Hostname: page-component-77c89778f8-9q27g Total loading time: 0 Render date: 2024-07-22T18:33:25.151Z Has data issue: false hasContentIssue false

Robustness and modular structure in networks

Published online by Cambridge University Press:  27 July 2015

Mathematics & Statistics, University of Vermont, Burlington, VT, USA and Center for Complex Network Research, Northeastern University, Boston, MA, USA (e-mail:
DTU Informatics, Technical University of Denmark, Kgs Lyngby, Denmark and College of Computer and Information Science, Northeastern University, Boston, MA, USA (e-mail:
School of Informatics & Computing, Indiana University, Bloomington IN, USA and Center for Complex Network Research, Northeastern University, Boston, MA, USA (e-mail:


Complex networks have recently attracted much interest due to their prevalence in nature and our daily lives (Vespignani, 2009; Newman, 2010). A critical property of a network is its resilience to random breakdown and failure (Albert et al., 2000; Cohen et al., 2000; Callaway et al., 2000; Cohen et al., 2001), typically studied as a percolation problem (Stauffer & Aharony, 1994; Achlioptas et al., 2009; Chen & D'Souza, 2011) or by modeling cascading failures (Motter, 2004; Buldyrev et al., 2010; Brummitt, et al. 2012). Many complex systems, from power grids and the Internet to the brain and society (Colizza et al., 2007; Vespignani, 2011; Balcan & Vespignani, 2011), can be modeled using modular networks comprised of small, densely connected groups of nodes (Girvan & Newman, 2002). These modules often overlap, with network elements belonging to multiple modules (Palla et al. 2005; Ahn et al. 2010). Yet existing work on robustness has not considered the role of overlapping, modular structure. Here we study the robustness of these systems to the failure of elements. We show analytically and empirically that it is possible for the modules themselves to become uncoupled or non-overlapping well before the network disintegrates. If overlapping modular organization plays a role in overall functionality, networks may be far more vulnerable than predicted by conventional percolation theory.

