  We study the problem of learning communities in the presence of modeling
errors and give robust recovery algorithms for the Stochastic Block Model
(SBM). This model, which is also known as the Planted Partition Model, is
widely used for community detection and graph partitioning in various fields,
including machine learning, statistics, and social sciences. Many algorithms
exist for learning communities in the Stochastic Block Model, but they do not
work well in the presence of errors.
  In this paper, we initiate the study of robust algorithms for partial
recovery in SBM with modeling errors or noise. We consider graphs generated
according to the Stochastic Block Model and then modified by an adversary. We
allow two types of adversarial errors, Feige---Kilian or monotone errors, and
edge outlier errors. Mossel, Neeman and Sly (STOC 2015) posed an open question
about whether an almost exact recovery is possible when the adversary is
allowed to add $o(n)$ edges. Our work answers this question affirmatively even
in the case of $k>2$ communities.
  We then show that our algorithms work not only when the instances come from
SBM, but also work when the instances come from any distribution of graphs that
is $\epsilon m$ close to SBM in the Kullback---Leibler divergence. This result
also works in the presence of adversarial errors. Finally, we present almost
tight lower bounds for two communities.
