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  3. The Shortest Path to Network Geometry
  • Cited by 14
  • M. Ángeles Serrano, University of Barcelona, University of Barcelona Institute of Complex Systems (UBICS) and Catalan Institution for Research and Advanced Studies (ICREA), Marián Boguñá, University of Barcelona and University of Barcelona Institute of Complex Systems (UBICS)
Publisher:
Cambridge University Press
Online publication date:
December 2021
Print publication year:
2022
Online ISBN:
9781108865791
DOI:
https://doi.org/10.1017/9781108865791
Subjects:
Algorithmics, Complexity, Computer Algebra, Computational Geometry, Physics and Astronomy, Computer Science, Statistical Physics
Series:
Elements in the Structure and Dynamics of Complex Networks
Information

Book description

Real networks comprise from hundreds to millions of interacting elements and permeate all contexts, from technology to biology to society. All of them display non-trivial connectivity patterns, including the small-world phenomenon, making nodes to be separated by a small number of intermediate links. As a consequence, networks present an apparent lack of metric structure and are difficult to map. Yet, many networks have a hidden geometry that enables meaningful maps in the two-dimensional hyperbolic plane. The discovery of such hidden geometry and the understanding of its role have become fundamental questions in network science giving rise to the field of network geometry. This Element reviews fundamental models and methods for the geometric description of real networks with a focus on applications of real network maps, including decentralized routing protocols, geometric community detection, and the self-similar multiscale unfolding of networks by geometric renormalization.

References

References

Abou-Rjeili, A. , & Karypis, G. (2006). Multilevel algorithms for partitioning power-law graphs. In Proceedings 20th IEEE International Parallel & Distributed Processing Symposium. doi: 10.1109/IPDPS.2006.1639360.
Alanis-Lobato, G., Mier, P., & Andrade-Navarro, M. A. (2016a). Efficient embedding of complex networks to hyperbolic space via their Laplacian. Sci. Rep., 6, 30108.
Alanis-Lobato, G., Mier, P., & Andrade-Navarro, M. A. (2016b, Nov. 15). Manifold learning and maximum likelihood estimation for hyperbolic network embedding. Applied Network Science, 1(1), 10. doi: https://doi.org/10.1007/s41109-016-0013-0
Allard, A., & Serrano, M. Á. (2020). Navigable maps of structural brain networks across species. PLOS Computational Biology, 16(2), e1007584.
Allard, A., Serrano, M. Á., García-Pérez, G., & Boguñá, M. (2017). The geometric nature of weights in real complex networks. Nat. Commun., 8, 14103.
Alvarez-Hamelin, J. I., Dall’Asta, L., Barrat, A., & Vespignani, A. (2008). K-core decomposition of internet graphs: hierarchies, selfsimilarity and measurement biases. Networks and Heterogeneous Media, 3(2), 371–393.
Amaral, L. A. N. (2008). A truer measure of our ignorance. Proc. Natl. Acad. Sci. USA, 105(19), 6795–6796.
Amaral, L. A. N., Scala, A., Barthélemy, M., & Stanley, H. E. (2000). Classes of small-world networks. Proc. Natl. Acad. Sci. USA, 97(21), 11149–11152.
Barabási, A. L., & Albert, R. (1999). Emergence of scaling in random networks. Science, 286(5439), 509–512.
Barrat, A., Barthélemy, M., Pastor-Satorras, R., & Vespignani, A. (2004). The architecture of complex weighted networks. Proc. Natl. Acad. Sci. USA, 101(11), 3747–3752.
Barrat, A., Barthélemy, M., & Vespignani, A. (2008). Dynamical processes on complex networks. Cambridge: Cambridge University Press.
Barthélemy, M. (2011). Spatial networks. Phys. Rep., 499(1–3), 1–101.
Bianconi, G. (2018). Multilayer networks: structure and function. Oxford: Oxford University Press.
Blasius, T., Friedrich, T., Krohmer, A. et al. (2018, Apr.). Efficient embedding of scale-free graphs in the hyperbolic plane. IEEE/ACM Trans. Netw., 26(2), 920–933. doi: https://doi.org/10.1109/TNET.2018.2810186
Blondel, V. D., Guillaume, J.-L., Lambiotte, R., & Lefebvre, É. (2008). Fast unfolding of communities in large networks. J. Stat. Mech., 2008(10), P10008.
