Convexity in complex networks

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.

[1]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[2]  F. Gavril.  The intersection graphs of subtrees in tree are exactly the chordal graphs , 1974 .

[3]  Igor Jurisica,et al.  Modeling interactome: scale-free or geometric? , 2004, Bioinform..

[4]  M. Newman,et al.  Random graphs with arbitrary degree distributions and their applications. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  Tom A. B. Snijders,et al.  Social Network Analysis , 2011, International Encyclopedia of Statistical Science.

[6]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[7]  Xiao Zhang,et al.  Identification of core-periphery structure in networks , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  R. Tsien,et al.  Specificity and Stability in Topology of Protein Networks , 2022 .

[9]  John Scott.  Social Network Analysis , 1988 .

[10]  Lada A. Adamic,et al.  The political blogosphere and the 2004 U.S. election: divided they blog , 2005, LinkKDD '05.

[11]  Vladimir Batagelj,et al.  Corrected overlap weight and clustering coefficient , 2019, Lecture Notes in Social Networks.

[12]  Martin G. Everett,et al.  Models of core/periphery structures , 2000, Soc. Networks.

[13]  Stephen B. Seidman,et al.  Network structure and minimum degree , 1983 .

[14]  T. Vicsek,et al.  Uncovering the overlapping community structure of complex networks in nature and society , 2005, Nature.

[15]  Mike Tyers,et al.  BioGRID: a general repository for interaction datasets , 2005, Nucleic Acids Res..

[16]  K. Sneppen,et al.  Specificity and Stability in Topology of Protein Networks , 2002, Science.

[17]  Neo D. Martinez,et al.  Simple rules yield complex food webs , 2000, Nature.

[18]  Vladimir Batagelj,et al.  Generalized blockmodeling , 2005, Structural analysis in the social sciences.

[19]  M. Newman,et al.  Hierarchical structure and the prediction of missing links in networks , 2008, Nature.

[20]  P. Holme.  Core-periphery organization of complex networks. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  Nitesh V. Chawla,et al.  Representing higher-order dependencies in networks , 2015, Science Advances.

[22]  Mark Newman,et al.  Networks: An Introduction , 2010 .

[23]  Jure Leskovec,et al.  Community-Affiliation Graph Model for Overlapping Network Community Detection , 2012, 2012 IEEE 12th International Conference on Data Mining.

[24]  John Scott.  What is social network analysis , 2010 .

[25]  Martin Rosvall,et al.  Maps of random walks on complex networks reveal community structure , 2007, Proceedings of the National Academy of Sciences.

[26]  Ignacio M. Pelayo,et al.  Geodesic Convexity in Graphs , 2013 .

[27]  M E J Newman,et al.  Community structure in social and biological networks , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[28]  Stephen P. Boyd,et al.  Network Lasso: Clustering and Optimization in Large Graphs , 2015, KDD.

[29]  김삼묘,et al.  “Bioinformatics” 특집을 내면서 , 2000 .

[30]  L. Christophorou.  Science , 2018, Emerging Dynamics: Science, Energy, Society and Values.

[31]  Natasa Przulj,et al.  Biological network comparison using graphlet degree distribution , 2007, Bioinform..

[33]  Jayme Luiz Szwarcfiter,et al.  On the computation of the hull number of a graph , 2009, Discret. Math..

[34]  S. Shen-Orr,et al.  Networks Network Motifs : Simple Building Blocks of Complex , 2002 .

[35]  J. F. F. Mendes,et al.  Critical Dynamics of thek-Core Pruning Process , 2015, 1505.05484.

[36]  E A Leicht,et al.  Mixture models and exploratory analysis in networks , 2006, Proceedings of the National Academy of Sciences.

[37]  A. Rényi,et al.  On the height of trees , 1967, Journal of the Australian Mathematical Society.

[38]  Marc Barthelemy,et al.  Spatial Networks , 2010, Encyclopedia of Social Network Analysis and Mining.

[39]  Albert-Lszl Barabsi,et al.  Network Science , 2016, Encyclopedia of Big Data.

[40]  Tim S. Evans,et al.  What is the dimension of citation space? , 2014, ArXiv.

[41]  Sergey N. Dorogovtsev,et al.  Critical phenomena in complex networks , 2007, ArXiv.

[42]  Frank Harary,et al.  Convexity in graphs , 1981 .

[43]  Sergey N. Dorogovtsev,et al.  k-core architecture and k-core percolation on complex networks , 2006 .

