Local Mixing Time: Distributed Computation and Applications

Abstract: The mixing time of a graph is an important metric, which is not only useful in analyzing connectivity and expansion properties of the network, but also serves as a key parameter in designing efficient algorithms. We introduce a new notion of mixing of a random walk on a (undirected) graph, called local mixing. Informally, the local mixing with respect to a given node $s$, is the mixing of a random walk probability distribution restricted to a large enough subset of nodes --- say, a subset of size at least $n/\beta$ for a given parameter $\beta$ --- containing $s$. The time to mix over such a subset by a random walk starting from a source node $s$ is called the local mixing time with respect to $s$. The local mixing time captures the local connectivity and expansion properties around a given source node and is a useful parameter that determines the running time of algorithms for partial information spreading, gossip etc. Our first contribution is formally defining the notion of local mixing time in an undirected graph. We then present an efficient distributed algorithm which computes a constant factor approximation to the local mixing time with respect to a source node $s$ in $\tilde{O}(\tau_s)$ rounds, where $\tau_s$ is the local mixing time w.r.t $s$ in an $n$-node regular graph. This bound holds when $\tau_s$ is significantly smaller than the conductance of the local mixing set (i.e., the set where the walk mixes locally); this is typically the interesting case where the local mixing time is significantly smaller than the mixing time (with respect to $s$). We also present a distributed algorithm that computes the exact local mixing time in $\tilde{O}(\tau_s \mathcal{D})$ rounds, where $\mathcal{D} =\min{\tau_s, D}$ and $D$ is the diameter of the graph. We further show that local mixing time tightly characterizes the complexity of partial information spreading.