Boettcher, S., & Brunson, C. (2011). Renormalization group for critical phenomena in complex networks. Frontiers in Physiology, 2, 102. doi: https://doi.org/10.3389/fphys.2011.00102
Boguñá, M., Bonamassa, I., Domenico, M. D. et al. (2020). Network geometry. Nat Rev Phys 3, 114–135. https://doi.org/10.1038/s42254-020-00264-4
Boguñá, M., & Krioukov, D. (2009). Navigating ultrasmall worlds in ultrashort time. Phys. Rev. Lett., 102(058701). (arXiv:0809.2995v1)
Boguñá, M., Krioukov, D., Almagro, P., & Serrano, M. Á. (2020, Apr.). Small worlds and clustering in spatial networks. Phys. Rev. Research, 2, 023040. doi: https://doi.org/10.1103/PhysRevResearch.2.023040
Boguñá, M., Krioukov, D. , & Claffy, K. (2009). Navigability of complex networks. Nat. Phys., 5(1), 74–80.
Boguñá, M., Papadopoulos, F., & Krioukov, D. (2010). Sustaining the Internet with hyperbolic mapping. Nat. Commun., 1, 62. doi: https://doi.org/10.1038/ncomms1063
Boguñá, M., Pastor-Satorras, R., & Vespignani, A. (2004). Cut-offs and finite size effects in scale-free networks. Eur. Phys. J. B, 38(2), 205–209.
Bringmann, K., Keusch, R., Lengler, J., Maus, Y., & Molla, A. R. (2017). Greedy routing and the algorithmic small-world phenomenon. In PODC ’17: Proceedings of the ACM Symposium on Principles of Distributed Computing. doi: https://doi.org/10.1145/3087801.3087829
Caldarelli, G., Capocci, A., De Los Rios, P., & Muñoz, M. A. (2002, December). Scale-free networks from varying vertex intrinsic fitness. Phys. Rev. Lett., 89(25), 258702. doi: https://doi.org/10.1103/PhysRevLett.89.258702
Claffy, K., Hyun, Y., Keys, K., Fomenkov, M., & Krioukov, D. (2009). Internet mapping: From art to science. In 2009 Cybersecurity Applications Technology Conference for Homeland Security (pp. 205–211). New York: IEEE. doi: https://doi.org/10.1109/CATCH.2009.38
Cohen, R., & Havlin, S. (2003). Scale-free networks are ultrasmall. Phys. Rev. Lett., 90(5), 058701.
Colizza, V., Pastor-Satorras, R. , & Vespignani, A. (2007). Reaction-diffusion processes and metapopulation models in heterogeneous networks. Nat. Phys., 3, 276–282.
Dall, J., & Christensen, M. (2002). Random geometric graphs. Phys. Rev. E, 66(1), 016121. doi: https://doi.org/10.1103/PhysRevE.66.016121
Dorogovtsev, S. N., & Mendes, J. F. F. (2003). Evolution of networks: From biological nets to the Internet and WWW. Oxford: Oxford University Press.
Dorogovtsev, S. N., Mendes, J. F. F., & Samukhin, A. N. (2001). A. size-dependent degree distribution of a scale-free growing network. Phys. Rev. E, 63(6), 062101.
D’Souza, R., Borgs, C., Chayes, J., Berger, N., & Kleinberg, R. (2007). Emergence of tempered preferential attachment from optimization. PNAS, 104(15), 6112–6117.
Fortunato, S., Flammini, A., & Menczer, F. (2006). Scale-free network growth by ranking. Phys. Rev. Lett., 96(21), 218701.
García-Pérez, G., Allard, A., Serrano, M. Á., & Boguñá, M. (2019, Dec.). Mercator: uncovering faithful hyperbolic embeddings of complex networks. New Journal of Physics, 21(12), 123033. doi: https://doi.org/10.1088%2F1367-2630%2Fab57d2
García-Pérez, G., Boguñá, M., Allard, A., & Serrano, M. Á. (2016, Sept.). The hidden hyperbolic geometry of international trade: World Trade Atlas 1870–2013. Sci. Rep., 6, 33441. doi: https://doi.org/10.1038/srep33441
García-Pérez, G., Boguñá, M., & Serrano, M. Á. (2018). Multiscale unfolding of real networks by geometric renormalization. Nat. Phys., 14(6), 583–589. doi: https://doi.org/10.1038/s41567-018-0072-5
García-Pérez, G., Serrano, M. Á., & Boguñá, M. (2018). Soft communities in similarity space. J. Stat. Phys., 173(3–4), 775–782. doi: https://doi.org/10.1007/s10955-018-2084-z
Goh, K. I., Salvi, G., Kahng, B., & Kim, D. (2006). Skeleton and fractal scaling in complex networks. Phys. Rev. Lett., 96, 018701.
