Scalable computation of the maximum flow in large brain connectivity networks

We are interested in computing an approximation of the maximum flow in large (brain) connectivity networks. The maximum flow in such networks is of interest in order to better understand the routing of information in the human brain. However, the runtime of $O(|V||E|^2)$ for the classic Edmonds-Karp algorithm renders computations of the maximum flow on networks with millions of vertices infeasible, where $V$ is the set of vertices and $E$ is the set of edges. In this contribution, we propose a new Monte Carlo algorithm which is capable of computing an approximation of the maximum flow in networks with millions of vertices via subsampling. Apart from giving a point estimate of the maximum flow, our algorithm also returns valid confidence bounds for the true maximum flow. Importantly, its runtime only scales as $O(B \cdot |\tilde{V}| |\tilde{E}|^2)$, where $B$ is the number of Monte Carlo samples, $\tilde{V}$ is the set of subsampled vertices, and $\tilde{E}$ is the edge set induced by $\tilde{V}$. Choosing $B \in O(|V|)$ and $|\tilde{V}| \in O(\sqrt{|V|})$ (implying $|\tilde{E}| \in O(|V|)$) yields an algorithm with runtime $O(|V|^{3.5})$ while still guaranteeing the usual "root-n" convergence of the confidence interval of the maximum flow estimate. We evaluate our proposed algorithm with respect to both accuracy and runtime on simulated graphs as well as graphs downloaded from the Brain Networks Data Repository (https://networkrepository.com).