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Convexity in complex networks

Published online by Cambridge University Press:  06 February 2018

TILEN MARC  and
LOVRO ŠUBELJ Open the ORCID record for LOVRO ŠUBELJ [Opens in a new window]
TILEN MARC
Affiliation:
Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia (e-mail: [email protected])
LOVRO ŠUBELJ
Affiliation:
University of Ljubljana, Faculty of Computer and Information Science, Ljubljana, Slovenia (e-mail: [email protected])
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Abstract

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Metric graph properties lie in the heart of the analysis of complex networks, while in this paper we study their convexity through mathematical definition of a convex subgraph. A subgraph is convex if every geodesic path between the nodes of the subgraph lies entirely within the subgraph. According to our perception of convexity, convex network is such in which every connected subset of nodes induces a convex subgraph. We show that convexity is an inherent property of many networks that is not present in a random graph. Most convex are spatial infrastructure networks and social collaboration graphs due to their tree-like or clique-like structure, whereas the food web is the only network studied that is truly non-convex. Core–periphery networks are regionally convex as they can be divided into a non-convex core surrounded by a convex periphery. Random graphs, however, are only locally convex meaning that any connected subgraph of size smaller than the average geodesic distance between the nodes is almost certainly convex. We present different measures of network convexity and discuss its applications in the study of networks.

Keywords

network convexityconvex subsetsconvex subgraphscore–periphery structure
Type
Research Article
Information
Network Science , Volume 6 , Issue 2 , June 2018 , pp. 176 - 203
Copyright
Copyright © Cambridge University Press 2018 

