38,909 research outputs found
Efficient and exact sampling of simple graphs with given arbitrary degree sequence
Uniform sampling from graphical realizations of a given degree sequence is a
fundamental component in simulation-based measurements of network observables,
with applications ranging from epidemics, through social networks to Internet
modeling. Existing graph sampling methods are either link-swap based
(Markov-Chain Monte Carlo algorithms) or stub-matching based (the Configuration
Model). Both types are ill-controlled, with typically unknown mixing times for
link-swap methods and uncontrolled rejections for the Configuration Model. Here
we propose an efficient, polynomial time algorithm that generates statistically
independent graph samples with a given, arbitrary, degree sequence. The
algorithm provides a weight associated with each sample, allowing the
observable to be measured either uniformly over the graph ensemble, or,
alternatively, with a desired distribution. Unlike other algorithms, this
method always produces a sample, without back-tracking or rejections. Using a
central limit theorem-based reasoning, we argue, that for large N, and for
degree sequences admitting many realizations, the sample weights are expected
to have a lognormal distribution. As examples, we apply our algorithm to
generate networks with degree sequences drawn from power-law distributions and
from binomial distributions.Comment: 8 pages, 3 figure
2.5K-Graphs: from Sampling to Generation
Understanding network structure and having access to realistic graphs plays a
central role in computer and social networks research. In this paper, we
propose a complete, and practical methodology for generating graphs that
resemble a real graph of interest. The metrics of the original topology we
target to match are the joint degree distribution (JDD) and the
degree-dependent average clustering coefficient (). We start by
developing efficient estimators for these two metrics based on a node sample
collected via either independence sampling or random walks. Then, we process
the output of the estimators to ensure that the target properties are
realizable. Finally, we propose an efficient algorithm for generating
topologies that have the exact target JDD and a close to the
target. Extensive simulations using real-life graphs show that the graphs
generated by our methodology are similar to the original graph with respect to,
not only the two target metrics, but also a wide range of other topological
metrics; furthermore, our generator is order of magnitudes faster than
state-of-the-art techniques
Fast counting with tensor networks
We introduce tensor network contraction algorithms for counting satisfying
assignments of constraint satisfaction problems (#CSPs). We represent each
arbitrary #CSP formula as a tensor network, whose full contraction yields the
number of satisfying assignments of that formula, and use graph theoretical
methods to determine favorable orders of contraction. We employ our heuristics
for the solution of #P-hard counting boolean satisfiability (#SAT) problems,
namely monotone #1-in-3SAT and #Cubic-Vertex-Cover, and find that they
outperform state-of-the-art solvers by a significant margin.Comment: v2: added results for monotone #1-in-3SAT; published versio
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
FLEET: Butterfly Estimation from a Bipartite Graph Stream
We consider space-efficient single-pass estimation of the number of
butterflies, a fundamental bipartite graph motif, from a massive bipartite
graph stream where each edge represents a connection between entities in two
different partitions. We present a space lower bound for any streaming
algorithm that can estimate the number of butterflies accurately, as well as
FLEET, a suite of algorithms for accurately estimating the number of
butterflies in the graph stream. Estimates returned by the algorithms come with
provable guarantees on the approximation error, and experiments show good
tradeoffs between the space used and the accuracy of approximation. We also
present space-efficient algorithms for estimating the number of butterflies
within a sliding window of the most recent elements in the stream. While there
is a significant body of work on counting subgraphs such as triangles in a
unipartite graph stream, our work seems to be one of the few to tackle the case
of bipartite graph streams.Comment: This is the author's version of the work. It is posted here by
permission of ACM for your personal use. Not for redistribution. The
definitive version was published in Seyed-Vahid Sanei-Mehri, Yu Zhang, Ahmet
Erdem Sariyuce and Srikanta Tirthapura. "FLEET: Butterfly Estimation from a
Bipartite Graph Stream". The 28th ACM International Conference on Information
and Knowledge Managemen
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