Distance-Dependent Kronecker Graphs for Modeling Social Networks

This paper focuses on a generalization of stochastic Kronecker graphs, introducing a Kronecker-like operator and defining a family of generator matrices H dependent on distances between nodes in a specified graph embedding. We prove that any lattice-based network model with sufficiently small distance-dependent connection probability will have a Poisson degree distribution and provide a general framework to prove searchability for such a network. Using this framework, we focus on a specific example of an expanding hypercube and discuss the similarities and differences of such a model with recently proposed network models based on a hidden metric space. We also prove that a greedy forwarding algorithm can find very short paths of length O((log log n)2) on the hypercube with n nodes, demonstrating that distance-dependent Kronecker graphs can generate searchable network models.

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

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

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

[4]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[5]  Marián Boguñá,et al.  Navigability of Complex Networks , 2007, ArXiv.

[6]  Amin Vahdat,et al.  On curvature and temperature of complex networks , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  Béla Bollobás,et al.  Random Graphs , 1985 .

[8]  Mark E. J. Newman,et al.  The Structure and Function of Complex Networks , 2003, SIAM Rev..

[9]  Jon M. Kleinberg,et al.  The small-world phenomenon: an algorithmic perspective , 2000, STOC '00.

[10]  Christos Faloutsos,et al.  Realistic, Mathematically Tractable Graph Generation and Evolution, Using Kronecker Multiplication , 2005, PKDD.

[11]  Amin Vahdat,et al.  Greedy forwarding in scale-free networks embedded in hyperbolic metric spaces , 2009, SIGMETRICS Perform. Evaluation Rev..

[12]  Christos Faloutsos,et al.  Dynamics of large networks , 2008 .

[13]  P. Bak,et al.  A forest-fire model and some thoughts on turbulence , 1990 .

[14]  Marián Boguñá,et al.  Navigating ultrasmall worlds in ultrashort time. , 2008, Physical review letters.

[15]  Marián Boguñá,et al.  Self-similarity of complex networks and hidden metric spaces , 2007, Physical review letters.

[16]  Adam Wierman,et al.  Generalizing Kronecker graphs in order to model searchable networks , 2009, 2009 47th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[17]  Mark Crovella,et al.  Hyperbolic Embedding and Routing for Dynamic Graphs , 2009, IEEE INFOCOM 2009.

[18]  Mohammad Mahdian,et al.  Stochastic Kronecker Graphs , 2007, WAW.

[19]  Aditya Akella,et al.  On the treeness of internet latency and bandwidth , 2009, SIGMETRICS '09.

[20]  Charles U. Martel,et al.  Analyzing Kleinberg's (and other) small-world Models , 2004, PODC '04.

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

[22]  Sharon L. Milgram,et al.  The Small World Problem , 1967 .