References

Adai, A. T., Date, S. V., Wieland, S., & Marcotte, E. M. (2004). LGL: Creating a map of protein function with an algorithm for visualizing very large biological networks. Journal of Molecular Biology, 340 (1), 179–190.Google Scholar
Adamic, L. A., & Glance, N. (2005). The political blogosphere and the 2004 U.S. election. Proceedings of the KDD Workshop on Link Discovery, pp. 36–43.Google Scholar
Albert, R., Jeong, H., & Barabasi, A. L. (2000). Error and attack tolerance of complex networks. Nature, 406 (6794), 378–382.CrossRefGoogle ScholarPubMed
Bandelt, H.-J., & Chepoi, V. (2008). Metric graph theory and geometry: A survey. Contemporary Mathematics, 453, 49–86.Google Scholar
Barabási, A.-L. (2016). Network science. Cambridge: Cambridge University Press.Google Scholar
Barabási, A.-L., & Albert, R. (1999). Emergence of scaling in random networks. Science, 286 (5439), 509–512.Google Scholar
Barthelemy, M. (2011). Spatial networks. Physics Reports: Review Section of Physics Letters, 499 (1–3), 1–101.Google Scholar
Batagelj, V., & Zaveršnik, M. (2011). An O(m) algorithm for cores decomposition of networks. Advances in Data Analysis and Classification, 5 (2), 129–145.Google Scholar
Batagelj, V. (1988). Similarity measures between structured objects. Proceedings of the International Conference on Interfaces between Mathematics, Chemistry and Computer Science, pp. 25–39.Google Scholar
Batagelj, V. (2016). Corrected overlap weight and clustering coefficient. Proceedings of the INSNA International Social Network Conference, pp. 16–17.Google Scholar
Baxter, G. J., Dorogovtsev, S. N., Lee, K.-E., Mendes, J. F. F., & Goltsev, A. V. (2015). Critical dynamics of the k-core pruning process. Physical Review X, 5 (3), 031017.CrossRefGoogle Scholar
Benson, A. R. G., David, F., & Leskovec, J. (2016). Higher-order organization of complex networks. Science, 353 (6295), 163–166.CrossRefGoogle ScholarPubMed
Borgatti, S. P., & Everett, M. G. (2000). Models of core/periphery structures. Social Networks, 21 (4), 375–395.Google Scholar
Boyd, S., & Vandenberghe, L. (2004). Convex optimization. Cambridge: Cambridge University Press.Google Scholar
Clauset, A., Moore, C., & Newman, M. E. J. (2008). Hierarchical structure and the prediction of missing links in networks. Nature, 453 (7191), 98–101.CrossRefGoogle ScholarPubMed
Clough, J. R., & Evans, T. S. (2016). What is the dimension of citation space? Physica A: Statistical Mechanics and its Applications, 448, 235–247.CrossRefGoogle Scholar
Clough, J. R., & Evans, T. S. (2017). Embedding graphs in Lorentzian spacetime. PLoS One, 12 (11), e0187301.Google Scholar
Cucuringu, M., Rombach, P., Lee, S. H., & Porter, M. A. (2016). Detection of core-periphery structure in networks using spectral methods and geodesic paths. European Journal of Applied Mathematics, 27 (6), 846–887.Google Scholar
Davis, A., Gardner, B. B., & Gardner, M. R. (1941). Deep South. Chicago: Chicago University Press.Google Scholar
Doreian, P., Batagelj, V., & Ferligoj, A. (2005). Generalized blockmodeling. Cambridge: Cambridge University Press.Google Scholar
Dorogovtsev, S. N., Goltsev, A. V., & Mendes, J. F. F. (2008). Critical phenomena in complex networks. Reviews of Modern Physics, 80 (4), 1275–1335.Google Scholar
Dourado, M. C., Gimbel, J. G., Kratochvíl, J., Protti, F., & Szwarcfiter, J. L. (2009). On the computation of the hull number of a graph. Discrete Mathematics, 309 (18), 5668–5674.Google Scholar
Erdős, P., & Rényi, A. (1959). On random graphs I. Publicationes Mathematicae Debrecen, 6, 290–297.Google Scholar
Estrada, E., & Knight, P. A. (2015). A first course in network theory. Oxford: Oxford University Press.Google Scholar
Everett, M. G, & Seidman, S. B. (1985). The hull number of a graph. Discrete Mathematics, 57 (3), 217–223.Google Scholar
Farber, M., & Jamison, R. (1986). Convexity in graphs and hypergraphs. SIAM Journal on Algebraic and Discrete Methods, 7 (3), 433–444.Google Scholar
Freeman, L. (1977). A set of measures of centrality based on betweenness. Sociometry, 40 (1), 35–41.CrossRefGoogle Scholar
Gavril, F. (1974). The intersection graphs of subtrees in trees are exactly the chordal graphs. Journal of Combinatorial Theory Series B, 16 (1), 47–56.CrossRefGoogle Scholar
Girvan, M., & Newman, M. E. J. (2002). Community structure in social and biological networks. Proceedings of the National Academy of Sciences of the United States of America, 99 (12), 7821–7826.Google Scholar
Guimerà, R., Sales-Pardo, M., & Amaral, L. A. N. (2007). Classes of complex networks defined by role-to-role connectivity profiles. Nature Physics, 3 (1), 63–69.Google Scholar
Hallac, D., Leskovec, J., & Boyd, S. (2015). Network lasso: Clustering and optimization in large graphs. Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 387–396.Google Scholar
Harary, F., & Nieminen, J. (1981). Convexity in graphs. Journal of Differential Geometry, 16 (2), 185–190.Google Scholar
Hébert-Dufresne, L., Grochow, J. A., & Allard, A. (2016). Multi-scale structure and topological anomaly detection via a new network statistic: The onion decomposition. Scientific Reports, 6, 31708.Google Scholar
Hočevar, T., & Demšar, J. (2014). A combinatorial approach to graphlet counting. Bioinformatics, 30 (4), 559–565.Google Scholar
