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B0530
**Title: **Dependence under random time-varying network distances with an application to Cox-Processes
**Authors: **Alexander Kreiss - KU Leuven (Belgium) **[presenting]**

**Abstract: **Multivariate stochastic processes indexed either by vertices or pairs of vertices of a dynamic network are considered. Under a dynamic network we understand a network with a fixed vertex set and an edge set which changes randomly over time. The aim is to conduct inference in models for this type of data. For real-world applications, it is important to allow that processes of adjacent pairs (or adjacent vertices) may be dependent. Since networks are often changing over time, the notion of adjacency is dynamic and as a consequence also the dependence should be dynamic. We will thus 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, beta-mixing with time-varying beta-coefficients and conditional independence. These concepts allow proving 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 Cox-type model for counting processes. This model is then applied to bike-sharing data.