Section: New Results
Recovering Asymmetric Communities in the Stochastic Block Model
In [22],
we consider the sparse stochastic block model in the case where the
degrees are uninformative. The case where the two communities have
approximately the same size has been extensively studied and we
concentrate here on the community detection problem in the case of
unbalanced communities. In this setting, spectral algorithms based
on the non-backtracking matrix are known to solve the community
detection problem (i.e. do strictly better than a random guess) when
the signal is sufficiently large namely above the so-called Kesten
Stigum threshold. In this regime and when the average degree tends
to infinity, we show that if the community of a vanishing fraction
of the vertices is revealed, then a local algorithm (belief
propagation) is optimal down to Kesten Stigum threshold and we
quantify explicitly its performance. Below the Kesten Stigum
threshold, we show that, in the large degree limit, there is a
second threshold called the spinodal curve below which, the
community detection problem is not solvable. The spinodal curve is
equal to the Kesten Stigum threshold when the fraction of vertices
in the smallest community is above