Systematic evaluation of a new combinatorial curvature for complex networks

Abstract We have recently introduced Forman’s discretization of Ricci curvature to the realm of complex networks. Forman curvature is an edge-based measure whose mathematical definition elegantly encapsulates the weights of nodes and edges in a complex network. In this contribution, we perform a comparative analysis of Forman curvature with other edge-based measures such as edge betweenness, embeddedness and dispersion in diverse model and real networks. We find that Forman curvature in comparison to embeddedness or dispersion is a better indicator of the importance of an edge for the large-scale connectivity of complex networks. Based on the definition of the Forman curvature of edges, there are two natural ways to define the Forman curvature of nodes in a network. In this contribution, we also examine these two possible definitions of Forman curvature of nodes in diverse model and real networks. Based on our empirical analysis, we find that in practice the unnormalized definition of the Forman curvature of nodes with the choice of combinatorial node weights is a better indicator of the importance of nodes in complex networks.

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

[2]  Robin Wilson,et al.  Modern Graph Theory , 2013 .

[3]  Jie Gao,et al.  Ricci curvature of the Internet topology , 2015, 2015 IEEE Conference on Computer Communications (INFOCOM).

[4]  L. Ward.  Social Forces , 1911, The Psychological Clinic.

[5]  Pablo M. Gleiser,et al.  Community Structure in Jazz , 2003, Adv. Complex Syst..

[6]  U. Brandes.  A faster algorithm for betweenness centrality , 2001 .

[7]  A. Châtelain,et al.  The European Physical Journal D , 1999 .

[8]  Anirban Banerjee,et al.  Graph spectra as a systematic tool in computational biology , 2007, Discret. Appl. Math..

[9]  Ginestra Bianconi,et al.  Network geometry with flavor: From complexity to quantum geometry. , 2015, Physical review. E.

[10]  S. Yau,et al.  Ricci curvature of graphs , 2011 .

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

[12]  P. V. Marsden,et al.  Measuring Tie Strength , 1984 .

[13]  R. McCann,et al.  Analysis and Geometry of Metric Measure Spaces , 2013 .

[14]  Frank Morgan,et al.  Manifolds with Density and Perelman's Proof of the Poincaré Conjecture , 2009, Am. Math. Mon..

[15]  U. Feige,et al.  Spectral Graph Theory , 2015 .

[16]  Jure Leskovec,et al.  Learning to Discover Social Circles in Ego Networks , 2012, NIPS.

[17]  ScienceDirect,et al.  Comptes rendus. Mathématique , 2002 .

[18]  Jean-Pierre Eckmann,et al.  Curvature of co-links uncovers hidden thematic layers in the World Wide Web , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[19]  林 鶴一,et al.  The Tôhoku mathematical journal = 東北數學雜誌 , .

[20]  Kathy P. Wheeler,et al.  Reviews of Modern Physics , 2013 .

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

[22]  Ginestra Bianconi,et al.  Emergent Hyperbolic Network Geometry , 2016, Scientific Reports.

[23]  Lucy Skrabanek,et al.  PDZBase: a protein?Cprotein interaction database for PDZ-domains , 2005, Bioinform..

[24]  Shiping Liu,et al.  Ollivier’s Ricci Curvature, Local Clustering and Curvature-Dimension Inequalities on Graphs , 2011, Discret. Comput. Geom..

[25]  Ed Reznik,et al.  Graph Curvature for Differentiating Cancer Networks , 2015, Scientific Reports.

[26]  Y. Ollivier.  A survey of Ricci curvature for metric spaces and Markov chains , 2010 .

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

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

[30]  October I.  Physical Review Letters , 2022 .

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

[32]  J. Jost,et al.  Forman curvature for complex networks , 2016, 1603.00386.

[33]  V Latora,et al.  Efficient behavior of small-world networks. , 2001, Physical review letters.

[34]  B. M. Fulk.  MATH , 1992 .

