89,304 research outputs found
The Critical Mean-Field Chayes-Machta Dynamics
The random-cluster model is a unifying framework for studying random graphs, spin systems and electrical networks that plays a fundamental role in designing efficient Markov Chain Monte Carlo (MCMC) sampling algorithms for the classical ferromagnetic Ising and Potts models. In this paper, we study a natural non-local Markov chain known as the Chayes-Machta dynamics for the mean-field case of the random-cluster model, where the underlying graph is the complete graph on n vertices. The random-cluster model is parametrized by an edge probability p and a cluster weight q. Our focus is on the critical regime: p = p_c(q) and q ? (1,2), where p_c(q) is the threshold corresponding to the order-disorder phase transition of the model. We show that the mixing time of the Chayes-Machta dynamics is O(log n ? log log n) in this parameter regime, which reveals that the dynamics does not undergo an exponential slowdown at criticality, a surprising fact that had been predicted (but not proved) by statistical physicists. This also provides a nearly optimal bound (up to the log log n factor) for the mixing time of the mean-field Chayes-Machta dynamics in the only regime of parameters where no non-trivial bound was previously known. Our proof consists of a multi-phased coupling argument that combines several key ingredients, including a new local limit theorem, a precise bound on the maximum of symmetric random walks with varying step sizes, and tailored estimates for critical random graphs. In addition, we derive an improved comparison inequality between the mixing time of the Chayes-Machta dynamics and that of the local Glauber dynamics on general graphs; this results in better mixing time bounds for the local dynamics in the mean-field setting
Green's Functions from Quantum Cluster Algorithms
We show that cluster algorithms for quantum models have a meaning independent
of the basis chosen to construct them. Using this idea, we propose a new method
for measuring with little effort a whole class of Green's functions, once a
cluster algorithm for the partition function has been constructed. To explain
the idea, we consider the quantum XY model and compute its two point Green's
function in various ways, showing that all of them are equivalent. We also
provide numerical evidence confirming the analytic arguments. Similar
techniques are applicable to other models. In particular, in the recently
constructed quantum link models, the new technique allows us to construct
improved estimators for Wilson loops and may lead to a very precise
determination of the glueball spectrum.Comment: 15 pages, LaTeX, with four figures. Added preprint numbe
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
Multi-level algorithms for modularity clustering
Modularity is one of the most widely used quality measures for graph
clusterings. Maximizing modularity is NP-hard, and the runtime of exact
algorithms is prohibitive for large graphs. A simple and effective class of
heuristics coarsens the graph by iteratively merging clusters (starting from
singletons), and optionally refines the resulting clustering by iteratively
moving individual vertices between clusters. Several heuristics of this type
have been proposed in the literature, but little is known about their relative
performance.
This paper experimentally compares existing and new coarsening- and
refinement-based heuristics with respect to their effectiveness (achieved
modularity) and efficiency (runtime). Concerning coarsening, it turns out that
the most widely used criterion for merging clusters (modularity increase) is
outperformed by other simple criteria, and that a recent algorithm by Schuetz
and Caflisch is no improvement over simple greedy coarsening for these
criteria. Concerning refinement, a new multi-level algorithm is shown to
produce significantly better clusterings than conventional single-level
algorithms. A comparison with published benchmark results and algorithm
implementations shows that combinations of coarsening and multi-level
refinement are competitive with the best algorithms in the literature.Comment: 12 pages, 10 figures, see
http://www.informatik.tu-cottbus.de/~rrotta/ for downloading the graph
clustering softwar
The Energy Complexity of Broadcast
Energy is often the most constrained resource in networks of battery-powered
devices, and as devices become smaller, they spend a larger fraction of their
energy on communication (transceiver usage) not computation. As an imperfect
proxy for true energy usage, we define energy complexity to be the number of
time slots a device transmits/listens; idle time and computation are free.
In this paper we investigate the energy complexity of fundamental
communication primitives such as broadcast in multi-hop radio networks. We
consider models with collision detection (CD) and without (No-CD), as well as
both randomized and deterministic algorithms. Some take-away messages from this
work include:
1. The energy complexity of broadcast in a multi-hop network is intimately
connected to the time complexity of leader election in a single-hop (clique)
network. Many existing lower bounds on time complexity immediately transfer to
energy complexity. For example, in the CD and No-CD models, we need
and energy, respectively.
2. The energy lower bounds above can almost be achieved, given sufficient
() time. In the CD and No-CD models we can solve broadcast using
energy and energy,
respectively.
3. The complexity measures of Energy and Time are in conflict, and it is an
open problem whether both can be minimized simultaneously. We give a tradeoff
showing it is possible to be nearly optimal in both measures simultaneously.
For any constant , broadcast can be solved in
time with
energy, where is the diameter of the network
Algorithmic and Statistical Perspectives on Large-Scale Data Analysis
In recent years, ideas from statistics and scientific computing have begun to
interact in increasingly sophisticated and fruitful ways with ideas from
computer science and the theory of algorithms to aid in the development of
improved worst-case algorithms that are useful for large-scale scientific and
Internet data analysis problems. In this chapter, I will describe two recent
examples---one having to do with selecting good columns or features from a (DNA
Single Nucleotide Polymorphism) data matrix, and the other having to do with
selecting good clusters or communities from a data graph (representing a social
or information network)---that drew on ideas from both areas and that may serve
as a model for exploiting complementary algorithmic and statistical
perspectives in order to solve applied large-scale data analysis problems.Comment: 33 pages. To appear in Uwe Naumann and Olaf Schenk, editors,
"Combinatorial Scientific Computing," Chapman and Hall/CRC Press, 201
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