scholarly journals Half-Graphs, Other Non-stable Degree Sequences, and the Switch Markov Chain

10.37236/9652 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Péter L. Erdős ◽  
Ervin Győri ◽  
Tamás Róbert Mezei ◽  
István Miklós ◽  
Dániel Soltész
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Rate Of Convergence ◽  
Uniform Distribution ◽  
Directed Graphs ◽  
Degree Sequence ◽  
Strong Stability ◽  
Degree Sequences ◽  
Markov Chain Method ◽  
Stable Degree

One of the simplest methods of generating a random graph with a given degree sequence is provided by the Monte Carlo Markov Chain method using switches. The switch Markov chain converges to the uniform distribution, but generally the rate of convergence is not known. After a number of results concerning various degree sequences, rapid mixing was established for so-called P-stable degree sequences (including that of directed graphs), which covers every previously known rapidly mixing region of degree sequences. In this paper we give a non-trivial family of degree sequences that are not P-stable and the switch Markov chain is still rapidly mixing on them. This family has an intimate connection to Tyshkevich-decompositions and strong stability as well.

scholarly journals New Classes of Degree Sequences with Fast Mixing Swap Markov Chain Sampling

2017 ◽  
Vol 27 (2) ◽  
pp. 186-207
Author(s):  
PÉTER L. ERDