A theory and algorithm for detecting and classifying weak, distributed patterns in network data is presented. The patterns we consider are anomalous temporal correlations between signals recorded at sensor nodes in a network. We use robust matrix completion and second order analysis to detect distributed patterns that are not discernible at the level of individual sensors. When viewed independently, the data at each node cannot provide a definitive determination of the underlying pattern, but when fused with data from across the network the relevant patterns emerge. We are specifically interested in detecting weak patterns in computer networks where the nodes (terminals, routers, servers, etc.) are sensors that provide measurements (of packet rates, user activity, central processing unit usage, etc.). The approach is applicable to many other types of sensor networks including wireless networks, mobile sensor networks, and social networks where correlated phenomena are of interest.

Abstract. The purpose of this paper is to apply particle methods to the numerical solution of the EPDiff equation. The weak solutions of EPDiff are contact discontinuities that carry momentum so that wavefront interactions represent collisions in which momentum is exchanged. This behavior allows for the description of many rich physical applications, but also introduces difficult numerical challenges. We present a particle method for the EPDiff equation that is well-suited for this class of solutions and for simulating collisions between wavefronts. Discretization by means of the particle method is shown to preserve the basic Hamiltonian, the weak and variational structure of the original problem, and to respect the conservation laws associated with symmetry under the Euclidean group. Numerical results illustrate that the particle method has superior features in both one and two dimensions, and can also be effectively implemented when the initial data of interest lies on a submanifold.

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