgiannis, M. (2013). Clustering and community detection in directed networks: a survey. Physics Reports, 533 (4), 95142.
Martin, T., Ball, B., & Newman, M. E. J. (2016). Structural inference for uncertain networks. Physical Review E, 93 (1),012306.
Masuda, N., & Lambiotte, R. (2020). A guide To Temporal Networks (vol. 6). World Scientific.
Masuda, N., Porter, M. A., & Lambiotte, R. (2017). Random walks and diffusion on networks. Physics Reports, 716, 158.
Mauroy, A., Susuki, Y., & Mezić, I. (2020). The Koopman Operator in Systems and Control. Springer.
McPherson, M., Smith-Lovin, L., & Cook, J. M. (2001). Birds of a feather: homophily in social networks. Annual Review of Sociology, 27(1), 415444.
Meilǎ, M. (2007). Comparing clusterings – an information based distance. Journal of Multivariate Analysis, 98 (5), 873895.
Menczer, F., Fortunato, S., & Davis, C. A. (2020). A First Course in Network Science. Cambridge University Press.
Meunier, D., Lambiotte, R., & Bullmore, E. T. (2010). Modular and hierarchically modular organization of brain networks. Frontiers in Neuroscience, 4, 200.
Milo, R., Shen-Orr, S., Itzkovitz, S. et al. (2002). Network motifs: simple building blocks of complex networks. Science, 298 (5594), 824827.
Mucha, P. J., Richardson, T., Macon, K., Porter, M. A., & Onnela, J.-P. (2010). Community structure in time-dependent, multiscale, and multiplex networks. Science, 328 (5980), 876878.
Newman, M. E., et al. (2003). Random graphs as models of networks. Handbook of Graphs and Networks, 1, 3568.
Newman, M. E. J. (2013). Spectral methods for community detection and graph partitioning. Physical Review E, 88 (4),042822.
Newman, M. E. J. (2016). Community detection in networks: modularity optimization and maximum likelihood are equivalent. arXiv preprint arXiv:1606.02319.
Newman, M. E. J. (2018a). Network. Oxford University Press.
Newman, M. E. J. (2018b). Network structure from rich but noisy data. Nature Physics, 14, 5.
Newman, M. E. J., & Girvan, M. (2004). Finding and evaluating community structure in networks. Physical Review E, 69(2), 026113.
Nicosia, V., Mangioni, G., Carchiolo, V., & Malgeri, M. (2009). Extending the definition of modularity to directed graphs with overlapping communities. Journal of Statistical Mechanics: Theory and Experiment, 2009(03),P03024.
O’Clery, N., Yuan, Y., Stan, G.-B., & Barahona, M. (2013). Observability and coarse graining of consensus dynamics through the external equitable partition. Physical Review E, 88(4). doi: https://doi.org/10.1103/physreve.88.042805
Park, J., & Newman, M. E. (2004). Statistical mechanics of networks. Physical Review E, 70 (6),066117.
Pastor-Satorras, R., Castellano, C., Van Mieghem, P., & Vespignani, A. (2015). Epidemic processes in complex networks. Reviews of Modern Physics, 87 (3), 925.
Pecora, L. M., Sorrentino, F., Hagerstrom, A. M., Murphy, T. E., & Roy, R. (2014). Cluster synchronization and isolated desynchronization in complex networks with symmetries. Nature Communications, 5, 4079.
Peel, L., Larremore, D. B., & Clauset, A. (2017). The ground truth about metadata and community detection in networks. Science Advances, 3 (5),e1602548.
Peixoto, T. P. (2018). Reconstructing networks with unknown and heterogeneous errors. Physical Review X, 8 (4),041011.
Peixoto, T. P. (2019). Bayesian stochastic blockmodeling. Advances in Network Clustering and Blockmodeling, 289332.
Perozzi, B., Al-Rfou, R., & Skiena, S. (2014). Deepwalk: online learning of social representations. In Proceedings of the 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (pp. 701–710).
Piccardi, C. (2011). Finding and testing network communities by lumped Markov chains. PLoS One, 6(11), e27028. doi: 10.1371/journal.pone.0027028
Pons, P., & Latapy, M. (2005). Computing communities in large networks using random walks. In: Yolum, P., Güngör, T., Gürgen, F., & Özturan, C. (eds.), International symposium on computer and information sciences, (pp. 284–293). Springer, Berlin, Heidelberg. doi: https://doi.org/10.1007/11569596_31
Porter, M., Onnela, J., & Mucha, P. (2009). Communities in networks. Notices of the AMS, 56 (9),10821097, 11641166.
