Distributed Testing of Conductance

We study the problem of testing conductance in the distributed computing model and give a two-sided tester that takes $\mathcal{O}(\log n)$ rounds to decide if a graph has conductance at least $\Phi$ or is $\epsilon$-far from having conductance at least $\Theta(\Phi^2)$ in the distributed CONGEST model. We also show that $\Omega(\log n)$ rounds are necessary for testing conductance even in the LOCAL model. In the case of a connected graph, we show that we can perform the test even when the number of vertices in the graph is not known a priori. This is the first two-sided tester in the distributed model we are aware of. The key idea in our algorithm is a way to perform a polynomial number of random walks from a set of vertices, avoiding the congestion on the edges.