An efficient alternative to Ollivier-Ricci curvature based on the Jaccard metric

We study Ollivier-Ricci curvature, a discrete version of Ricci curvature, which has gained popularity over the past several years and has found applications in diverse fields. However, the Ollivier-Ricci curvature requires an optimal mass transport problem to be solved, which can be computationally expensive for large networks. In view of this, we propose two alternative measures of curvature to Ollivier-Ricci which are motivated by the Jaccard coefficient and are demonstrably less computationally intensive, a cheaper Jaccard (JC) and a more expensive generalized Jaccard (gJC) curvature metric. We show theoretically that the gJC closely matches the Ollivier-Ricci curvature for Erdos-Renyi graphs in the asymptotic regime of large networks. Furthermore, we study the goodness of approximation between the proposed curvature metrics and Ollivier-Ricci curvature for several network models and real networks. Our results suggest that in comparison to an alternative curvature metric for graphs, the Forman-Ricci curvature, the gJC exhibits a reasonably good fit to the Ollivier-Ricci curvature for a wide range of networks, while the JC is shown to be a good proxy only for certain scenarios.

[1]  David Liben-Nowell,et al.  The link-prediction problem for social networks , 2007 .

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

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

[4]  Emil Saucan,et al.  Discrete curvatures and network analysis , 2017 .

[5]  Sumit Mukherjee,et al.  Exact and asymptotic results on coarse Ricci curvature of graphs , 2013, Discret. Math..

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

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

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

[9]  Emil Saucan,et al.  Forman-Ricci Flow for Change Detection in Large Dynamic Data Sets , 2016, Axioms.

[10]  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.

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

[12]  Emil Saucan,et al.  Systematic evaluation of a new combinatorial curvature for complex networks , 2016, 1610.01507.

[13]  Jure Leskovec,et al.  {SNAP Datasets}: {Stanford} Large Network Dataset Collection , 2014 .

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

[15]  Robert E. Tarjan,et al.  Dynamic trees as search trees via euler tours, applied to the network simplex algorithm , 1997, Math. Program..

[16]  Mathew D. Penrose,et al.  Random Geometric Graphs , 2003 .

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

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

[19]  Chi Wang,et al.  Wireless network capacity versus Ollivier-Ricci curvature under Heat-Diffusion (HD) protocol , 2014, 2014 American Control Conference.

[20]  Yusuke Higuchi.  Combinatorial curvature for planar graphs , 2001 .

[21]  I. Foster,et al.  1 Mapping the Gnutella Network : Macroscopic Properties of Large-Scale Peer-to-Peer Systems , 2002 .

[22]  A. Arenas,et al.  Models of social networks based on social distance attachment. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

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

[25]  Emil Saucan,et al.  Characterizing complex networks with Forman-Ricci curvature and associated geometric flows , 2016, J. Complex Networks.

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

[27]  A. Tannenbaum,et al.  Ricci curvature: An economic indicator for market fragility and systemic risk , 2016, Science Advances.

[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]  Maziar Nekovee,et al.  Worm epidemics in wireless ad hoc networks , 2007, ArXiv.

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