Limit laws for random spatial graphical models

We consider spatial graphical models on random Euclidean points, applicable for data in sensor and social networks. We establish limit laws for general functions of the graphical model such as the mean value, the entropy rate etc. as the number of nodes goes to infinity under certain conditions. These conditions require the corresponding Gibbs measure to be spatially mixing and for the random graph of the model to satisfy a certain localization property known as stabilization. Graphs such the k nearest neighbor graph and the geometric disc graph belong to the class of stabilizing graphs. Intuitively, these conditions require the data at each node not to have strong dependence on data and positions of nodes far away. Finally, it is shown that spatial mixing of the Gibbs measure on a random graph holds when a suitably defined degree-dependent (but otherwise independent) node percolation does not have a giant component.

[1]  Hans-Otto Georgii,et al.  Gibbs Measures and Phase Transitions , 1988 .

[2]  Van den Berg,et al.  Disagreement percolation in the study of Markov fields , 1994 .

[3]  Gary L. Miller,et al.  Separators for sphere-packings and nearest neighbor graphs , 1997, JACM.

[4]  The random geometry of equilibrium phases , 1999, math/9905031.

[5]  John Odentrantz,et al.  Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues , 2000, Technometrics.

[6]  J. Yukich,et al.  Central limit theorems for some graphs in computational geometry , 2001 .

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

[8]  Sekhar Tatikonda,et al.  Loopy Belief Propogation and Gibbs Measures , 2002, UAI.

[9]  J. Yukich,et al.  Weak laws of large numbers in geometric probability , 2003 .

[10]  Dror Weitz.  Combinatorial criteria for uniqueness of Gibbs measures , 2005 .

[11]  Thomas P. Hayes.  A simple condition implying rapid mixing of single-site dynamics on spin systems , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[12]  Zhenning Kong,et al.  Percolation processes and wireless network resilience , 2008, 2008 Information Theory and Applications Workshop.

[13]  Michael I. Jordan,et al.  Graphical Models, Exponential Families, and Variational Inference , 2008, Found. Trends Mach. Learn..

[14]  A. Swami,et al.  Energy Scaling Laws for Distributed Inference in Random Networks , 2008, ArXiv.

[15]  Ananthram Swami,et al.  Energy scaling laws for distributed inference in random fusion networks , 2008, IEEE Journal on Selected Areas in Communications.

[16]  Dmitriy Katz,et al.  Sequential cavity method for computing limits of the log-partition function for lattice models , 2009, SODA.

[17]  S. Fortunato,et al.  Statistical physics of social dynamics , 2007, 0710.3256.