Element contents
Percolation in Spatial Networks
Spatial Network Models Beyond Nearest Neighbours Structures
Published online by Cambridge University Press: 15 June 2022
Summary
Percolation theory is a well studied process utilized by networks theory to understand the resilience of networks under random or targeted attacks. Despite their importance, spatial networks have been less studied under the percolation process compared to the extensively studied non-spatial networks. In this Element, the authors will discuss the developments and challenges in the study of percolation in spatial networks ranging from the classical nearest neighbors lattice structures, through more generalized spatial structures such as networks with a distribution of edge lengths or community structure, and up to spatial networks of networks.
Keywords
- Type
- Element
- Information
- Online ISBN: 9781009168076Publisher: Cambridge University PressPrint publication: 14 July 2022
References
Albert, R., Jeong, H., & Barabási, A.-L. (2000). Error and attack tolerance of complex networks. Nature, 406(6794), 378–382.Google Scholar
Barabási, A.-L., & Albert, R. (1999). Emergence of scaling in random networks. Science, 286(5439), 509–512. doi: https://doi.org/10.1126/science.286.5439.509.CrossRefGoogle ScholarPubMed
Bashan, A., et al. (2013). The extreme vulnerability of interdependent spatially embedded networks. Nature Physics, 9(10), 667–672.CrossRefGoogle Scholar
Bell, M. G., & Iida, Y. (1997). Transportation network analysis. Wiley Online Library.CrossRefGoogle Scholar
Ben-Avraham, D., et al. (2003). Geographical embedding of scale-free networks. PhysicaA: Statistical Mechanics and Its Applications, 330(1–2), 107–116.CrossRefGoogle Scholar
Berezin, Y., et al. (2015). Localized attacks on spatially embedded networks with dependencies. Scientific Reports, 5(1),1–5.Google Scholar
Bianconi, G., Pin, P., & Marsili, M. (2009). Assessing the relevance of node features for network structure. Proceedings of the National Academy of Sciences, 106(28), 11433–11438.CrossRefGoogle Scholar
Bogu ná, M., et al. (2021). Network geometry. Nature Reviews Physics, Nature Publishing Group 3(2), 114–135.Google Scholar
Bollobás, B. (1985). Random Graphs. London Mathematical Society Monographs, Academic Press, London.Google Scholar
Bonamassa, I., et al. (2019). Critical stretching of mean-field regimes in spatial networks. Physical Review Letters, 123(8), 088301.CrossRefGoogle ScholarPubMed
Buldyrev, S. V., et al. (2010). Catastrophic cascade of failures in interdependent networks. Nature, 464(7291), 1025–1028.CrossRefGoogle ScholarPubMed
Bullmore, E., & Sporns, O. (2012). The economy of brain network organization. Nature Reviews Neuroscience, 13(5), 336–349.CrossRefGoogle ScholarPubMed
Bunde, A., & Havlin, S. (1991). Fractals and disordered systems. Springer, New York.CrossRefGoogle Scholar
Cohen, R., Havlin, S., & Ben-Avraham, D. (2003). Efficient immunization strategies for computer networks and populations. Physical Review Letters, 91(24), 247901.Google Scholar
Cohen, R., et al. (2000). Resilience of the internet to random breakdowns. Physical Review Letters, 85(21), 4626.CrossRefGoogle ScholarPubMed
Cohen, R., et al. (2001). Breakdown of the internet under intentional attack. Physical Review Letters, 86(16), 3682.Google Scholar
Colizza, V., Barrat, A., Barthélemy, M., & Vespignani, A. (2006). The role of the airline transportation network in the prediction and predictability of global epidemics. Proceedings of the National Academy of Sciences, 103(7), 2015–2020.Google Scholar
Danziger, M. M., et al. (2013). Interdependent spatially embedded networks: dynamics at percolation threshold. In 2013 International Conference on Signal-Image Technology & Internet-Based Systems (pp. 619–625). doi: https://doi.org/10.1109/SITIS.2013.101Google Scholar
Danziger, M. M., et al. (2016). The effect of spatiality on multiplex networks. EPL (Europhysics Letters), 115(3), 36002.CrossRefGoogle Scholar
Danziger, M. M., et al. (2020). Faster calculation of the percolation correlation length on spatial networks. Physical Review E, 101(1), 013306.CrossRefGoogle ScholarPubMed
Daqing, L., et al. (2011). Dimension of spatially embedded networks. Nature Physics, 7(6), 481–484.CrossRefGoogle Scholar
