1,753 research outputs found
Wedge Sampling for Computing Clustering Coefficients and Triangle Counts on Large Graphs
Graphs are used to model interactions in a variety of contexts, and there is
a growing need to quickly assess the structure of such graphs. Some of the most
useful graph metrics are based on triangles, such as those measuring social
cohesion. Algorithms to compute them can be extremely expensive, even for
moderately-sized graphs with only millions of edges. Previous work has
considered node and edge sampling; in contrast, we consider wedge sampling,
which provides faster and more accurate approximations than competing
techniques. Additionally, wedge sampling enables estimation local clustering
coefficients, degree-wise clustering coefficients, uniform triangle sampling,
and directed triangle counts. Our methods come with provable and practical
probabilistic error estimates for all computations. We provide extensive
results that show our methods are both more accurate and faster than
state-of-the-art alternatives.Comment: Full version of SDM 2013 paper "Triadic Measures on Graphs: The Power
of Wedge Sampling" (arxiv:1202.5230
Graph Sample and Hold: A Framework for Big-Graph Analytics
Sampling is a standard approach in big-graph analytics; the goal is to
efficiently estimate the graph properties by consulting a sample of the whole
population. A perfect sample is assumed to mirror every property of the whole
population. Unfortunately, such a perfect sample is hard to collect in complex
populations such as graphs (e.g. web graphs, social networks etc), where an
underlying network connects the units of the population. Therefore, a good
sample will be representative in the sense that graph properties of interest
can be estimated with a known degree of accuracy. While previous work focused
particularly on sampling schemes used to estimate certain graph properties
(e.g. triangle count), much less is known for the case when we need to estimate
various graph properties with the same sampling scheme. In this paper, we
propose a generic stream sampling framework for big-graph analytics, called
Graph Sample and Hold (gSH). To begin, the proposed framework samples from
massive graphs sequentially in a single pass, one edge at a time, while
maintaining a small state. We then show how to produce unbiased estimators for
various graph properties from the sample. Given that the graph analysis
algorithms will run on a sample instead of the whole population, the runtime
complexity of these algorithm is kept under control. Moreover, given that the
estimators of graph properties are unbiased, the approximation error is kept
under control. Finally, we show the performance of the proposed framework (gSH)
on various types of graphs, such as social graphs, among others
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
On Counting Triangles through Edge Sampling in Large Dynamic Graphs
Traditional frameworks for dynamic graphs have relied on processing only the
stream of edges added into or deleted from an evolving graph, but not any
additional related information such as the degrees or neighbor lists of nodes
incident to the edges. In this paper, we propose a new edge sampling framework
for big-graph analytics in dynamic graphs which enhances the traditional model
by enabling the use of additional related information. To demonstrate the
advantages of this framework, we present a new sampling algorithm, called Edge
Sample and Discard (ESD). It generates an unbiased estimate of the total number
of triangles, which can be continuously updated in response to both edge
additions and deletions. We provide a comparative analysis of the performance
of ESD against two current state-of-the-art algorithms in terms of accuracy and
complexity. The results of the experiments performed on real graphs show that,
with the help of the neighborhood information of the sampled edges, the
accuracy achieved by our algorithm is substantially better. We also
characterize the impact of properties of the graph on the performance of our
algorithm by testing on several Barabasi-Albert graphs.Comment: A short version of this article appeared in Proceedings of the 2017
IEEE/ACM International Conference on Advances in Social Networks Analysis and
Mining (ASONAM 2017
Triadic Measures on Graphs: The Power of Wedge Sampling
Graphs are used to model interactions in a variety of contexts, and there is
a growing need to quickly assess the structure of a graph. Some of the most
useful graph metrics, especially those measuring social cohesion, are based on
triangles. Despite the importance of these triadic measures, associated
algorithms can be extremely expensive. We propose a new method based on wedge
sampling. This versatile technique allows for the fast and accurate
approximation of all current variants of clustering coefficients and enables
rapid uniform sampling of the triangles of a graph. Our methods come with
provable and practical time-approximation tradeoffs for all computations. We
provide extensive results that show our methods are orders of magnitude faster
than the state-of-the-art, while providing nearly the accuracy of full
enumeration. Our results will enable more wide-scale adoption of triadic
measures for analysis of extremely large graphs, as demonstrated on several
real-world examples
Estimating Graphlet Statistics via Lifting
Exploratory analysis over network data is often limited by the ability to
efficiently calculate graph statistics, which can provide a model-free
understanding of the macroscopic properties of a network. We introduce a
framework for estimating the graphlet count---the number of occurrences of a
small subgraph motif (e.g. a wedge or a triangle) in the network. For massive
graphs, where accessing the whole graph is not possible, the only viable
algorithms are those that make a limited number of vertex neighborhood queries.
We introduce a Monte Carlo sampling technique for graphlet counts, called {\em
Lifting}, which can simultaneously sample all graphlets of size up to
vertices for arbitrary . This is the first graphlet sampling method that can
provably sample every graphlet with positive probability and can sample
graphlets of arbitrary size . We outline variants of lifted graphlet counts,
including the ordered, unordered, and shotgun estimators, random walk starts,
and parallel vertex starts. We prove that our graphlet count updates are
unbiased for the true graphlet count and have a controlled variance for all
graphlets. We compare the experimental performance of lifted graphlet counts to
the state-of-the art graphlet sampling procedures: Waddling and the pairwise
subgraph random walk
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