Research Article
Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


Achlioptas, D., D'Souza, R. M., & Spencer, J. (2009). Explosive percolation in random networks. Science, 323 (5920), 14531455.Google Scholar
Ahn, Y.-Y., Bagrow, J. P., & Lehmann, S. (2010). Link communities reveal multiscale complexity in networks. Nature, 466 (7307), 761764.Google Scholar
Albert, R., Jeong, H., & Barabási, A.-L. (2000). Error and attack tolerance of complex networks. Nature, 406 (6794), 378382.Google Scholar
Arenas, A., Díaz-Guilera, A., Kurths, J., Moreno, Y., & Zhou, C. (2008). Synchronization in complex networks. Physics Reports, 469 (3), 93153.Google Scholar
Balcan, D., & Vespignani, A. (2011). Phase transitions in contagion processes mediated by recurrent mobility patterns. Nature Physics, 7 (7), 581586.Google Scholar
Barabási, A.-L., & Albert, R. (1999). Emergence of scaling in random networks. Science, 286 (5439), 509512.Google Scholar
Brummitt, C. D., D'Souza, R. M., & Leicht, E. A. (2012). Suppressing cascades of load in interdependent networks. Proceedings of the National Academy of Sciences USA, 109 (12), E680E689.Google Scholar
Buldyrev, S. V., Parshani, R., Paul, G., Stanley, H. E., & Havlin, S. (2010). Catastrophic cascade of failures in interdependent networks. Nature, 464 (7291), 10251028.Google Scholar
Callaway, D. S., Newman, M. E. J., Strogatz, Steven H., & Watts, Duncan J. (2000). Network robustness and fragility: Percolation on random graphs. Physical Review Letters, 85 (25), 54685471.Google Scholar
Chen, W. & D'Souza, R. M. (2011). Explosive percolation with multiple giant components. Physical Review Letters, 106 (11), 115701.Google Scholar
Cohen, R., Erez, K., ben-Avraham, D., & Havlin, S. (2000). Resilience of the internet to random breakdowns. Physical Review Letters, 85 (21), 46264628.Google Scholar
Cohen, R., Erez, K., ben-Avraham, D., & Havlin, S. (2001). Breakdown of the internet under intentional attack. Physical Review Letters, 86 (16), 36823685.CrossRefGoogle ScholarPubMed
Colizza, V., Pastor-Satorras, R., & Vespignani, A. (2007). Reaction–diffusion processes and metapopulation models in heterogeneous networks. Nature Physics, 3 (4), 276282.CrossRefGoogle Scholar
de Reus, M. A., Saenger, V. M., Kahn, R. S., & van den Heuvel, M. P. (2014). An edge-centric perspective on the human connectome: Link communities in the brain. Philosophical Transactions of the Royal Society B: Biological Sciences, 369 (1653), 20130527.Google Scholar
Dorogovtsev, S. N., Goltsev, A. V., & Mendes, J. F. F. (2008). Critical phenomena in complex networks. Reviews of Modern Physics, 80 (4), 12751335.Google Scholar
Galbraith, J. R. (1974). Organization design: An information processing view. Interfaces, 4 (3), 2836.Google Scholar
Girvan, M., & Newman, M. E. J. (2002). Community structure in social and biological networks. Proceedings of the National Academy of Sciences USA, 99 (12), 78217826.CrossRefGoogle ScholarPubMed
Kaiser, K. M., & King, W. R. (1982). The manager-analyst interface in systems development. MIS Quarterly, 6 (1), 4959.Google Scholar
Kaiser, M. (2011). A tutorial in connectome analysis: Topological and spatial features of brain networks. Neuroimage, 57 (3), 892907.Google Scholar
Mørup, M., Madsen, K., Dogonowski, A.-M., Siebner, H., & Hansen, L. K. (2010). Infinite relational modeling of functional connectivity in resting state fmri. Advances in Neural Information Processing Systems, Vol. 23, pp. 1750–1758.Google Scholar
Motter, A. E. (2004). Cascade control and defense in complex networks. Physical Review Letters, 93 (9), 098701.Google Scholar
Newman, M. E. J. (2003). Properties of highly clustered networks. Physical Review E, 68 (2), 026121.Google Scholar
Newman, M. E. J. (2006). Modularity and community structure in networks. Proceedings of the National Academy of Sciences USA, 103 (23), 85778582.Google Scholar
Newman, M. E. J. (2010). Networks: An introduction. USA: Oxford University Press.CrossRefGoogle Scholar
Newman, M. E. J., & Park, J. (2003). Why social networks are different from other types of networks. Physical Review E, 68 (3), 036122.Google Scholar
Newman, M. E. J., Strogatz, S. H., & Watts, D. J. (2001). Random graphs with arbitrary degree distributions and their applications. Physical Review E, 64 (Jul), 026118.Google Scholar
Newman, M. E. J., Watts, D. J., & Strogatz, S. H. (2002). Random graph models of social networks. Proceedings of the National Academy of Sciences USA, 99 (Suppl 1), 2566.Google Scholar
Palla, G., Derenyi, I., Farkas, I., & Vicsek, T. (2005). Uncovering the overlapping community structure of complex networks in nature and society. Nature, 435 (7043), 814818.Google Scholar
Pu, S., Wong, J., Turner, B., Cho, E., & Wodak, S. J. (2009). Up-to-date catalogues of yeast protein complexes. Nucleic Acids Research, 37 (3), 825831.Google Scholar
Serrano, M., Boguñá, M., & Vespignani, A. (2009). Extracting the multiscale backbone of complex weighted networks. Proceedings of the National Academy of Sciences USA, 106 (16), 6483.Google Scholar
Sood, V., & Redner, S. (2005). Voter model on heterogeneous graphs. Physical Review Letters, 94 (17), 178701.Google Scholar
Stauffer, D., & Aharony, A. (1994). Introduction to percolation theory. London, UK: Taylor & Francis.Google Scholar
Stearns, F. W. (2010). One hundred years of pleiotropy: A retrospective. Genetics, 186 (3), 767773.Google Scholar
Vedres, B., & Stark, D. (2010). Structural folds: Generative disruption in overlapping groups. American Journal of Sociology, 115 (4), 11501190.Google Scholar
Vego, M. N. (2009). Joint operational warfare: Theory and practice. Washington, DC: Government Printing Office.Google Scholar
Vespignani, A. (2009). Predicting the behavior of techno-social systems. Science, 325 (5939), 425428.Google Scholar
Vespignani, A. (2011). Modelling dynamical processes in complex socio-technical systems. Nature Physics, 8 (1), 3239.CrossRefGoogle Scholar
Wasserman, S., & Faust, K. (1994). Social network analysis: Methods and applications. Vol. 506, Cambridge, UK: Cambridge University Press.Google Scholar