[44]  Janez Demsar,et al.  A combinatorial approach to graphlet counting , 2014, Bioinform..

[45]  Albert-László Barabási,et al.  Error and attack tolerance of complex networks , 2000, Nature.

[46]  Ludo Waltman,et al.  Clustering Scientific Publications Based on Citation Relations: A Systematic Comparison of Different Methods , 2015, PloS one.

[47]  Sang Hoon Lee,et al.  Detection of core–periphery structure in networks using spectral methods and geodesic paths , 2014, European Journal of Applied Mathematics.

[48]  Ernesto Estrada,et al.  A First Course in Network Theory , 2015 .

[49]  Vladimir Batagelj,et al.  Fast algorithms for determining (generalized) core groups in social networks , 2011, Adv. Data Anal. Classif..

[50]  Jérôme Kunegis,et al.  KONECT: the Koblenz network collection , 2013, WWW.

[51]  R. Luce,et al.  Connectivity and generalized cliques in sociometric group structure , 1950, Psychometrika.

[52]  Vladimir Batagelj,et al.  Similarity measures between structured objects , 1989 .

[53]  Sharon L. Milgram,et al.  The Small World Problem , 1967 .

[54]  公庄 庸三.  Discrete math = 離散数学 , 2004 .

[55]  Van de M. L. J. Vel.  Theory of convex structures , 1993 .

[56]  Antoine Allard,et al.  Multi-scale structure and topological anomaly detection via a new network statistic: The onion decomposition , 2015, Scientific Reports.

[57]  Christos Faloutsos,et al.  Graph evolution: Densification and shrinking diameters , 2006, TKDD.

[58]  Martin G. Everett,et al.  The hull number of a graph , 1985, Discret. Math..

[59]  Edward M Marcotte,et al.  LGL: creating a map of protein function with an algorithm for visualizing very large biological networks. , 2004, Journal of molecular biology.

[60]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[61]  Jure Leskovec,et al.  Community Structure in Large Networks: Natural Cluster Sizes and the Absence of Large Well-Defined Clusters , 2008, Internet Math..

[62]  S. Shen-Orr,et al.  Network motifs: simple building blocks of complex networks. , 2002, Science.

[63]  H. Bandelt,et al.  Metric graph theory and geometry: a survey , 2006 .

[64]  M. Farber,et al.  Convexity in graphs and hypergraphs , 1986 .

[65]  R. Guimerà,et al.  Classes of complex networks defined by role-to-role connectivity profiles. , 2007, Nature physics.

[66]  Santo Fortunato,et al.  Community detection in networks: A user guide , 2016, ArXiv.

[67]  Zoran Levnajic,et al.  Revealing the Hidden Language of Complex Networks , 2014, Scientific Reports.

[68]  M E J Newman,et al.  Finding and evaluating community structure in networks. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[69]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[70]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[71]  B. M. Fulk.  MATH , 1992 .

[72]  R. Rosenfeld.  Nature , 2009, Otolaryngology--head and neck surgery : official journal of American Academy of Otolaryngology-Head and Neck Surgery.

[73]  Harry Eugene Stanley,et al.  $k$-core percolation on complex networks: Comparing random, localized and targeted attacks , 2016, Physical review. E.

[74]  Leonard M. Freeman,et al.  A set of measures of centrality based upon betweenness , 1977 .

[75]  Mason A. Porter,et al.  Core-Periphery Structure in Networks , 2012, SIAM J. Appl. Math..

[76]  Vladimir Batagelj,et al.  An O(m) Algorithm for Cores Decomposition of Networks , 2003, ArXiv.

[77]  Tim S. Evans,et al.  Embedding graphs in Lorentzian spacetime , 2016, PloS one.

[78]  Jure Leskovec,et al.  Higher-order organization of complex networks , 2016, Science.

[79]  Stephen Chadwick,et al.  The Deep South , 2012 .

[80]  S. Shen-Orr,et al.  Superfamilies of Evolved and Designed Networks , 2004, Science.

[81]  M. Newman,et al.  Finding community structure in networks using the eigenvectors of matrices. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[82]  Douglas R. White,et al.  Role models for complex networks , 2007, 0708.0958.

[83]  Qiang Yang,et al.  Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining: Foreword , 2008, Knowledge Discovery and Data Mining.

[84]  A. Meir,et al.  The distance between points in random trees , 1970 .

[85]  Marko Bajec,et al.  Robust network community detection using balanced propagation , 2011, ArXiv.

[86]  Ulrik Brandes,et al.  What is network science? , 2013, Network Science.