Online Barycenter Estimation of Large Weighted Graphs

In this paper, we propose a new method to compute the barycenter of large weighted graphs endowed with probability measures on their nodes. We suppose that the edge weights are distances between the nodes and that the probability measure on the nodes is related to events observed there. For instance, a graph can represent a subway network: its edge weights are the distance between two stations, and the observed events at each node are the subway users getting in or leaving the subway network at this station. The probability measure on the nodes does not need to be explicitly known. Our strategy only uses observed node related events to give more or less emphasis to the different nodes. Furthermore, the barycenter estimation can be updated in real time with each new event. We propose a multiscale extension of \cite{arXiv:1605.04148} where the decribed strategy is valid only for medium-sized graphs due to memory costs. Our multiscale approach is inspired from the geometrical decomposition of the barycenter in a Euclidean space: we apply a heuristic \textit{divide et impera} strategy based on a preliminary clustering. Our strategy is finally assessed on road- and social-networks of up to $10^6$ nodes. We show that its results compare well with \cite{arXiv:1605.04148} in terms of accuracy and stability on small graphs, and that it can additionally be used on large graphs even on standard laptops