Guimerà, R., Mossa, S., Turtschi, A., & Amaral, L. A. N. (2005). The worldwide air transportation network: Anomalous centrality, community structure, and cities. Proc. Natl. Acad. Sci. USAs, 10(22), 7794–7799.
Gulyás, A., Bíró, J. J., Kőrösi, A., Rétvári, G., & Krioukov, D. (2015). Navigable networks as Nash equilibria of navigation games. Nat. Commun., 6(1), 7651. doi: https://doi.org/10.1038/ncomms8651
Kadanoff, Leo P. (2000). Statistical physics: Statics, dynamics and renormalization. Singapore: World Scientific.
Karypis, G., & Kumar, V. (1999). A fast and highly quality multilevel scheme for partitioning irregular graphs. SIAM Journal on Scientific Computing, 20(1), 359–392.
Kim, J. S., Goh, K. I., Hahng, B., & Kim, D. (2007). Fractality and self-similarity in scale-free networks. New J. Phys., 9, 177.
Kleineberg, K.-K., Boguñá, M., Serrano, M. Á., & Papadopoulos, F. (2016). Hidden geometric correlations in real multiplex networks. Nat. Phys., 12(11), 1076–1081.
Kleineberg, K.-K., Buzna, L., Papadopoulos, F., Boguñá, M., & Serrano, M. Á. (2017). Geometric correlations mitigate the extreme vulnerability of multiplex networks against targeted attacks. Phys. Rev. Lett., 118, 218301.
Klimt, B., & Yang, Y. (2004). Introducing the Enron Corpus. In CEAS 2004 – First Conference on Email and Anti-Spam, July 30–31, 2004, Mountain View, CA. Accessed at http://dblp.uni-trier.de/db/conf/ceas/ceas2004.html#KlimtY04
Korman, A., & Peleg, D. (2006). Dynamic routing schemes for general graphs. In Bugliesi, M., Preneel, B., Sassone, V., & Wegener, I. (eds.), ICALP: Proceedings of Int. Colloquium on Automata, Languages and Programming (vol. 4051, pp. 619–630). Springer. doi: https://doi.org/10.1007/11786986_54
Krioukov, D., Kitsak, M., Sinkovits, et al. (2012). Network cosmology. Sci. Rep., 2. Accessed at doi: https://doi.org/10.1038/srep00793
Krioukov, D., Papadopoulos, F., Kitsak, M., Vahdat, A., & Boguñá, M. (2010). Hyperbolic geometry of complex networks. Phys. Rev. E, 82(3), 036106.
Krioukov, D., Papadopoulos, F., Vahdat, A., & Boguñá, M. (2009, Sep.). Curvature and temperature of complex networks. Phys. Rev. E, 80(3), 035101. Accessed at http://link.aps.org/doi/10.1103/PhysRevE.80.035101 doi: https://doi.org/10.1103/PhysRevE.80.035101
Kunegis, J. (2013). KONECT – The Koblenz Network Collection. In Proceedings of the International Conference on World Wide Web Companion (pp. 1343–1350). Accessed at http://konect.cc/
Leskovec, J., Lang, K. J., Dasgupta, A., & Mahoney, M. W. (2009). Community structure in large networks: Natural cluster sizes and the absence of large well-defined clusters. Internet Mathematics, 6(1), 29–123. doi: https:http://doi.org/10.1080/15427951.2009.10129177
Maldacena, J. (1998). The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys., 2(2), 231–252. doi: https://doi.org/10.4310/ATMP.1998.v2.n2.a1
Mandelbrot, B. (1961). On the theory of word frequencies and on related Markovian models of discourse. In Proceedings of the Twelve Symposia in Applied Mathematics, Roman Jakobson Editor. Structure of Language and Its Mathematical Aspects, New York, USA (pp. 190–219). Providence, RI: American Mathematical Society.
