5,592 research outputs found
Belief propagation for optimal edge cover in the random complete graph
We apply the objective method of Aldous to the problem of finding the
minimum-cost edge cover of the complete graph with random independent and
identically distributed edge costs. The limit, as the number of vertices goes
to infinity, of the expected minimum cost for this problem is known via a
combinatorial approach of Hessler and W\"{a}stlund. We provide a proof of this
result using the machinery of the objective method and local weak convergence,
which was used to prove the limit of the random assignment problem.
A proof via the objective method is useful because it provides us with more
information on the nature of the edge's incident on a typical root in the
minimum-cost edge cover. We further show that a belief propagation algorithm
converges asymptotically to the optimal solution. This can be applied in a
computational linguistics problem of semantic projection. The belief
propagation algorithm yields a near optimal solution with lesser complexity
than the known best algorithms designed for optimality in worst-case settings.Comment: Published in at http://dx.doi.org/10.1214/13-AAP981 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Spectral redemption: clustering sparse networks
Spectral algorithms are classic approaches to clustering and community
detection in networks. However, for sparse networks the standard versions of
these algorithms are suboptimal, in some cases completely failing to detect
communities even when other algorithms such as belief propagation can do so.
Here we introduce a new class of spectral algorithms based on a
non-backtracking walk on the directed edges of the graph. The spectrum of this
operator is much better-behaved than that of the adjacency matrix or other
commonly used matrices, maintaining a strong separation between the bulk
eigenvalues and the eigenvalues relevant to community structure even in the
sparse case. We show that our algorithm is optimal for graphs generated by the
stochastic block model, detecting communities all the way down to the
theoretical limit. We also show the spectrum of the non-backtracking operator
for some real-world networks, illustrating its advantages over traditional
spectral clustering.Comment: 11 pages, 6 figures. Clarified to what extent our claims are
rigorous, and to what extent they are conjectures; also added an
interpretation of the eigenvectors of the 2n-dimensional version of the
non-backtracking matri
Scaling hypothesis for the Euclidean bipartite matching problem
We propose a simple yet very predictive form, based on a Poisson's equation,
for the functional dependence of the cost from the density of points in the
Euclidean bipartite matching problem. This leads, for quadratic costs, to the
analytic prediction of the large limit of the average cost in dimension
and of the subleading correction in higher dimension. A non-trivial
scaling exponent, , which differs from the
monopartite's one, is found for the subleading correction. We argue that the
same scaling holds true for a generic cost exponent in dimension .Comment: 11 page
On the number of -cycles in the assignment problem for random matrices
We continue the study of the assignment problem for a random cost matrix. We
analyse the number of -cycles for the solution and their dependence on the
symmetry of the random matrix. We observe that for a symmetric matrix one and
two-cycles are dominant in the optimal solution. In the antisymmetric case the
situation is the opposite and the one and two-cycles are suppressed. We solve
the model for a pure random matrix (without correlations between its entries)
and give analytic arguments to explain the numerical results in the symmetric
and antisymmetric case. We show that the results can be explained to great
accuracy by a simple ansatz that connects the expected number of -cycles to
that of one and two cycles.Comment: To appear in Journal of Statistical Mechanic
Phase transition in the Countdown problem
Here we present a combinatorial decision problem, inspired by the celebrated
quiz show called the countdown, that involves the computation of a given target
number T from a set of k randomly chosen integers along with a set of
arithmetic operations. We find that the probability of winning the game
evidences a threshold phenomenon that can be understood in the terms of an
algorithmic phase transition as a function of the set size k. Numerical
simulations show that such probability sharply transitions from zero to one at
some critical value of the control parameter, hence separating the algorithm's
parameter space in different phases. We also find that the system is maximally
efficient close to the critical point. We then derive analytical expressions
that match the numerical results for finite size and permit us to extrapolate
the behavior in the thermodynamic limit.Comment: Submitted for publicatio
Spectral Detection on Sparse Hypergraphs
We consider the problem of the assignment of nodes into communities from a
set of hyperedges, where every hyperedge is a noisy observation of the
community assignment of the adjacent nodes. We focus in particular on the
sparse regime where the number of edges is of the same order as the number of
vertices. We propose a spectral method based on a generalization of the
non-backtracking Hashimoto matrix into hypergraphs. We analyze its performance
on a planted generative model and compare it with other spectral methods and
with Bayesian belief propagation (which was conjectured to be asymptotically
optimal for this model). We conclude that the proposed spectral method detects
communities whenever belief propagation does, while having the important
advantages to be simpler, entirely nonparametric, and to be able to learn the
rule according to which the hyperedges were generated without prior
information.Comment: 8 pages, 5 figure
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