Identifying network structure similarity using spectral graph theory

Most real networks are too large or they are not available for real time analysis. Therefore, in practice, decisions are made based on partial information about the ground truth network. It is of great interest to have metrics to determine if an inferred network (the partial information network) is similar to the ground truth. In this paper we develop a test for similarity between the inferred and the true network. Our research utilizes a network visualization tool, which systematically discovers a network, producing a sequence of snapshots of the network. We introduce and test our metric on the consecutive snapshots of a network, and against the ground truth.To test the scalability of our metric we use a random matrix theory approach while discovering Erdös-Rényi graphs. This scaling analysis allows us to make predictions about the performance of the discovery process.

[1]  B. McKay,et al.  Constructing cospectral graphs , 1982 .

[2]  Gary Chartrand,et al.  A First Course in Graph Theory , 2012 .

[3]  Christos H. Papadimitriou,et al.  On the Eigenvalue Power Law , 2002, RANDOM.

[4]  Hisashi Kashima,et al.  Kernels for graph classification , 2002 .

[5]  W. Haemers,et al.  Which graphs are determined by their spectrum , 2003 .

[6]  G. J. Rodgers,et al.  Density of states of sparse random matrices , 1990 .

[7]  A. Jackson,et al.  Spectral ergodicity and normal modes in ensembles of sparse matrices , 2001 .

[8]  Koujin Takeda,et al.  Cavity approach to the spectral density of sparse symmetric random matrices. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  J. A. Méndez-Bermúdez,et al.  Scaling properties of multilayer random networks. , 2016, Physical review. E.

[10]  L. Collatz,et al.  Spektren endlicher grafen , 1957 .

[11]  Piet Van Mieghem,et al.  Graph Spectra for Complex Networks , 2010 .

[12]  G. J. Rodgers,et al.  Eigenvalue distribution of large dilute random matrices , 1997 .

[13]  Fan Chung,et al.  Spectral Graph Theory , 1996 .

[14]  George C. Verghese,et al.  Graph similarity scoring and matching , 2008, Appl. Math. Lett..

[15]  Ioana Dumitriu,et al.  Sparse regular random graphs: Spectral density and eigenvectors , 2009, 0910.5306.

[16]  R. Kuehn.  Spectra of sparse random matrices , 2008, 0803.2886.

[17]  Pantelimon Stanica,et al.  The spectrum of generalized Petersen graphs , 2010, Australas. J Comb..

[18]  Erik C. Rye,et al.  The Marginal Benefit of Monitor Placement on Networks , 2016, CompleNet.

[19]  Jeannette C. M. Janssen,et al.  Model Selection for Social Networks Using Graphlets , 2012, Internet Math..

[20]  Xiao Liu,et al.  Integrating Local Features into Discriminative Graphlets for Scene Classification , 2011, ICONIP.

[21]  Colin Cooper,et al.  Randomization and Approximation Techniques in Computer Science , 1999, Lecture Notes in Computer Science.

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

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

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

[25]  J. J. Seidel,et al.  Graphs and their spectra , 1989 .

[26]  Y. Kabashima,et al.  First eigenvalue/eigenvector in sparse random symmetric matrices: influences of degree fluctuation , 2012, 1204.5534.

[27]  Spectral density singularities, level statistics, and localization in a sparse random matrix ensemble. , 1992, Physical review letters.

[28]  Mohammad Al Hasan,et al.  GUISE: a uniform sampler for constructing frequency histogram of graphlets , 2013, Knowledge and Information Systems.

[29]  F. Slanina.  Equivalence of replica and cavity methods for computing spectra of sparse random matrices. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  Moo K. Chung,et al.  Computing the Shape of Brain Networks Using Graph Filtration and Gromov-Hausdorff Metric , 2011, MICCAI.

[31]  Jafar Habibi,et al.  Distance metric learning for complex networks: towards size-independent comparison of network structures. , 2015, Chaos.

[32]  Yonatan Aumann,et al.  Optimization of probe coverage for high-resolution oligonucleotide aCGH , 2007, Bioinform..

[33]  Christos Faloutsos,et al.  Fast best-effort pattern matching in large attributed graphs , 2007, KDD '07.

[34]  J. A. Méndez-Bermúdez,et al.  Diluted banded random matrices: scaling behavior of eigenfunction and spectral properties , 2017, 1701.01484.

[35]  Rodgers,et al.  Density of states of a sparse random matrix. , 1988, Physical review. B, Condensed matter.

[36]  Spectral density singularities, level statistics, and localization in a sparse random matrix ensemble. , 1992 .

[37]  Stanley F. Florkowski.  Spectral Graph Theory of the Hypercube , 2008 .

[38]  Y. Kabashima,et al.  Cavity approach to the first eigenvalue problem in a family of symmetric random sparse matrices , 2010, 1001.3935.

[39]  Guilhem Semerjian,et al.  Sparse random matrices: the eigenvalue spectrum revisited , 2002 .

[40]  Francisco A Rodrigues,et al.  Universality in the spectral and eigenfunction properties of random networks. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[41]  Y. Fyodorov,et al.  Universality of level correlation function of sparse random matrices , 1991 .

[42]  Ralucca Gera,et al.  Optimizing Network Discovery with Clever Walks , 2017, ASONAM.

[43]  L. Trefethen,et al.  Numerical linear algebra , 1997 .

[44]  F L Metz,et al.  Spectra of sparse non-hermitian random matrices: an analytical solution. , 2012, Physical review letters.

[45]  Christopher T. Workman,et al.  DASS: efficient discovery and p-value calculation of substructures in unordered data , 2007, Bioinform..

[46]  Fyodorov,et al.  Localization in ensemble of sparse random matrices. , 1991, Physical review letters.

[47]  Vinko Zlatic,et al.  Complex Networks VIII , 2017 .

[48]  Ryan Miller,et al.  Graph Structure Similarity using Spectral Graph Theory , 2016, COMPLEX NETWORKS.

[49]  Frank Harary,et al.  Cospectral Graphs and Digraphs , 1971 .

[50]  Yi Yang,et al.  Discovering Discriminative Graphlets for Aerial Image Categories Recognition , 2013, IEEE Transactions on Image Processing.

[51]  H. Günthard,et al.  Zusammenhang von Graphentheorie und MO-Theorie von Molekeln mit Systemen konjugierter Bindungen , 1956 .

[52]  T. Rogers,et al.  Cavity approach to the spectral density of non-Hermitian sparse matrices. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[53]  J. A. Méndez-Bermúdez,et al.  Scattering and transport properties of tight-binding random networks. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[54]  Robert A. Koyak,et al.  Nonparametric Tests for Homogeneity Based on Non-Bipartite Matching , 2011 .

[55]  S. Evangelou.  A numerical study of sparse random matrices , 1992 .

[56]  Pivithuru Wijegunawardana,et al.  Seeing Red: Locating People of Interest in Networks , 2017 .