  We propose a simple and time-optimal algorithm for property testing a graph
for its conductance in the CONGEST model. Our algorithm takes only $O(\log n)$
rounds of communication (which is known to be optimal), and consists of simply
running multiple random walks of $O(\log n)$ length from a certain number of
random sources, at the end of which nodes can decide if the underlying network
is a good conductor or far from it. Unlike previous algorithms, no aggregation
is required even with a smaller number of walks. Our main technical
contribution involves a tight analysis of this process for which we use
spectral graph theory. We introduce and leverage the concept of sticky vertices
which are vertices in a graph with low conductance such that short random walks
originating from these vertices end in a region around them.
  The present state-of-the-art distributed CONGEST algorithm for the problem by
Fichtenberger and Vasudev [MFCS 2018], runs in $O(\log n)$ rounds using three
distinct phases : building a rooted spanning tree (\emph{preprocessing}),
running $O(n^{100})$ random walks to generate statistics (\emph{Phase~1}), and
then convergecasting to the root to make the decision (\emph{Phase~2}). The
whole of our algorithm is, however, similar to their Phase~1 running only
$O(m^2) = O(n^4)$ walks.
  Note that aggregation (using spanning trees) is a popular technique but
spanning tree(s) are sensitive to node/edge/root failures, hence, we hope our
work points to other more distributed, efficient and robust solutions for
suitable problems.