Porter, M. A., & Gleeson, J. P. (2016). Dynamical systems on networks. Frontiers in Applied Dynamical Systems: Reviews and Tutorials, 4. doi: https://doi.org/10.1101/2021.01.21.427609
Proskurnikov, A. V., & Tempo, R. (2017). A tutorial on modeling and analysis of dynamic social networks. Part i. Annual Reviews in Control, 43, 6579. doi: https://doi.org/10.1016/j.arcontrol.2017.03.002
Putra, P., Thompson, T. B., & Goriely, A. (2021). Braiding braak and braak: staging patterns and model selection in network neurodegeneration. bioRxiv. Accessed at www.biorxiv.org.
Read, K. E. (1954). Cultures of the Central Highlands, New Guinea. Southwestern Journal of Anthropology, 10 (1), 143.
Reichardt, J., & Bornholdt, S. (2006). Statistical mechanics of community detection. Physical Review E, 74 (1),016110.
Reichardt, J., & White, D. R. (2007). Role models for complex networks. The European Physical Journal B, 60 (2), 217224.
Rényi, A., et al. (1961). On measures of entropy and information. In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability Volume 1: contributions to the theory of statistics.
Rohe, K., Chatterjee, S., Yu, B., et al. (2011). Spectral clustering and the high-dimensional stochastic blockmodel. The Annals of Statistics, 39 (4), 18781915.
Rombach, M. P., Porter, M. A., Fowler, J. H., & Mucha, P. J. (2014). Core-periphery structure in networks. SIAM Journal on Applied Mathematics, 74 (1), 167190.
Rossetti, G., & Cazabet, R. (2018). Community discovery in dynamic networks: a survey. ACM Computing Surveys (CSUR), 51(2), 137.
Rossi, R. A., Jin, D., Kim, S. et al. (2020). On proximity and structural role-based embeddings in networks: misconceptions, techniques, and applications. ACM Transactions on Knowledge Discovery from Data (TKDD), 14 (5), 137.
Rosvall, M., & Bergstrom, C. T. (2008). Maps of random walks on complex networks reveal community structure. Proceedings of the National Academy of Sciences, 105 (4), 11181123.
Rosvall, M., Esquivel, A. V., Lancichinetti, A., West, J. D., & Lambiotte, R. (2014). Memory in network flows and its effects on spreading dynamics and community detection. Nature Communications, 5, 4630. doi: https://doi.org/10.1038/ncomms5630
Ruggeri, N., & De Bacco, C. (2019, August). Sampling on networks: estimating eigenvector centrality on incomplete graphs. arXiv:1908.00388v1 [cs.SI].
Salnikov, V., Schaub, M. T., & Lambiotte, R. (2016). Using higher-order Markov models to reveal flow-based communities in networks. Scientific Reports, 6, 23194. doi: https://doi.org/10.1038/srep23194
Sanchez-Garcia, R. J. (2018). Exploiting symmetry in network analysis. arXiv preprint arXiv:1803.06915.
Schaeffer, S. E. (2007). Graph clustering. Computer Science Review, 1(1), 2764. doi: https://doi.org/10.1016/j.cosrev.2007.05.001
Schaub, M. T. (2014). Unraveling complex networks under the prism of dynamical processes: relations between structure and dynamics. (Doctoral dissertation). Imperial College London.
Schaub, M. T., Benson, A. R., Horn, P., Lippner, G., & Jadbabaie, A. (2020). Random walks on simplicial complexes and the normalized Hodge 1-Laplacian. SIAM Review, 62 (2), 353391.
Schaub, M. T., Billeh, Y. N., Anastassiou, C. A., Koch, C., & Barahona, M. (2015). Emergence of slow-switching assemblies in structured neuronal networks. PLOS Computational Biology, 11 (7),e1004196.
Schaub, M. T., Delvenne, J.-C., Lambiotte, R., & Barahona, M. (2019a). Multiscale dynamical embeddings of complex networks. Physical Review E, 99 (6),062308. doi: https://doi.org/10.1103/PhysRevE.99.062308
Schaub, M. T., Delvenne, J.-C., Lambiotte, R., & Barahona, M. (2019b). Structured networks and coarse-grained descriptions: a dynamical perspective. Advances in Network Clustering and Blockmodeling, pp. 333361.
Schaub, M. T., Delvenne, J.-C., Rosvall, M., & Lambiotte, R. (2017, 02). The many facets of community detection in complex networks. Applied Network Science, 2(1), 4. doi: 10.1007/s41109-017-0023-6
Schaub, M. T., Delvenne, J.-C., Yaliraki, S. N., & Barahona, M. (2012a). Markov dynamics as a zooming lens for multiscale community detection: non clique-like communities and the field-of-view limit. PloS One, 7(2), e32210.
Schaub, M. T., Lambiotte, R., & Barahona, M. (2012b). Encoding dynamics for multiscale community detection: Markov time sweeping for the map equation. Physical Review E, 86 (2),026112.
