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

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 is linked with the network in the following way: Two neighbouring vertices (or two adjacent pairs of vertices) are dependent, while we assume that the dependence decreases as the distance in the network increases. We make this intuition mathematically precise by considering three concepts based on correlation, beta-mixing with time-varying beta-coefficients and conditional independence. Then, we will use these concepts in order to prove weak-dependence results, e.g. 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 multiplicative Cox-type hazard model. This model is then applied to bike-sharing data.