Holme, P. (2005). Core-periphery organization of complex networks. Physical Review E, 72 (4), 046111.Google Scholar
Kunegis, J. (2013). KONECT: The Koblenz network collection. Proceedings of the International World Wide Web Conference, pp. 1343–1350.Google Scholar
Leskovec, J., Kleinberg, J., & Faloutsos, C. (2007). Graph evolution: Densification and shrinking diameters. ACM Transactions on Knowledge Discovery From Data, 1 (1), 1–41.Google Scholar
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.Google Scholar
Luce, R. D. (1950). Connectivity and generalized cliques in sociometric group structure. Psychometrika, 15 (2), 169–190.Google Scholar
Maslov, S., & Sneppen, K. (2002). Specificity and stability in topology of protein networks. Science, 296 (5569), 910–913.Google Scholar
Meir, A., & Moon, J. W. (1970). The distance between points in random trees. Journal of Combinatorial Theory, 8 (1), 99–103.Google Scholar
Milgram, S. (1967). The small world problem. Psychology Today, 1 (1), 60–67.Google Scholar
Milo, R., Shen-Orr, S., Itzkovitz, S., Kashtan, N., Chklovskii, D., & Alon, U. (2002). Network motifs: Simple building blocks of complex networks. Science, 298 (5594), 824–827.Google Scholar
Milo, R., Itzkovitz, S., Kashtan, N., Levitt, R., Shen-Orr, S., Ayzenshtat, I., Sheffer, M., & Alon, U. (2004). Superfamilies of evolved and designed networks. Science, 303 (5663), 1538–1542.Google Scholar
Newman, M. E. J. (2006). Finding community structure in networks using the eigenvectors of matrices. Physical Review E, 74 (3), 036104.Google Scholar
Newman, M. E. J., & Girvan, M. (2004). Finding and evaluating community structure in networks. Physical Review E, 69 (2), 026113.Google Scholar
Newman, M. E. J, & Leicht, E. A. (2007). Mixture models and exploratory analysis in networks. Proceedings of the National Academy of Sciences of the United States of America, 104 (23), 9564–9569.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 (2), 026118.Google Scholar
Newman, M. E. J. (2010). Networks: An introduction. Oxford: Oxford University Press.Google Scholar
Palla, G., Derényi, I., Farkas, I., & Vicsek, T. (2005). Uncovering the overlapping community structure of complex networks in nature and society. Nature, 435 (7043), 814–818.Google Scholar
Pelayo, I. M. (2013). Geodesic convexity in graphs. New York, NY: Springer.Google Scholar
Pržulj, N., Corneil, D. G., & Jurisica, I. (2004). Modeling interactome: Scale-free or geometric? Bioinformatics, 20 (18), 3508–3515.Google Scholar
Pržulj, N. (2007). Biological network comparison using graphlet degree distribution. Bioinformatics, 23 (2), e177–e183.Google Scholar
Reichardt, J., & White, D. R. (2007). Role models for complex networks. European Physical Journal B, 60 (2), 217–224.Google Scholar
Rényi, A., & Szekeres, G. (1967). On the height of trees. Journal of the Australian Mathematical Society, 7 (4), 497–507.Google Scholar
Rombach, M., Porter, M., Fowler, J., & Mucha, P. (2014). Core-periphery structure in networks. SIAM Journal on Applied Mathematics, 74 (1), 167–190.Google Scholar
Rosvall, M., & Bergstrom, C. T. (2008). Maps of random walks on complex networks reveal community structure. Proceedings of the National Academy of Sciences of the United States of America, 105 (4), 1118–1123.Google Scholar
Seidman, S. B. (1983). Network structure and minimum degree. Social Networks, 5 (3), 269–287.Google Scholar
Stark, C., Breitkreutz, B.-J., Reguly, T., Boucher, L., Breitkreutz, A., & Tyers, M. (2006). BioGRID: A general repository for interaction datasets. Nucleic Acids Research, 34 (1), 535–539.Google Scholar
Šubelj, L., & Bajec, M. (2011). Robust network community detection using balanced propagation. European Physical Journal B, 81 (3), 353–362.Google Scholar
Šubelj, L., Van Eck, N. J., & Waltman, L. (2016). Clustering scientific publications based on citation relations: A systematic comparison of different methods. PLoS One, 11 (4), e0154404.Google Scholar
Van de Vel, M. L. J. (1993). Theory of convex structures. Amsterdam: North-Holland.Google Scholar
Wasserman, S., & Faust, K. (1994). Social network analysis. Cambridge: Cambridge University Press.Google Scholar
Watts, D. J., & Strogatz, S. H. (1998). Collective dynamics of ‘small-world’ networks. Nature, 393 (6684), 440–442.Google Scholar
Williams, R. J., & Martinez, N. D. (2000). Simple rules yield complex food webs. Nature, 404 (6774), 180–183.CrossRefGoogle ScholarPubMed
Xu, J., Wickramarathne, T. L., & Chawla, N. V. (2016). Representing higher-order dependencies in networks. Science Advances, 2 (5), e1600028.Google Scholar
Yang, J., & Leskovec, J. (2012). Community-affiliation graph model for overlapping network community detection. Proceedings of the IEEE International Conference on Data Mining, pp. 1170–1175.CrossRefGoogle Scholar
Yaveroğlu, Ö. N., Malod-Dognin, N., Davis, D., Levnajić, Z., Janjic, V., Karapandza, R., Stojmirovic, A., & Pržulj, N. (2014). Revealing the hidden language of complex networks. Scientific Reports, 4, 4547.Google Scholar
Yuan, X., Dai, Y., Stanley, H. E., & Havlin, S. (2016). k-core percolation on complex networks: Comparing random, localized, and targeted attacks. Physical Review E, 93 (6), 062302.Google Scholar
Zhang, X., Martin, T., & Newman, M. E. J. (2015). Identification of core-periphery structure in networks. Physical Review E, 91 (3), 032803.Google Scholar