Forman-Ricci Flow for Change Detection in Large Dynamic Data Sets

We present a viable geometric solution for the detection of dynamic effects in complex networks. Building on Forman’s discretization of the classical notion of Ricci curvature, we introduce a novel geometric method to characterize different types of real-world networks with an emphasis on peer-to-peer networks. We study the classical Ricci-flow in a network-theoretic setting and introduce an analytic tool for characterizing dynamic effects. The formalism suggests a computational method for change detection and the identification of fast evolving network regions and yields insights into topological properties and the structure of the underlying data.

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

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

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

[4]  A. Barabasi,et al.  Network biology: understanding the cell's functional organization , 2004, Nature Reviews Genetics.

[5]  Anirban Banerjee,et al.  Spectral plot properties: Towards a qualitative classification of networks , 2008, Networks Heterog. Media.

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

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

[8]  Yuval Shavitt,et al.  On the curvature of the Internet and its usage for overlay construction and distance estimation , 2004, IEEE INFOCOM 2004.

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

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

[11]  Sara Ballouz,et al.  EGAD: Ultra-fast functional analysis of gene networks , 2016, bioRxiv.

[12]  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.

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

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

[15]  Ian T. Foster,et al.  Mapping the Gnutella Network , 2002, IEEE Internet Comput..

[16]  T. Georgiou,et al.  Market Fragility, Systemic Risk, and Ricci Curvature , 2015, 1505.05182.

[17]  Yehoshua Y. Zeevi,et al.  Combinatorial Ricci Curvature and Laplacians for Image Processing , 2009, 2009 2nd International Congress on Image and Signal Processing.

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

[19]  M. Kochen,et al.  Contacts and influence , 1978 .

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

[21]  Emil Saucan,et al.  Can one see the shape of a network? , 2016, ArXiv.

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

[23]  B. Chow,et al.  The Ricci flow on surfaces , 2004 .

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

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

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

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

[28]  Przemyslaw Kazienko,et al.  Matching Organizational Structure and Social Network Extracted from Email Communication , 2011, BIS.

[29]  Emil Saucan,et al.  Combinatorial Ricci Curvature for Image Processing , 2008, The MIDAS Journal.

[30]  G. Perelman.  Ricci flow with surgery on three-manifolds , 2003, math/0303109.

[31]  G. Petri,et al.  Homological scaffolds of brain functional networks , 2014, Journal of The Royal Society Interface.

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

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

[34]  Richard S. Hamilton,et al.  The Ricci flow on surfaces , 1986 .

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

[36]  Yehoshua Y. Zeevi,et al.  Ricci curvature and flow for image denoising and super-resolution , 2012, 2012 Proceedings of the 20th European Signal Processing Conference (EUSIPCO).

[37]  Cliff Lampe,et al.  The Benefits of Facebook "Friends: " Social Capital and College Students' Use of Online Social Network Sites , 2007, J. Comput. Mediat. Commun..

[38]  Ian T. Foster,et al.  Mapping the Gnutella Network: Properties of Large-Scale Peer-to-Peer Systems and Implications for System Design , 2002, ArXiv.

[39]  Г.Я. Перельман,et al.  Ricci flow with surgery on three-manifolds , 2003 .

[40]  B. Chow,et al.  COMBINATORIAL RICCI FLOWS ON SURFACES , 2002, math/0211256.

[41]  Guoliang Xu.  Discrete Laplace-Beltrami operators and their convergence , 2004, Comput. Aided Geom. Des..

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

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

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

[45]  Emil Saucan,et al.  A metric Ricci flow for surfaces and its applications , 2014 .

[46]  G. Perelman.  The entropy formula for the Ricci flow and its geometric applications , 2002, math/0211159.

[47]  Y. Zeevi,et al.  Ricci flow for image processing , 2014, 2014 IEEE 28th Convention of Electrical & Electronics Engineers in Israel (IEEEI).

[48]  Konstantin Mischaikow,et al.  Morse Theory for Filtrations and Efficient Computation of Persistent Homology , 2013, Discret. Comput. Geom..

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

[50]  Jie Gao,et al.  Greedy routing with guaranteed delivery using Ricci flows , 2009, 2009 International Conference on Information Processing in Sensor Networks.