[35]  J. Herskowitz,et al.  Proceedings of the National Academy of Sciences, USA , 1996, Current Biology.

[36]  Karsten Grove.  Book Review: Metric structures for Riemannian and non-Riemannian spaces , 2001 .

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

[38]  Iraj Saniee,et al.  Large-scale curvature of networks. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[39]  Y. Ollivier.  A visual introduction to Riemannian curvatures and some discrete generalizations , 2012 .

[40]  Jon M. Kleinberg,et al.  Romantic partnerships and the dispersion of social ties: a network analysis of relationship status on facebook , 2013, CSCW.

[41]  S. Yau,et al.  Ricci curvature and eigenvalue estimate on locally finite graphs , 2010 .

[42]  J. van Leeuwen,et al.  Discrete and Computational Geometry , 2002, Lecture Notes in Computer Science.

[43]  S. L. Wong,et al.  Towards a proteome-scale map of the human protein–protein interaction network , 2005, Nature.

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

[45]  Emil Saucan,et al.  Curvature Based Clustering for DNA Microarray Data Analysis , 2005, IbPRIA.

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

[47]  Anirban Banerjee,et al.  Effect on normalized graph Laplacian spectrum by motif attachment and duplication , 2012, Appl. Math. Comput..

[48]  Proceedings of the 17th ACM conference on Computer supported cooperative work & social computing , 2014 .

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

[50]  A Díaz-Guilera,et al.  Self-similar community structure in a network of human interactions. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[51]  Y. Ollivier.  Ricci curvature of Markov chains on metric spaces , 2007, math/0701886.

[52]  O. Bagasra,et al.  Proceedings of the National Academy of Sciences , 1914, Science.

[53]  Dmitri V. Krioukov,et al.  Clustering Implies Geometry in Networks. , 2016, Physical review letters.

[54]  Y. Ollivier.  Ricci curvature of metric spaces , 2007 .

[55]  Peter L. Hammer,et al.  Discrete Applied Mathematics , 1993 .

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

[57]  M. Gromov.  Metric Structures for Riemannian and Non-Riemannian Spaces , 1999 .

[58]  Minping Qian,et al.  Networks: From Biology to Theory , 2007 .

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

[60]  Eugene Agichtein,et al.  Proceedings of the 26th International Conference on World Wide Web Companion , 2017, WWW 2017.

[61]  Santo Fortunato,et al.  Community detection in graphs , 2009, ArXiv.

[62]  A. Barabasi,et al.  Lethality and centrality in protein networks , 2001, Nature.

[63]  Stanley Wasserman,et al.  Social Network Analysis: Methods and Applications , 1994, Structural analysis in the social sciences.

[65]  J. Jost.  Riemannian geometry and geometric analysis , 1995 .

[66]  Rajmohan Rajaraman,et al.  Topology control and routing in ad hoc networks: a survey , 2002, SIGA.

[67]  J. Jost,et al.  Ollivier-Ricci curvature and the spectrum of the normalized graph Laplace operator , 2011, 1105.3803.

[68]  P. Cochat,et al.  Et al , 2008, Archives de pediatrie : organe officiel de la Societe francaise de pediatrie.

[69]  Frank Morgan,et al.  Manifolds with Density , 2005 .

[70]  Ke Chen,et al.  Applied Mathematics and Computation , 2022 .

[71]  B. Bollobás.  The evolution of random graphs , 1984 .

[72]  Frank Harary,et al.  Graph Theory , 2016 .

[73]  P. Erdos,et al.  On the evolution of random graphs , 1984 .

[74]  Robin Forman,et al.  Bochner's Method for Cell Complexes and Combinatorial Ricci Curvature , 2003, Discret. Comput. Geom..

[75]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[76]  Ginestra Bianconi,et al.  Emergent Complex Network Geometry , 2014, Scientific Reports.

[77]  P. Holland,et al.  Transitivity in Structural Models of Small Groups , 1971 .