Correlation bounds, mixing and m-dependence under random time-varying network distances with an application to Cox-processes

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We will consider multivariate stochastic processes indexed either by vertices or pairs of vertices of a dynamic network. Under a dynamic network, we understand a network with a fixed vertex set and an edge set which changes randomly over time. We will assume that the spatial dependence-structure of the processes conditional on the network behaves in the following way: Close vertices (or pairs of vertices) are dependent, while we assume that the dependence decreases conditionally on that the distance in the network increases. We make this intuition mathematically precise by considering three concepts based on correlation, β-mixing with time-varying β-coefficients and conditional independence. These concepts allow proving weak-dependence results, for example, an exponential inequality, which might be of independent interest. In order to demonstrate the use of these concepts in an application, we study the asymptotics (for growing networks) of a goodness of fit test in a dynamic interaction network model based on a Cox-type model for counting processes. This model is then applied to bike-sharing data.