# Discovering Nested Communities

## TL;DR

This paper introduces a method for discovering nested communities in graphs, addressing the challenge of ambiguous community structures by finding a sequence of increasingly dense communities containing a starting set.

## Contribution

It proposes a novel approach to identify nested communities, dividing the problem into ordering and community detection, with empirical and theoretical validation of the heuristic used.

## Key findings

- Efficient algorithm for fixed vertex order
- Heuristic for ordering shows good empirical performance
- Theoretical support for the ordering heuristic

## Abstract

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.