Milo, R., Itzkovitz, S., Kashtan, N. et al. (2004, March). Superfamilies of evolved and designed networks. Science, 303(5663), 1538–1542.
Min, B., Yi, S. D., Lee, K.-M., & Goh, K.-I. (2014). Network robustness of multiplex networks with interlayer degree correlations. Phys. Rev. E, 89(4), 042811.
Muscoloni, A., & Cannistraci, C. V. (2018). A nonuniform popularity-similarity optimization (nPSO) model to efficiently generate realistic complex networks with communities. New J. Phys., 20(5), 052002. doi: https://doi.org/10.1088/1367-2630/aac06f
Muscoloni, A., & Cannistraci, C. V. (2019). Navigability evaluation of complex networks by greedy routing efficiency. Proc. Natl. Acad. Sci., 116(5), 1468–1469. doi: https://doi.org/10.1073/pnas.1817880116
Muscoloni, A., Thomas, J. M., Ciucci, S., Bianconi, G., & Cannistraci, C. V. (2017). Machine learning meets complex networks via coalescent embedding in the hyperbolic space. Nature Communications, 8(1), 1615. https://doi.org/10.1038/s41467-017-01825-5
Newman, M., & Watts, D. (1999). Renormalization group analysis of the small-world network model. Physics Letters A, 263(4–6), 341–346. Accessed at www.sciencedirect.com/science/article/pii/S0375960199007574 doi: https://doi.org/10.1016/S0375-9601(99)00757-4
Newman, M. E. J. (2010). Networks: An introduction. Oxford: Oxford University Press.
Newman, M. E. J., & Girvan, M. (2004, Feb.). Finding and evaluating community structure in networks. Phys. Rev. E, 69(2), 026113. doi: https://doi.org/10.1103/PhysRevE.69.026113
Nicosia, V., & Latora, V. (2015). Measuring and modeling correlations in multiplex networks. Phys. Rev. E, 92, 032805.
Openflights network dataset – KONECT. (2016, Sept.). Accessed at http://konect.uni-koblenz.de/networks/openflights
Ortiz, E., Starnini, M., & Serrano, M. Á. (2017). Navigability of temporal networks in hyperbolic space. Sci. Rep., 7, 15054.
Papadopoulos, F., Aldecoa, R., & Krioukov, D. (2015, Aug.). Network geometry inference using common neighbors. Phys. Rev. E, 92(2). 022807. doi: https://doi.org/10.1103/PhysRevE.92.022807
Papadopoulos, F., Kitsak, M., Serrano, M. Á., Boguñá, M., & Krioukov, D. (2012). Popularity versus similarity in growing networks. Nature, 489(7417), 537–540. doi: https://doi.org/10.1038/nature11459
Papadopoulos, F., Krioukov, D., Boguñá, M., & Vahdat, A. (2010). Greedy forwarding in dynamic scale-free networks embedded in hyperbolic metric spaces. In 2010 Proceedings IEEE Infocom (pp. 1–9).
Papadopoulos, F., Psomas, C., & Krioukov, D. (2015). Network mapping by replaying hyperbolic growth. IEEE/ACM Trans. Netw., 23(1), 198–211. doi: https://doi.org/10.1109/TNET.2013.2294052
Papadopoulos, F., & Psounis, K. (2007, Oct.). Efficient identification of uncongested internet links for topology downscaling. SIG-COMM Comput. Commun. Rev., 37(5), 39–52. doi: https://doi.org/10.1145/1290168.1290173
Papadopoulos, F., Psounis, K., & Govindan, R. (2006, Dec.). Performance preserving topological downscaling of internet-like networks. IEEE Journal on Selected Areas in Communications, 24(12), 2313–2326. doi: https://doi.org/10.1109/JSAC.2006.884029
Pastor-Satorras, R., Smith, E., & Sole, R. V. (2003). Evolving protein interaction networks through gene duplication. J. Theor. Biol., 222(2), 199–210.
