Hyperbolicity, degeneracy, and expansion of random intersection graphs

We establish the conditions under which several algorithmically exploitable structural features hold for random intersection graphs, a natural model for many real-world networks where edges correspond to shared attributes. Specifically, we fully characterize the degeneracy of random intersection graphs, and prove that the model asymptotically almost surely produces graphs with hyperbolicity at least $$\log {n}$$logn. Further, we prove that when degenerate, the graphs generated by this model belong to a bounded-expansion graph class with high probability, a property particularly suitable for the design of linear time algorithms.

[1]  Vladimir Batagelj,et al.  An O(m) Algorithm for Cores Decomposition of Networks , 2003, ArXiv.

[2]  Iraj Saniee,et al.  Lack of spectral gap and hyperbolicity in asymptotic Erdo¨s-Renyi sparse random graphs , 2012, 2012 5th International Symposium on Communications, Control and Signal Processing.

[3]  Mindaugas Bloznelis,et al.  Degree and clustering coefficient in sparse random intersection graphs , 2013, 1303.3388.

[4]  Walter A. Kosters,et al.  Computing the Eccentricity Distribution of Large Graphs , 2013, Algorithms.

[5]  M. Newman,et al.  Random graphs with arbitrary degree distributions and their applications. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  Yilun Shang,et al.  Lack of Gromov-hyperbolicity in small-world networks , 2012 .

[7]  A. O. Houcine.  On hyperbolic groups , 2006 .

[8]  J. Nesetril,et al.  Grad and classes with bounded expansion III. restricted dualities , 2005, math/0508325.

[9]  Joel H. Spencer,et al.  Sudden Emergence of a Giantk-Core in a Random Graph , 1996, J. Comb. Theory, Ser. B.

[10]  Antoine Vigneron,et al.  Computing the Gromov hyperbolicity of a discrete metric space , 2012, Inf. Process. Lett..

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

[12]  Jaroslav Nesetril,et al.  Characterisations and examples of graph classes with bounded expansion , 2009, Eur. J. Comb..

[13]  Jaroslav Nesetril,et al.  Grad and classes with bounded expansion I. Decompositions , 2008, Eur. J. Comb..

[14]  Eric Jones,et al.  SciPy: Open Source Scientific Tools for Python , 2001 .

[15]  Willemien Kets,et al.  RANDOM INTERSECTION GRAPHS WITH TUNABLE DEGREE DISTRIBUTION AND CLUSTERING , 2009, Probability in the Engineering and Informational Sciences.

[16]  Matthieu Latapy,et al.  Fast computation of empirically tight bounds for the diameter of massive graphs , 2009, JEAL.

[17]  Yilun Shang.  Non-Hyperbolicity of Random Graphs with Given Expected Degrees , 2013 .

[18]  Jun Zhao,et al.  On k-Connectivity and Minimum Vertex Degree in Random s-Intersection Graphs , 2015, ANALCO.

[19]  Edmond A. Jonckheere,et al.  Scaled Gromov hyperbolic graphs , 2008, J. Graph Theory.

[20]  David Coudert,et al.  On Computing the Hyperbolicity of Real-World Graphs , 2015, ESA.

[21]  Valentas Kurauskas,et al.  Assortativity and clustering of sparse random intersection graphs , 2012, 1209.4675.

[22]  Blair D. Sullivan,et al.  Tree-Like Structure in Large Social and Information Networks , 2013, 2013 IEEE 13th International Conference on Data Mining.

[23]  Jaroslav Nesetril,et al.  Sparsity - Graphs, Structures, and Algorithms , 2012, Algorithms and combinatorics.

[24]  Roberto Grossi,et al.  Finding the Diameter in Real-World Graphs - Experimentally Turning a Lower Bound into an Upper Bound , 2010, ESA.

[25]  Robert D. Kleinberg.  Geographic Routing Using Hyperbolic Space , 2007, IEEE INFOCOM 2007 - 26th IEEE International Conference on Computer Communications.

[26]  Edward R. Scheinerman,et al.  On Random Intersection Graphs: The Subgraph Problem , 1999, Combinatorics, Probability and Computing.

[27]  Jaroslav Nesetril,et al.  On nowhere dense graphs , 2011, Eur. J. Comb..

[28]  Jaroslav Nesetril,et al.  Grad and classes with bounded expansion II. Algorithmic aspects , 2008, Eur. J. Comb..

