  We study the problem of recovering a known cluster structure in a sparse
network, also known as the planted partitioning problem, by means of
statistical mechanics. We find a sharp transition from un-recoverable to
recoverable structure as a function of the separation of the clusters. For
multivariate data, such transitions have been observed frequently, but always
as a function of the number of data points provided, i.e. given a large enough
data set, two point clouds can always be recognized as different clusters, as
long as their separation is non-zero. In contrast, for the sparse networks
studied here, a cluster structure remains undetectable even in an infinitely
large network if a critical separation is not exceeded. We give analytic
formulas for this critical separation as a function of the degree distribution
of the network and calculate the shape of the recoverability-transition. Our
findings have implications for unsupervised learning and data-mining in
relational data bases and provide bounds on the achievable performance of graph
clustering algorithms.
