Graph Comparison via the Non-backtracking Spectrum

The comparison of graphs is a vitally important, yet difficult task which arises across a number of diverse research areas including biological and social networks. There have been a number of approaches to define graph distance, however, often these are not metrics (rendering standard data-mining techniques infeasible) or are computationally infeasible for large graphs. In this work we define a new pseudometric based on the spectrum of the nonbacktracking graph operator and show that it cannot only be used to compare graphs generated through different mechanisms but can reliably compare graphs of varying size. We observe that the family of Watts-Strogatz graphs lie on a manifold in the nonbacktracking spectral embedding and show how this metric can be used in a standard classification problem of empirical graphs.

[1]  Jukka-Pekka Onnela,et al.  Flexible Model Selection for Mechanistic Network Models via Super Learner , 2018, 1804.00237.

[2]  Tina Eliassi-Rad,et al.  Graph Distance from the Topological View of Non-backtracking Cycles , 2018, ArXiv.

[3]  Michael Batty,et al.  A long-time limit of world subway networks , 2011, 1105.5294.

[4]  D. Constantine,et al.  Marked length rigidity for one-dimensional spaces , 2003, Journal of Topology and Analysis.

[5]  Manlio De Domenico,et al.  Spectral entropies as information-theoretic tools for complex network comparison , 2016, 1609.01214.

[6]  W. Zachary,et al.  An Information Flow Model for Conflict and Fission in Small Groups , 1977, Journal of Anthropological Research.

[7]  Florent Krzakala,et al.  Spectral density of the non-backtracking operator on random graphs , 2014, ArXiv.

[8]  Jukka-Pekka Onnela,et al.  Taxonomies of networks from community structure. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  Christos Faloutsos,et al.  Graphs over time: densification laws, shrinking diameters and possible explanations , 2005, KDD '05.

[10]  Danai Koutra,et al.  DELTACON: A Principled Massive-Graph Similarity Function , 2013, SDM.

[11]  Florent Krzakala,et al.  Spectral Clustering of graphs with the Bethe Hessian , 2014, NIPS.

[12]  Petter Holme,et al.  Currency and commodity metabolites: their identification and relation to the modularity of metabolic networks. , 2006, IET systems biology.

[13]  Edwin R. Hancock,et al.  Measuring Graph Similarity Using Spectral Geometry , 2008, ICIAR.

[14]  L. Trefethen,et al.  Spectra and Pseudospectra , 2020 .

[15]  Kaspar Riesen,et al.  Graph Embedding in Vector Spaces by Means of Prototype Selection , 2007, GbRPR.

[16]  J. Friedman,et al.  THE NON-BACKTRACKING SPECTRUM OF THE UNIVERSAL COVER OF A GRAPH , 2007, 0712.0192.

[17]  K. Hashimoto.  Zeta functions of finite graphs and representations of p-adic groups , 1989 .

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

[19]  Audry Terras.  What are zeta functions of graphs and what are they good for ? , 2005 .

[20]  Mason A. Porter,et al.  Social Structure of Facebook Networks , 2011, ArXiv.

[21]  H. Bass.  THE IHARA-SELBERG ZETA FUNCTION OF A TREE LATTICE , 1992 .

[22]  Edwin R. Hancock,et al.  Graph Characterization via Ihara Coefficients , 2011, IEEE Transactions on Neural Networks.

[23]  F. Allen.  The Cambridge Structural Database: a quarter of a million crystal structures and rising. , 2002, Acta crystallographica. Section B, Structural science.

[24]  Stratis Ioannidis,et al.  A Family of Tractable Graph Distances , 2018, SDM.

[25]  M. A. Muñoz,et al.  Journal of Statistical Mechanics: An IOP and SISSA journal Theory and Experiment Detecting network communities: a new systematic and efficient algorithm , 2004 .

[26]  Ping Zhu,et al.  A Study of Graph Spectra for Comparing Graphs , 2005, BMVC.

[27]  Ernest Valveny,et al.  Generalized median graph computation by means of graph embedding in vector spaces , 2010, Pattern Recognit..

[28]  B. McKay.  The expected eigenvalue distribution of a large regular graph , 1981 .

[29]  Peter Grindrod,et al.  The Deformed Graph Laplacian and Its Applications to Network Centrality Analysis , 2018, SIAM J. Matrix Anal. Appl..

[30]  Gesine Reinert,et al.  Identifying networks with common organizational principles , 2017, J. Complex Networks.

[31]  Ping Zhu,et al.  A study of graph spectra for comparing graphs and trees , 2008, Pattern Recognit..

[32]  G. Chartrand,et al.  Graph similarity and distance in graphs , 1998 .

[33]  King-Sun Fu,et al.  A distance measure between attributed relational graphs for pattern recognition , 1983, IEEE Transactions on Systems, Man, and Cybernetics.

[34]  P. Pattison,et al.  Cumulated social roles: The duality of persons and their algebras☆ , 1986 .

[35]  Burleigh B. Gardner,et al.  Deep South: A Social Anthropological Study of Caste and Class , 1942 .

[36]  S. Holmes,et al.  Tracking network dynamics: A survey using graph distances , 2018, The Annals of Applied Statistics.

[37]  Vladimír Kvasnička,et al.  Reaction and chemical distances and reaction graphs , 1991 .

[38]  Edwin R. Hancock,et al.  Spectral embedding of graphs , 2003, Pattern Recognit..

[39]  Guillermo Sapiro,et al.  Graph Matching: Relax at Your Own Risk , 2014, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[40]  Elchanan Mossel,et al.  Spectral redemption in clustering sparse networks , 2013, Proceedings of the National Academy of Sciences.

[41]  Mario Vento,et al.  Thirty Years Of Graph Matching In Pattern Recognition , 2004, Int. J. Pattern Recognit. Artif. Intell..

[42]  László Babai,et al.  Graph isomorphism in quasipolynomial time [extended abstract] , 2015, STOC.

[43]  N. Altman.  An Introduction to Kernel and Nearest-Neighbor Nonparametric Regression , 1992 .

[44]  Geoffrey E. Hinton,et al.  Visualizing Data using t-SNE , 2008 .

[45]  Konstantin Avrachenkov,et al.  Cooperative Game Theory Approaches for Network Partitioning , 2017, COCOON.

[46]  Gunnar W. Klau,et al.  A new graph-based method for pairwise global network alignment , 2009, BMC Bioinformatics.