  Mining dense subgraphs is an important primitive across a spectrum of
graph-mining tasks. In this work, we formally establish that two recurring
characteristics of real-world graphs, namely heavy-tailed degree distributions
and large clustering coefficients, imply the existence of substantially large
vertex neighborhoods with high edge-density. This observation suggests a very
simple approach for extracting large quasi-cliques: simply scan the vertex
neighborhoods, compute the clustering coefficient of each vertex, and output
the best such subgraph. The implementation of such a method requires counting
the triangles in a graph, which is a well-studied problem in graph mining. When
empirically tested across a number of real-world graphs, this approach reveals
a surprise: vertex neighborhoods include maximal cliques of non-trivial sizes,
and the density of the best neighborhood often compares favorably to subgraphs
produced by dedicated algorithms for maximizing subgraph density. For graphs
with small clustering coefficients, we demonstrate that small vertex
neighborhoods can be refined using a local-search method to ``grow'' larger
cliques and near-cliques. Our results indicate that contrary to worst-case
theoretical results, mining cliques and quasi-cliques of non-trivial sizes from
real-world graphs is often not a difficult problem, and provides motivation for
further work geared towards a better explanation of these empirical successes.
