Spectral statistics of random geometric graphs

We use random matrix theory to study the spectrum of random geometric graphs, a fundamental model of spatial networks. Considering ensembles of random geometric graphs we look at short-range correlations in the level spacings of the spectrum via the nearest-neighbour and next-nearest-neighbour spacing distribution and long-range correlations via the spectral rigidity statistic. These correlations in the level spacings give information about localisation of eigenvectors, level of community structure and the level of randomness within the networks. We find a parameter-dependent transition between Poisson and Gaussian orthogonal ensemble statistics. That is the spectral statistics of spatial random geometric graphs fits the universality of random matrix theory found in other models such as Erdős-Rényi, Barabási-Albert and Watts-Strogatz random graphs.

[1]  Xinghuo Yu,et al.  Finding the Most Influential Nodes in Pinning Controllability of Complex Networks , 2017, IEEE Transactions on Circuits and Systems II: Express Briefs.

[2]  Shlomo Havlin,et al.  The effect of spatiality on multiplex networks , 2016 .

[3]  Ernesto Estrada,et al.  Consensus dynamics on random rectangular graphs , 2016 .

[4]  Yukio Hayashi,et al.  Spatially self-organized resilient networks by a distributed cooperative mechanism , 2016, ArXiv.

[5]  Hai-Jun Zhou,et al.  Identifying optimal targets of network attack by belief propagation , 2016, Physical review. E.

[6]  C. Dettmann,et al.  Connectivity of networks with general connection functions , 2014, Physical review. E.

[7]  Igor Rivin,et al.  Spectral Experiments+ , 2014, Exp. Math..

[8]  Hao Liao,et al.  Enhancing speed of pinning synchronizability: low-degree nodes with high feedback gains , 2015, Scientific Reports.

[9]  Ernesto Estrada,et al.  Synchronizability of random rectangular graphs. , 2015, Chaos.

[10]  Hernán A. Makse,et al.  Influence maximization in complex networks through optimal percolation , 2015, Nature.

[11]  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.

[12]  SARIKA JALAN,et al.  Importance of randomness in biological networks: A random matrix analysis , 2015 .

[13]  Kevin E. Bassler,et al.  Mesoscopic structures and the Laplacian spectra of random geometric graphs , 2014, J. Complex Networks.

[14]  Ernesto Estrada,et al.  Graph and Network Theory , 2013, 1302.4378.

[15]  Xinghuo Yu,et al.  Optimal pinning controllability of complex networks: dependence on network structure. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Yukio Hayashi,et al.  Growing Self-Organized Design of Efficient and Robust Complex Networks , 2014, 2014 IEEE Eighth International Conference on Self-Adaptive and Self-Organizing Systems.

[17]  Jean-Charles Delvenne,et al.  Random Walks, Markov Processes and the Multiscale Modular Organization of Complex Networks , 2014, IEEE Transactions on Network Science and Engineering.

[18]  Sarika Jalan,et al.  Randomness and preserved patterns in cancer network , 2014, Scientific Reports.

[19]  Shlomo Havlin,et al.  Conditions for viral influence spreading through multiplex correlated social networks , 2014, 1404.3114.

[20]  Sang Hoon Lee,et al.  Density-Based and Transport-Based Core-Periphery Structures in Networks , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  Mason A. Porter,et al.  Core-Periphery Structure in Networks , 2012, SIAM J. Appl. Math..

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

[23]  Fabio Della Rossa,et al.  Profiling core-periphery network structure by random walkers , 2013, Scientific Reports.

[24]  H. Yau,et al.  Spectral statistics of Erdős–Rényi graphs I: Local semicircle law , 2011, 1103.1919.

[25]  Qing Zhang,et al.  Stochastic Analysis, Control, Optimization and Applications , 2012 .

[26]  Harry Eugene Stanley,et al.  Robustness of onion-like correlated networks against targeted attacks , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  H. Yau,et al.  Spectral Statistics of Erdős-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues , 2011, 1103.3869.

[28]  Shlomo Havlin,et al.  How people interact in evolving online affiliation networks , 2011, ArXiv.

[29]  Petter Holme,et al.  Onion structure and network robustness , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  Albert-László Barabási,et al.  Controllability of complex networks , 2011, Nature.

[31]  Hans J. Herrmann,et al.  Mitigation of malicious attacks on networks , 2011, Proceedings of the National Academy of Sciences.

[32]  Edmund M. Yeh,et al.  Cascading Link Failure in the Power Grid: A Percolation-Based Analysis , 2011, 2011 IEEE International Conference on Communications Workshops (ICC).