Radicchi, F., Ramasco, J. J., Barrat, A., & Fortunato, S. (2008, Oct.). Complex networks renormalization: Flows and fixed points. Phys. Rev. Lett., 101(14), 148701. doi: https://doi.org/10.1103/PhysRevLett.101.148701
Rolland, T., Taşan, M., Charloteaux, B. et al. (2014). A proteome-scale map of the human interactome network. Cell, 159(5), 1212–1226. Accessed at www.sciencedirect.com/science/article/pii/S0092867414014226 doi: https://doi.org/10.1016/j.cell.2014.10.050
Rozenfeld, H. D., Song, C., & Makse, H. A. (2010, Jan.). Small-world to fractal transition in complex networks: A renormalization group approach. Phys. Rev. Lett., 104(2), 025701. doi: https://doi.org/10.1103/PhysRevLett.104.025701.
Sarveniazi, A. (2014). An actual survey of dimensionality reduction. American Journal of Computational Mathematics, 4(2), 55–72.
Serrà, J., Corral, A., Boguñá, M., Haro, M., & Arcos, J. L. (2012). Measuring the evolution of contemporary western popular music. Sci. Rep., 2. Accessed at www.nature.com/srep/2012/120726/srep00521/full/srep00521.html doi: https://doi.org/10.1038/srep00521
Serrano, M. Á., Boguñá, M., & Sagues, F. (2012). Uncovering the hidden geometry behind metabolic networks. Mol. BioSyst., 8(3), 843–850. doi: https://doi.org/10.1039/C2MB05306C
Serrano, M. Á., Buzna, L., & Boguñá, M. (2015). Escaping the avalanche collapse in self-similar multiplexes. New J. Phys., 17, 053033.
Serrano, M. Á., Krioukov, D., & Boguñá, M. (2008). Self-similarity of complex networks and hidden metric spaces. Phys. Rev. Lett., 100(7), 078701.
Serrano, M. Á., Krioukov, D., & Boguñá, M. (2011, Jan.). Percolation in self-similar networks. Phys. Rev. Lett., 106(4), 048701. doi: https://doi.org/10.1103/PhysRevLett.106.048701
Song, C., Havlin, S., & Makse, H. A. (2005). Self-similarity of complex networks. Nature, 433(7024), 392–395.
Song, C., Havlin, S., & Makse, H. A. (2006). Origins of fractality in the growth of complex networks. Nat. Phys., 2(4), 275–281.
Stanley, H. E. (1971). Introduction to phase transitions and critical phenomena. Oxford: Oxford University Press.
Starnini, M., Ortiz, E., & Serrano, M. Á. (2019). Geometric randomization of real networks with prescribed degree sequence. New J. Phys., 21(5s), 053039.
Takemura, S.-y., Bharioke, A., Lu, Z. et al. (2013). A visual motion detection circuit suggested by drosophila connectomics. Nature, 500(7461), 175–181.
Watts, D. J., & Strogatz, S. H. (1998). Collective dynamics of “small-world” networks. Nature, 393(6684), 440–442.
Wilson, K. G. (1975). The renormalization group: Critical phenomena and the kondo problem. Rev. Mod. Phys., 47(4), 773–840.
Wilson, K. G. (1983, Jul.). The renormalization group and critical phenomena. Rev. Mod. Phys., 55(3), 583–600. doi: https://doi.org/10.1103/RevModPhys.55.583
Yao, W. M., & Fahmy, S. (2008, Apr.). Downscaling network scenarios with Denial of Service (DoS) attacks. In 2008 IEEE Sarnoff Symposium (pp. 1–6). doi: https://doi.org/10.1109/SARNOF.2008.4520099
Yao, W. M., & Fahmy, S. (2011, June). Partitioning network testbed experiments. In 2011 31st International Conference on Distributed Computing Systems (pp. 299–309). doi: https://doi.org/10.1109/ICDCS.2011.22
Zheng, M., Allard, A., Hagmann, P., Alemán-Gómez, Y., & Serrano, M. Á. (2020). Geometric renormalization unravels self-similarity of the multiscale human connectome. Proc. Natl. Acad. Sci., 117(33), 20244–20253. doi: https://doi.org/10.1073/pnas.1922248117
Zheng, M., García-Pérez, G., Boguñá, & M., Serrano, M. Á. (2021). Scaling-up real networks by geometric branching growth. Proc. Natl. Acad. Sci., 118(21), e2018994118. doi: https://doi.org/10.1073/pnas.2018994118
Zuev, K., Boguñá, M., Bianconi, G., & Krioukov, D. (2015). Emergence of soft communities from geometric preferential attachment. Sci. Rep., 5, 9421. doi: https://doi.org/10.1038/srep09421

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