6,979 research outputs found
Beyond Triangles: A Distributed Framework for Estimating 3-profiles of Large Graphs
We study the problem of approximating the -profile of a large graph.
-profiles are generalizations of triangle counts that specify the number of
times a small graph appears as an induced subgraph of a large graph. Our
algorithm uses the novel concept of -profile sparsifiers: sparse graphs that
can be used to approximate the full -profile counts for a given large graph.
Further, we study the problem of estimating local and ego -profiles, two
graph quantities that characterize the local neighborhood of each vertex of a
graph.
Our algorithm is distributed and operates as a vertex program over the
GraphLab PowerGraph framework. We introduce the concept of edge pivoting which
allows us to collect -hop information without maintaining an explicit
-hop neighborhood list at each vertex. This enables the computation of all
the local -profiles in parallel with minimal communication.
We test out implementation in several experiments scaling up to cores
on Amazon EC2. We find that our algorithm can estimate the -profile of a
graph in approximately the same time as triangle counting. For the harder
problem of ego -profiles, we introduce an algorithm that can estimate
profiles of hundreds of thousands of vertices in parallel, in the timescale of
minutes.Comment: To appear in part at KDD'1
Proof of a conjecture on induced subgraphs of Ramsey graphs
An n-vertex graph is called C-Ramsey if it has no clique or independent set
of size C log n. All known constructions of Ramsey graphs involve randomness in
an essential way, and there is an ongoing line of research towards showing that
in fact all Ramsey graphs must obey certain "richness" properties
characteristic of random graphs. More than 25 years ago, Erd\H{o}s, Faudree and
S\'{o}s conjectured that in any C-Ramsey graph there are
induced subgraphs, no pair of which have the same
numbers of vertices and edges. Improving on earlier results of Alon, Balogh,
Kostochka and Samotij, in this paper we prove this conjecture
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