[29]  Blair D. Sullivan,et al.  Structural Sparsity of Complex Networks: Random Graph Models and Linear Algorithms , 2014, ArXiv.

[30]  Feodor F. Dragan,et al.  Diameters, centers, and approximating trees of delta-hyperbolicgeodesic spaces and graphs , 2008, SCG '08.

[31]  Virgil D. Gligor,et al.  On the strengths of connectivity and robustness in general random intersection graphs , 2014, 53rd IEEE Conference on Decision and Control.

[32]  Victor Chepoi,et al.  Diameters , centers , and approximating trees of δ-hyperbolic geodesic spaces and graphs ∗ [ Extended Abstract ] , 2007 .

[33]  Wei Chen,et al.  On the Hyperbolicity of Small-World and Treelike Random Graphs , 2013, Internet Math..

[34]  Rae Baxter,et al.  Acknowledgments.-The authors would like to , 1982 .

[35]  Andreas N. Lagerås,et al.  A Note on the Component Structure in Random Intersection Graphs with Tunable Clustering , 2008, Electron. J. Comb..

[36]  Jun Zhao,et al.  Connectivity in secure wireless sensor networks under transmission constraints , 2014, 2014 52nd Annual Allerton Conference on Communication, Control, and Computing (Allerton).

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

[38]  Michael Behrisch,et al.  Component Evolution in Random Intersection Graphs , 2007, Electron. J. Comb..

[39]  F. Ball,et al.  Epidemics on random intersection graphs , 2010, 1011.4242.

[40]  Jaroslav Nesetril,et al.  First order properties on nowhere dense structures , 2010, The Journal of Symbolic Logic.

[41]  Douglas R. Stinson,et al.  On the complexity of the herding attack and some related attacks on hash functions , 2012, Des. Codes Cryptogr..

[42]  Jerzy Jaworski,et al.  The degree of a typical vertex in generalized random intersection graph models , 2006, Discret. Math..

[43]  Valentas Kurauskas,et al.  Large Cliques in Sparse Random Intersection Graphs , 2013, Electron. J. Comb..

[44]  D. Marquardt.  An Algorithm for Least-Squares Estimation of Nonlinear Parameters , 1963 .

[45]  Felix Reidl,et al.  Structural sparseness and complex networks , 2016 .

[46]  Erhard Godehardt,et al.  Clustering Coefficients of Random Intersection Graphs , 2010, GfKl.

[47]  Graham A. Niblo.  METRIC SPACES OF NON‐POSITIVE CURVATURE (Grundlehren der Mathematischen Wissenschaften 319) , 2001 .

[48]  Katarzyna Rybarczyk,et al.  Diameter, connectivity, and phase transition of the uniform random intersection graph , 2011, Discret. Math..

[49]  M. Bridson,et al.  Metric Spaces of Non-Positive Curvature , 1999 .

[50]  Robin Thomas,et al.  Testing first-order properties for subclasses of sparse graphs , 2011, JACM.

[51]  Kenneth Levenberg.  A METHOD FOR THE SOLUTION OF CERTAIN NON – LINEAR PROBLEMS IN LEAST SQUARES , 1944 .

[52]  Amin Vahdat,et al.  Hyperbolic Geometry of Complex Networks , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[53]  Katarzyna Rybarczyk.  The Coupling Method for Inhomogeneous Random Intersection Graphs , 2017, Electron. J. Comb..

[54]  Wei Chen,et al.  On the Hyperbolicity of Small-World and Treelike Random Graphs , 2012, Internet Math..

[55]  Zdenek Dvorak,et al.  Constant-factor approximation of domination number in sparse graphs , 2011, ArXiv.

[56]  Stephan Kreutzer,et al.  Parameterized Complexity of First-Order Logic , 2009, Electron. Colloquium Comput. Complex..

[57]  David Coudert,et al.  On Computing the Gromov Hyperbolicity , 2015, ACM J. Exp. Algorithmics.

[58]  Jun Zhao.  Minimum node degree and k-connectivity in wireless networks with unreliable links , 2014, 2014 IEEE International Symposium on Information Theory.

[59]  Aric Hagberg,et al.  Exploring Network Structure, Dynamics, and Function using NetworkX , 2008, Proceedings of the Python in Science Conference.

[60]  Roberto Grossi,et al.  On computing the diameter of real-world undirected graphs , 2013, Theor. Comput. Sci..