[33]  Robin Chapman.  Surveys in Combinatorics 2011 , 2011 .

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

[35]  Daisuke Watanabe,et al.  A Study on Analyzing the Grid Road Network Patterns using Relative Neighborhood Graph , 2010 .

[36]  Jeffrey G. Andrews,et al.  Stochastic geometry and random graphs for the analysis and design of wireless networks , 2009, IEEE Journal on Selected Areas in Communications.

[37]  Marta C. González,et al.  Understanding spatial connectivity of individuals with non-uniform population density , 2009, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[38]  S. Jalan.  Spectral analysis of deformed random networks. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[39]  Sarika Jalan,et al.  Randomness of random networks: A random matrix analysis , 2009 .

[40]  Yamir Moreno,et al.  Synchronization in Random Geometric Graphs , 2009, Int. J. Bifurc. Chaos.

[41]  Baowen Li,et al.  Localizations on Complex Networks , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[42]  Desmond J. Higham,et al.  Fitting a geometric graph to a protein-protein interaction network , 2008, Bioinform..

[43]  Maziar Nekovee,et al.  Worm epidemics in wireless ad hoc networks , 2007, ArXiv.

[44]  Z. Toroczkai,et al.  Proximity networks and epidemics , 2007, physics/0701255.

[45]  Sarika Jalan,et al.  Random matrix analysis of complex networks. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[46]  Sarika Jalan,et al.  Universality in complex networks: random matrix analysis. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[47]  G. Palla,et al.  Spectral transitions in networks , 2006, physics/0701054.

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

[49]  Richard H. Scheuermann,et al.  Application of random matrix theory to biological networks , 2005, q-bio/0503035.

[50]  Paul Blackwell,et al.  Spectra of adjacency matrices of random geometric graphs , 2006 .

[51]  George L. Trigg,et al.  Mathematical Tools for Physicists , 2005 .

[52]  M. Stephanov,et al.  Random Matrices , 2005, hep-ph/0509286.

[53]  S. Havlin,et al.  Localization transition on complex networks via spectral statistics. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[54]  Mathew D. Penrose,et al.  Random Geometric Graphs , 2003 .

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

[56]  M. Newman.  Assortative mixing in networks. , 2002, Physical review letters.

[57]  J. Dall,et al.  Random geometric graphs. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[58]  K. Goh,et al.  Universal behavior of load distribution in scale-free networks. , 2001, Physical review letters.

[59]  A. Barabasi,et al.  Spectra of "real-world" graphs: beyond the semicircle law. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[60]  Gregory J. Pottie,et al.  Wireless integrated network sensors , 2000, Commun. ACM.

[61]  Z. Rudnick,et al.  Eigenvalue Spacings for Regular Graphs , 2003, hep-th/0310002.

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

[63]  Satish Kumar,et al.  Next century challenges: scalable coordination in sensor networks , 1999, MobiCom.

[64]  A. Barabasi,et al.  Mean-field theory for scale-free random networks , 1999, cond-mat/9907068.

[65]  M. Mézard,et al.  Spectra of euclidean random matrices , 1999, cond-mat/9906135.

[66]  Andrew B. Kahng,et al.  Spectral Partitioning with Multiple Eigenvectors , 1999, Discret. Appl. Math..

[67]  Piyush Gupta,et al.  Critical Power for Asymptotic Connectivity in Wireless Networks , 1999 .

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

[69]  Yoshua Bengio,et al.  Gradient-based learning applied to document recognition , 1998, Proc. IEEE.

[70]  T. Guhr,et al.  RANDOM-MATRIX THEORIES IN QUANTUM PHYSICS : COMMON CONCEPTS , 1997, cond-mat/9707301.

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

[72]  G. Mitchell,et al.  Small sample size effects in statistical analyses of eigenvalue distributions , 1992 .

[73]  Cheon.  Eigenvalue statistics of distorted random matrices. , 1990, Physical review letters.

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

[75]  O. Bohigas,et al.  Chaotic motion and random matrix theories , 1984 .

[76]  B. Bollobás.  Surveys in Combinatorics , 1979 .

[77]  O. Bohigas,et al.  Level density fluctuations and random matrix theory , 1975 .

[78]  T. A. Brody.  A statistical measure for the repulsion of energy levels , 1973 .

[79]  G. Arfken,et al.  Mathematical Methods for Physicists (Second Edition) , 1971 .

[80]  F. Dyson.  Statistical Theory of the Energy Levels of Complex Systems. I , 1962 .

[81]  E. N. Gilbert,et al.  Random Plane Networks , 1961 .