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

## TL;DR

This paper develops new correlation and mixing bounds for stochastic processes on dynamic networks, and applies these results to analyze a Cox-process model for bike-sharing data, demonstrating asymptotic properties of a goodness-of-fit test.

## Contribution

It introduces novel correlation and mixing bounds for processes on time-varying networks and applies them to Cox-process models, advancing understanding of dependence in dynamic network data.

## Key findings

- Established exponential inequalities for weak dependence on dynamic networks.
- Proved asymptotic properties of a goodness-of-fit test in Cox-process models.
- Applied the theoretical results to real bike-sharing data.

## Abstract

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, 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.