Den Nijs, M. (1979). A relation between the temperature exponents of the eight-vertex and q-state Potts model. Journal of Physics A: Mathematical and General, 12(10), 1857.Google Scholar
Dong, G., et al. (2018). Resilience of networks with community structure behaves as if under an external field. Proceedings of the National Academy of Sciences, 115(27), 6911–6915.Google Scholar
Donges, J. F., et al. (2009). Complex networks in climate dynamics. The European Physical Journal Special Topics, 174(1), 157–179.Google Scholar
Dosenbach, N. U., et al. (2007). Distinct brain networks for adaptive and stable task control in humans. Proceedings of the National Academy of Sciences, 104(26), 11073–11078.Google Scholar
Emmerich, T., et al. (2013). Complex networks embedded in space: dimension and scaling relations between mass, topological distance, and Euclidean distance. Physical Review E, 87(3), 032802.Google Scholar
Ercsey-Ravasz, M., et al. (2013). A predictive network model of cerebral cortical connectivity based on a distance rule. Neuron, 80(1), 184–197.Google Scholar
Erdős, P., & Rényi, A. (1959). On random graphs I. Publicationes Mathematicae Debrecen, 6, 290.CrossRefGoogle Scholar
Erdős, P., & Rényi, A. (1960). On the evolution of random graphs. Publications of the Mathematical Institute of the Hungarian Academy of Sciences, 5(1), 17–60.Google Scholar
Faloutsos, M., Faloutsos, P., & Faloutsos, C. (1999). On power-law relationships of the internet topology. ACM SIGCOMM Computer Communication Review, 29(4), 251–262.Google Scholar
Fan, J., et al. (2017). Network analysis reveals strongly localized impacts of El Ni no. Proceedings of the National Academy of Sciences, 114(29), 7543–7548.Google Scholar
Fan, J., et al. (2018). Structural resilience of spatial networks with inter-links behaving as an external field. New Journal of Physics, 20(9), 093003.CrossRefGoogle Scholar
Gallos, L. K., et al. (2012). A small world of weak ties provides optimal global integration of self-similar modules in functional brain networks. Proceedings of the National Academy of Sciences, 109(8), 2825–2830.Google Scholar
Gao, J., Li, D., & Havlin, S. (2014). From a single network to a network of networks. National Science Review, 1(3), 346–356.Google Scholar
Gao, J., et al. (2011). Robustness of a network of networks. Physical Review Letters, 107(19), 195701.Google Scholar
Gao, J., et al. (2012). Networks formed from interdependent networks. Nature Physics, 8(1), 40.CrossRefGoogle Scholar
Gastner, M. T., & Newman, M. E. (2006). The spatial structure of networks. The European Physical Journal B-Condensed Matter and Complex Systems, 49(2), 247–252.Google Scholar
Gibson, T. E., et al. (2016). On the origins and control of community types in the human microbiome. PLoS Computational Biology, 12(2), e1004688.Google Scholar
Goldenberg, J., & Levy, M. (2009). Distance is not dead: social interaction and geographical distance in the internet era. arXiv preprint arXiv:0906.3202.Google Scholar
Gross, B., & Havlin, S. (2020). Epidemic spreading and control strategies in spatial modular network. Applied Network Science, 5(1), 1–14.CrossRefGoogle ScholarPubMed
Gross, B., et al. (2017). Multi-universality and localized attacks in spatially embedded networks. In Proceedings of the Asia-Pacific Econophysics Conference 2016 – Big Data Analysis and Modeling toward Super Smart Society – (APEC-SSS2016) JPS Conference Proceedings, 16, 011002. doi: https://doi.org/10.7566/JPSCP.16.01100Google Scholar
Gross, B., et al. (2020a). Interconnections between networks acting like an external field in a first-order percolation transition. Physical Review E, 101(2), 022316.Google Scholar
Gross, B., et al. (2020b). Two transitions in spatial modular networks. New Journal of Physics, 22(5), 053002.Google Scholar
Gross, B., et al. (2021). Interdependent transport via percolation backbones in spatial networks. Physica A: Statistical Mechanics and Its Applications, 567(9), 125644.Google Scholar
Guimera, R., Mossa, S., Turtschi, A., & Amaral, L. N. (2005). The worldwide air transportation network: anomalous centrality, community structure, and cities’ global roles. Proceedings of the National Academy of Sciences, 102(22), 7794–7799.Google Scholar
Hajdu, L., Bóta, A., Krész, M., Khani, A., & Gardner, L. M. (2019). Discovering the hidden community structure of public transportation networks. Networks and Spatial Economics, 20(1), 209–231.Google Scholar
