91,201 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
Counting and Sampling Small Structures in Graph and Hypergraph Data Streams
In this thesis, we explore the problem of approximating the number of elementary substructures called simplices in large k-uniform hypergraphs. The hypergraphs are assumed to be too large to be stored in memory, so we adopt a data stream model, where the hypergraph is defined by a sequence of hyperedges.
First we propose an algorithm that (ε, δ)-estimates the number of simplices using O(m1+1/k / T) bits of space. In addition, we prove that no constant-pass streaming algorithm can (ε, δ)- approximate the number of simplices using less than O( m 1+1/k / T ) bits of space. Thus we resolve the space complexity of the simplex counting problem by providing an algorithm that matches the lower bound.
Second, we examine the triangle counting question –a hypergraph where k = 2. We develop and analyze an almost optimal O (n+m 3/2 / T) triangle-counting algorithm based on ideas introduced in [KMPT12]. The proposed algorithm is subsequently used to establish a method for uniformly sampling triangles in a graph stream using O(m 3/2 / T) bits of space, which beats the state-of-the-art O(mn / T) algorithm given by [PTTW13
A Fast Counting Method for 6-motifs with Low Connectivity
A -motif (or graphlet) is a subgraph on nodes in a graph or network.
Counting of motifs in complex networks has been a well-studied problem in
network analysis of various real-word graphs arising from the study of social
networks and bioinformatics. In particular, the triangle counting problem has
received much attention due to its significance in understanding the behavior
of social networks. Similarly, subgraphs with more than 3 nodes have received
much attention recently. While there have been successful methods developed on
this problem, most of the existing algorithms are not scalable to large
networks with millions of nodes and edges.
The main contribution of this paper is a preliminary study that genaralizes
the exact counting algorithm provided by Pinar, Seshadhri and Vishal to a
collection of 6-motifs. This method uses the counts of motifs with smaller size
to obtain the counts of 6-motifs with low connecivity, that is, containing a
cut-vertex or a cut-edge. Therefore, it circumvents the combinatorial explosion
that naturally arises when counting subgraphs in large networks
On Approximating the Number of -cliques in Sublinear Time
We study the problem of approximating the number of -cliques in a graph
when given query access to the graph.
We consider the standard query model for general graphs via (1) degree
queries, (2) neighbor queries and (3) pair queries. Let denote the number
of vertices in the graph, the number of edges, and the number of
-cliques. We design an algorithm that outputs a
-approximation (with high probability) for , whose
expected query complexity and running time are
O\left(\frac{n}{C_k^{1/k}}+\frac{m^{k/2}}{C_k}\right)\poly(\log
n,1/\varepsilon,k).
Hence, the complexity of the algorithm is sublinear in the size of the graph
for . Furthermore, we prove a lower bound showing that
the query complexity of our algorithm is essentially optimal (up to the
dependence on , and ).
The previous results in this vein are by Feige (SICOMP 06) and by Goldreich
and Ron (RSA 08) for edge counting () and by Eden et al. (FOCS 2015) for
triangle counting (). Our result matches the complexities of these
results.
The previous result by Eden et al. hinges on a certain amortization technique
that works only for triangle counting, and does not generalize for larger
cliques. We obtain a general algorithm that works for any by
designing a procedure that samples each -clique incident to a given set
of vertices with approximately equal probability. The primary difficulty is in
finding cliques incident to purely high-degree vertices, since random sampling
within neighbors has a low success probability. This is achieved by an
algorithm that samples uniform random high degree vertices and a careful
tradeoff between estimating cliques incident purely to high-degree vertices and
those that include a low-degree vertex
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