31,016 research outputs found
Approximating the Spectrum of a Graph
The spectrum of a network or graph with adjacency matrix ,
consists of the eigenvalues of the normalized Laplacian . This set of eigenvalues encapsulates many aspects of the structure
of the graph, including the extent to which the graph posses community
structures at multiple scales. We study the problem of approximating the
spectrum , of in the regime where the graph is too
large to explicitly calculate the spectrum. We present a sublinear time
algorithm that, given the ability to query a random node in the graph and
select a random neighbor of a given node, computes a succinct representation of
an approximation , such that . Our algorithm has query complexity and running time ,
independent of the size of the graph, . We demonstrate the practical
viability of our algorithm on 15 different real-world graphs from the Stanford
Large Network Dataset Collection, including social networks, academic
collaboration graphs, and road networks. For the smallest of these graphs, we
are able to validate the accuracy of our algorithm by explicitly calculating
the true spectrum; for the larger graphs, such a calculation is computationally
prohibitive.
In addition we study the implications of our algorithm to property testing in
the bounded degree graph model
Finding Non-overlapping Clusters for Generalized Inference Over Graphical Models
Graphical models use graphs to compactly capture stochastic dependencies
amongst a collection of random variables. Inference over graphical models
corresponds to finding marginal probability distributions given joint
probability distributions. In general, this is computationally intractable,
which has led to a quest for finding efficient approximate inference
algorithms. We propose a framework for generalized inference over graphical
models that can be used as a wrapper for improving the estimates of approximate
inference algorithms. Instead of applying an inference algorithm to the
original graph, we apply the inference algorithm to a block-graph, defined as a
graph in which the nodes are non-overlapping clusters of nodes from the
original graph. This results in marginal estimates of a cluster of nodes, which
we further marginalize to get the marginal estimates of each node. Our proposed
block-graph construction algorithm is simple, efficient, and motivated by the
observation that approximate inference is more accurate on graphs with longer
cycles. We present extensive numerical simulations that illustrate our
block-graph framework with a variety of inference algorithms (e.g., those in
the libDAI software package). These simulations show the improvements provided
by our framework.Comment: Extended the previous version to include extensive numerical
simulations. See http://www.ima.umn.edu/~dvats/GeneralizedInference.html for
code and dat
Subsampled Power Iteration: a Unified Algorithm for Block Models and Planted CSP's
We present an algorithm for recovering planted solutions in two well-known
models, the stochastic block model and planted constraint satisfaction
problems, via a common generalization in terms of random bipartite graphs. Our
algorithm matches up to a constant factor the best-known bounds for the number
of edges (or constraints) needed for perfect recovery and its running time is
linear in the number of edges used. The time complexity is significantly better
than both spectral and SDP-based approaches.
The main contribution of the algorithm is in the case of unequal sizes in the
bipartition (corresponding to odd uniformity in the CSP). Here our algorithm
succeeds at a significantly lower density than the spectral approaches,
surpassing a barrier based on the spectral norm of a random matrix.
Other significant features of the algorithm and analysis include (i) the
critical use of power iteration with subsampling, which might be of independent
interest; its analysis requires keeping track of multiple norms of an evolving
solution (ii) it can be implemented statistically, i.e., with very limited
access to the input distribution (iii) the algorithm is extremely simple to
implement and runs in linear time, and thus is practical even for very large
instances
Partitioning and modularity of graphs with arbitrary degree distribution
We solve the graph bi-partitioning problem in dense graphs with arbitrary
degree distribution using the replica method. We find the cut-size to scale
universally with . In contrast, earlier results studying the problem in
graphs with a Poissonian degree distribution had found a scaling with ^1/2
[Fu and Anderson, J. Phys. A: Math. Gen. 19, 1986]. The new results also
generalize to the problem of q-partitioning. They can be used to find the
expected modularity Q [Newman and Grivan, Phys. Rev. E, 69, 2004] of random
graphs and allow for the assessment of statistical significance of the output
of community detection algorithms.Comment: Revised version including new plots and improved discussion of some
mathematical detail
Towards a property graph generator for benchmarking
The use of synthetic graph generators is a common practice among
graph-oriented benchmark designers, as it allows obtaining graphs with the
required scale and characteristics. However, finding a graph generator that
accurately fits the needs of a given benchmark is very difficult, thus
practitioners end up creating ad-hoc ones. Such a task is usually
time-consuming, and often leads to reinventing the wheel. In this paper, we
introduce the conceptual design of DataSynth, a framework for property graphs
generation with customizable schemas and characteristics. The goal of DataSynth
is to assist benchmark designers in generating graphs efficiently and at scale,
saving from implementing their own generators. Additionally, DataSynth
introduces novel features barely explored so far, such as modeling the
correlation between properties and the structure of the graph. This is achieved
by a novel property-to-node matching algorithm for which we present preliminary
promising results
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