Halu, A., Mukherjee, S., & Bianconi, G. (2014). Emergence of overlap in ensembles of spatial multiplexes and statistical mechanics of spatial interacting network ensembles. Physical Review E, 89(1), 012806.Google Scholar
Hamedmoghadam, H., etal. (2021). Percolation of heterogeneous flows uncovers the bottlenecks of infrastructure networks. Nature Communications, 12(1), 1–10.Google Scholar
Havlin, S., & Nossal, R. (1984). Topological properties of percolation clusters. Journal ofPhysics A: Mathematical and General, 17(8), L427.Google Scholar
Horvát, S., et al. (2016). Spatial embedding and wiring cost constrain the functional layout of the cortical network of rodents and primates. PLoS biology, 14(7), e1002512.Google Scholar
Hu, Y., et al. (2011). Possible origin of efficient navigation in small worlds. Physical Review Letters, 106(10), 108701.Google Scholar
Huang, W., et al. (2014). Navigation in spatial networks: a survey. Physica A: Statistical Mechanics and Its Applications, 393, 132–154.CrossRefGoogle Scholar
Jeong, H., et al. (2000). The large-scale organization of metabolic networks. Nature, 407(6804), 651–654.Google Scholar
Kirkpatrick, S. (1973). Percolation and conduction. Reviews of Modern Physics, 45(4), 574.Google Scholar
Kovács, I. A., et al. (2019). Network-based prediction of protein interactions. Nature Communications, 10(1), 1–8.CrossRefGoogle ScholarPubMed
Lambiotte, R., et al. (2008). Geographical dispersal of mobile communication networks. Physica A: Statistical Mechanics and Its Applications, 387(21), 5317–5325.Google Scholar
Latora, V., & Marchiori, M. (2005). Vulnerability and protection of infrastructure networks. Physical Review E, 71(1), 015103.Google Scholar
Li, D., et al. (2011). Percolation of spatially constraint networks. EPL (Europhysics Letters), 93(6), 68004.Google Scholar
Li, D., et al. (2015). Percolation transition in dynamical traffic network with evolving critical bottlenecks. Proceedings of the National Academy of Sciences, 112(3), 669–672. Retrieved from www.pnas.org/content/112/3/669.abstract doi: https://doi.org/10.1073/pnas.1419185112.Google Scholar
Li, Z., et al. (2017). The OncoPPi network of cancer-focused protein-protein interactions to inform biological insights and therapeutic strategies. Nature Communications, 8(1), 1–14.Google Scholar
Liben-Nowell, D., Novak, J., Kumar, R., Raghavan, P., & Tomkins, A. (2005). Geographic routing in social networks. Proceedings of the National Academy of Sciences, 102(33), 11623–11628.Google Scholar
Liu, Y., et al. (2021). Efficient network immunization under limited knowledge. National Science Review, 8(1), nwaa229.Google Scholar
Ludescher, J., et al. (2014). Very early warning of next El Ni no. Proceedings of the National Academy of Sciences, 111(6), 2064–2066.Google Scholar
Markov, N. T., et al. (2014). A weighted and directed interareal connectivity matrix for macaque cerebral cortex. Cerebral Cortex, 24(1), 17–36.Google Scholar
Menck, P. J., Heitzig, J., Kurths, J., & Schellnhuber, H. J. (2014). How dead ends undermine power grid stability. Nature Communications, 5(1), 1–8.Google Scholar
Milo, R., et al. (2002). Network motifs: simple building blocks of complex networks. Science, 298(5594), 824–827.Google Scholar
Moretti, P., & Mu noz, M. A. (2013). Griffiths phases and the stretching of criticality in brain networks. Nature Communications, 4, 2521.CrossRefGoogle ScholarPubMed
Newman, M. E. (2003). The structure and function of complex networks. SIAM Review, 45(2), 167–256.Google Scholar
Newman, M. E. J., Strogatz, S. H., & Watts, D. J. (2001, July). Random graphs with arbitrary degree distributions and their applications. Phys. Rev. E, 64, 026118.Google Scholar
Nienhuis, B. (1982). Analytical calculation of two leading exponents of the dilute Potts model. Journal of Physics A: Mathematical and General, 15(1), 199.Google Scholar
Paine, R. T. (1966). Food web complexity and species diversity. The American Naturalist, 100(910), 65–75.Google Scholar
Parshani, R., etal. (2010). Interdependent networks: reducing the coupling strength leads to a change from a first to second order percolation transition. Physical Review Letters, 105(4), 048701.Google Scholar
Pastor-Satorras, R., et al. (2015). Epidemic processes in complex networks. Reviews of Modern Physics, 87(3), 925.CrossRefGoogle Scholar
