34 research outputs found
Wang-Landau Algorithm: a Theoretical Analysis of the Saturation of the Error
In this work we present a theoretical analysis of the convergence of the
Wang-Landau algorithm [Phys. Rev. Lett. 86, 2050 (2001)] which was introduced
years ago to calculate the density of states in statistical models. We study
the dynamical behavior of the error in the calculation of the density of
states.We conclude that the source of the saturation of the error is due to the
decreasing variations of the refinement parameter. To overcome this limitation,
we present an analytical treatment in which the refinement parameter is scaled
down as a power law instead of exponentially. An extension of the analysis to
the N-fold way variation of the method is also discussed.Comment: 7 pages, 5 figure
Performance Limitations of Flat Histogram Methods and Optimality of Wang-Landau Sampling
We determine the optimal scaling of local-update flat-histogram methods with
system size by using a perfect flat-histogram scheme based on the exact density
of states of 2D Ising models.The typical tunneling time needed to sample the
entire bandwidth does not scale with the number of spins N as the minimal N^2
of an unbiased random walk in energy space. While the scaling is power law for
the ferromagnetic and fully frustrated Ising model, for the +/- J
nearest-neighbor spin glass the distribution of tunneling times is governed by
a fat-tailed Frechet extremal value distribution that obeys exponential
scaling. We find that the Wang-Landau algorithm shows the same scaling as the
perfect scheme and is thus optimal.Comment: 5 pages, 6 figure
Efficiency of the Wang-Landau algorithm: a simple test case
We analyze the efficiency of the Wang-Landau algorithm to sample a multimodal
distribution on a prototypical simple test case. We show that the exit time
from a metastable state is much smaller for the Wang Landau dynamics than for
the original standard Metropolis-Hastings algorithm, in some asymptotic regime.
Our results are confirmed by numerical experiments on a more realistic test
case
Sampling motif-constrained ensembles of networks
The statistical significance of network properties is conditioned on null
models which satisfy spec- ified properties but that are otherwise random.
Exponential random graph models are a principled theoretical framework to
generate such constrained ensembles, but which often fail in practice, either
due to model inconsistency, or due to the impossibility to sample networks from
them. These problems affect the important case of networks with prescribed
clustering coefficient or number of small connected subgraphs (motifs). In this
paper we use the Wang-Landau method to obtain a multicanonical sampling that
overcomes both these problems. We sample, in polynomial time, net- works with
arbitrary degree sequences from ensembles with imposed motifs counts. Applying
this method to social networks, we investigate the relation between
transitivity and homophily, and we quantify the correlation between different
types of motifs, finding that single motifs can explain up to 60% of the
variation of motif profiles.Comment: Updated version, as published in the journal. 7 pages, 5 figures, one
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