Limit laws for random spatial graphical models

Abstract

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.

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