Schaub, M. T., O’Clery, N., Billeh, Y. N. et al. (2016). Graph partitions and cluster synchronization in networks of oscillators. Chaos: An Interdisciplinary Journal of Nonlinear Science, 26 (9),094821.
Schaub, M. T., & Peel, L. (2020). Hierarchical community structure in networks. arXiv preprint arXiv:2009.07196.
Schaub, M. T., Zhu, Y., Seby, J.-B., Roddenberry, T. M., & Segarra, S. (2021). Signal processing on higher-order networks: Livin’ on the edge... and beyond. Signal Processing, 187, 108149. doi: https://doi.org/10.1016/j.sigpro.2021.108149
Serrano, M. A., & Boguná, M. (2021). The shortest path to network geometry. Cambridge University Press.
Serrano, M. A., Krioukov, D., & Boguná, M. (2008). Self-similarity of complex networks and hidden metric spaces. Physical Review Letters, 100 (7),078701.
Shi, J., & Malik, J. (1997). Normalized cuts and image segmentation. In Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition (pp. 731737).
Shuman, D. I., Narang, S. K., Frossard, P., Ortega, A., & Vandergheynst, P. (2013). The emerging field of signal processing on graphs: extending high-dimensional data analysis to networks and other irregular domains. IEEE Signal Processing Magazine, 30 (3), 8398.
Simon, H. A. (1962). The architecture of complexity. Proceedings of the American Philosophical Society, 106 (6), 467482.
Simon, H. A., & Ando, A. (1961). Aggregation of variables in dynamic systems. Econometrica: Journal of the Econometric Society, 111138.
Simpson, E. H. (1949). Measurement of diversity. Nature, 163 (4148),688.
Smiljanić, J., Edler, D., & Rosvall, M. (2020). Mapping flows on sparse networks with missing links. Physical Review E, 102 (1),012302.
Spielman, D. A., & Teng, S.-H. (2011). Spectral sparsification of graphs. SIAM Journal on Computing, 40(4), 9811025. doi: https://doi.org/10.1137/08074489X
Stamm, F. I., Neuhäuser, L., Lemmerich, F., Schaub, M. T., & Strohmaier, M. (2020). Systematic edge uncertainty in attributed social networks and its effects on rankings of minority nodes. arXiv:2010.11546v2 [cs.SI]
Stewart, G. W. (2001). Matrix algorithms volume 2: eigensystems. SIAM.
Stewart, I., Golubitsky, M., & Pivato, M. (2003). Symmetry groupoids and patterns of synchrony in coupled cell networks. SIAM Journal on Applied Dynamical Systems, 2 (4), 609646.
Strogatz, S. (2004). Sync: the emerging science of spontaneous order. Penguin UK.
Stumpf, M. P., Wiuf, C., & May, R. M. (2005). Subnets of scale-free networks are not scale-free: sampling properties of networks. Proceedings of the National Academy of Sciences, 102 (12), 42214224.
Tian, F., Gao, B., Cui, Q., Chen, E., & Liu, T.-Y. (2014). Learning deep representations for graph clustering. In Proceedings of the AAAI Conference on Artificial Intelligence (vol. 28).
Traag, V. A., Waltman, L., & Van Eck, N. J. (2019). From Louvain to Leiden: guaranteeing well-connected communities. Scientific Reports, 9 (1), 112.
Trefethen, L. N., & Embree, M. (2005). Spectra and pseudospectra: the behavior of nonnormal matrices and operators. Princeton University Press.
Van Lierde, H., Chow, T. W., & Delvenne, J.-C. (2019). Spectral clustering algorithms for the detection of clusters in block-cyclic and block-acyclic graphs. Journal of Complex Networks, 7 (1), 153.
Von Luxburg, U. (2007). A tutorial on spectral clustering. Statistics and Computing, 17(4), 395416.
Wagner, C., Singer, P., & Karimi, F. (2017). Sampling from social networks with attributes, pp. 11811190. doi: https://doi.org/10.1145/3038912.3052665
Wainwright, M. J. (2019). High-dimensional statistics: a non-asymptotic viewpoint, (vol. 48). Cambridge University Press.
Wasserman, S., & Faust, K. (1994). Social network analysis: methods and applications, (vol. 8). Cambridge University Press.
Wu, Z., Pan, S., Chen, F. et al. (2020). A comprehensive survey on graph neural networks. IEEE Transactions on Neural Networks and Learning Systems. arXiv:1901.00596v4 [cs.LG]
Xu, R., & Wunsch, D. (2008). Clustering (vol. 10). John Wiley & Sons.
Young, J.-G., Cantwell, G. T., & Newman, M. E. J. (2020). Robust Bayesian inference of network structure from unreliable data. arXiv:2008.03334v2 [cs.SI]
Yu, Y., Wang, T., & Samworth, R. J. (2015). A useful variant of the Davis–Kahan theorem for statisticians. Biometrika, 102 (2), 315323.

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