  Community detection is considered for a stochastic block model graph of n
vertices, with K vertices in the planted community, edge probability p for
pairs of vertices both in the community, and edge probability q for other pairs
of vertices.
  The main focus of the paper is on weak recovery of the community based on the
graph G, with o(K) misclassified vertices on average, in the sublinear regime
$n^{1-o(1)} \leq K \leq o(n).$ A critical parameter is the effective
signal-to-noise ratio $\lambda=K^2(p-q)^2/((n-K)q)$, with $\lambda=1$
corresponding to the Kesten-Stigum threshold. We show that a belief propagation
algorithm achieves weak recovery if $\lambda>1/e$, beyond the Kesten-Stigum
threshold by a factor of $1/e.$ The belief propagation algorithm only needs to
run for $\log^\ast n+O(1) $ iterations, with the total time complexity $O(|E|
\log^*n)$, where $\log^*n$ is the iterated logarithm of $n.$ Conversely, if
$\lambda \leq 1/e$, no local algorithm can asymptotically outperform trivial
random guessing. Furthermore, a linear message-passing algorithm that
corresponds to applying power iteration to the non-backtracking matrix of the
graph is shown to attain weak recovery if and only if $\lambda>1$. In addition,
the belief propagation algorithm can be combined with a linear-time voting
procedure to achieve the information limit of exact recovery (correctly
classify all vertices with high probability) for all $K \ge \frac{n}{\log n}
\left( \rho_{\rm BP} +o(1) \right),$ where $\rho_{\rm BP}$ is a function of
$p/q$.
