  Identifying communities (or clusters), namely groups of nodes with
comparatively strong internal connectivity, is a fundamental task for deeply
understanding the structure and function of a network. Yet, there is a lack of
formal criteria for defining communities and for testing their significance. We
propose a sharp definition which is based on a significance threshold. By means
of a lumped Markov chain model of a random walker, a quality measure called
"persistence probability" is associated to a cluster. Then the cluster is
defined as an "$\alpha$-community" if such a probability is not smaller than
$\alpha$. Consistently, a partition composed of $\alpha$-communities is an
"$\alpha$-partition". These definitions turn out to be very effective for
finding and testing communities. If a set of candidate partitions is available,
setting the desired $\alpha$-level allows one to immediately select the
$\alpha$-partition with the finest decomposition. Simultaneously, the
persistence probabilities quantify the significance of each single community.
Given its ability in individually assessing the quality of each cluster, this
approach can also disclose single well-defined communities even in networks
which overall do not possess a definite clusterized structure.
