  Finding dense subgraphs in a graph is a fundamental graph mining task, with
applications in several fields. Algorithms for identifying dense subgraphs are
used in biology, in finance, in spam detection, etc. Standard formulations of
this problem such as the problem of finding the maximum clique of a graph are
hard to solve. However, some tractable formulations of the problem have also
been proposed, focusing mainly on optimizing some density function, such as the
degree density and the triangle density. However, maximization of degree
density usually leads to large subgraphs with small density. In this paper, we
introduce the k-clique-graph densest subgraph problem, k >= 3, a novel
formulation for the discovery of dense subgraphs. Given an input graph, its
k-clique-graph is a new graph created from the input graph where each vertex of
the new graph corresponds to a k-clique of the input graph and two vertices are
connected with an edge if they share a common (k-1)-clique. We define a simple
density function, the k-clique-graph density, which gives compact and at the
same time dense subgraphs, and we project its resulting subgraphs back to the
input graph. In this paper we focus on the triangle-graph densest subgraph
problem obtained for k=3. To optimize the proposed function, we present an
efficient greedy approximation algorithm that scales well to larger graphs. We
evaluate the proposed algorithm on real datasets and compare it with other
algorithms in terms of the size and the density of the extracted subgraphs. The
results verify the ability of the proposed algorithm in finding high-quality
subgraphs in terms of size and density. Finally, we apply the proposed method
to the important problem of keyword extraction from textual documents.