Pocock, M. J., et al. (2012). The robustness and restoration of a network of ecological networks. Science, 555(6071), 973–977.CrossRefGoogle Scholar
Polis, G. A., & Strong, D. R. (1996). Food web complexity and community dynamics. The American Naturalist, 147(5), 813–846.Google Scholar
Reis, S. D., et al. (2014). Avoiding catastrophic failure in correlated networks of networks. Nature Physics, 10(10), 762–767.Google Scholar
Reynolds, P., Stanley, H., & Klein, W. (1977). Ghost fields, pair connectedness, and scaling: exact results in one-dimensional percolation. Journal of Physics A: Mathematical and General, 10(11), L203.Google Scholar
Rozenfeld, A. F., et al. (2002). Scale-free networks on lattices. Physical Review Letters, 89(21), 218701.Google Scholar
Schmeltzer, C., Soriano, J., Sokolov, I. M., & Rudiger, S. (2014). Percolation of spatially constrained Erdős-Rényi networks with degree correlations. Physical Review E, 89(1), 012116.Google Scholar
Shai, S., et al. (2015). Critical tipping point distinguishing two types of transitions in modular network structures. Physical Review E, 92(6), 062805.Google Scholar
Shao, S., et al. (2015). Percolation of localized attack on complex networks. New Journal of Physics, 17(2), 023049.Google Scholar
Shekhtman, L. M., et al. (2014). Robustness of a network formed of spatially embedded networks. Physical Review E, 90(1), 012809.Google Scholar
Smillie, C. S., et al. (2011). Ecology drives a global network of gene exchange connecting the human microbiome. Nature, 480(7376), 241–244.Google Scholar
Sporns, O. (2010). Networks of the brain. Massachusetts Institute of Technology Press.Google Scholar
Stanley, H. (1971). Introduction to phase transitions and critical phenomena. Oxford University Press.Google Scholar
Stauffer, D., & Sornette, D. (1999). Self-organized percolation model for stock market fluctuations. Physica A: Statistical Mechanics and Its Applications, 271(3–4), 496–506.Google Scholar
Stippinger, M., & Kertész, J. (2014). Enhancing resilience of interdependent networks by healing. Physica A: Statistical Mechanics and Its Applications, 416, 481–487.Google Scholar
Stork, D., & Richards, W. D. (1992). Nonrespondents in communication network studies: problems and possibilities. Group & Organization Management, 17(2), 193–209.Google Scholar
Suki, B., Bates, J. H., & Frey, U. (2011). Complexity and emergent phenomena. Comprehensive Physiology, 1(2), 995–1029.Google Scholar
Sykes, M. F., & Essam, J. W. (1964). Exact critical percolation probabilities for site and bond problems in two dimensions. Journal of Mathematical Physics, 5(8), 1117–1127.Google Scholar
Vaknin, D., et al. (2017). Spreading of localized attacks in spatial multiplex networks. New Journal of Physics, 19(7), 073037.Google Scholar
Vaknin, D., et al. (2020). Spreading of localized attacks on spatial multiplex networks with a community structure. Physical Review Research, 2(4), 043005.Google Scholar
Viswanathan, G. M., et al. (1999). Optimizing the success of random searches. Nature, 401(6756), 911–914.Google Scholar
Wang, W., et al. (2017). Unification of theoretical approaches for epidemic spreading on complex networks. Reports on Progress in Physics, 80(3), 036603.Google Scholar
Waxman, B. M. (1988). Routing of multipoint connections. IEEE Journal on Selected Areas in Communications, 6(9), 1617–1622.Google Scholar
Wei, L., et al. (2012). Cascading failures in interdependent lattice networks: the critical role of the length of dependency links. Physical Review Letters, 108(22), 228702.Google Scholar
Wei, L., et al. (2014). Ranking the economic importance of countries and industries. Journal of Network Theory in Finance, 3(3), 1–17.Google Scholar
Yang, Y., Nishikawa, T., & Motter, A. E. (2017). Small vulnerable sets determine large network cascades in power grids. Science, American Association for the Advancement of Science 358(6365), eaan3184.Google Scholar
Yuan, X., Shao, S., Stanley, H. E., & Havlin, S. (2015). How breadth of degree distribution influences network robustness: comparing localized and random attacks. Physical Review E, 92(3), 032122.Google Scholar
Zhou, D., et al. (2014). Simultaneous first- and second-order percolation transitions in interdependent networks. Physical Review E, 90(1), 012803.Google Scholar
- 7
- Cited by