  Finding communities in graphs is one of the most well-studied problems in
data mining and social-network analysis. In many real applications, the
underlying graph does not have a clear community structure. In those cases,
selecting a single community turns out to be a fairly ill-posed problem, as the
optimization criterion has to make a difficult choice between selecting a tight
but small community or a more inclusive but sparser community.
  In order to avoid the problem of selecting only a single community we propose
discovering a sequence of nested communities. More formally, given a graph and
a starting set, our goal is to discover a sequence of communities all
containing the starting set, and each community forming a denser subgraph than
the next. Discovering an optimal sequence of communities is a complex
optimization problem, and hence we divide it into two subproblems: 1) discover
the optimal sequence for a fixed order of graph vertices, a subproblem that we
can solve efficiently, and 2) find a good order. We employ a simple heuristic
for discovering an order and we provide empirical and theoretical evidence that
our order is good.
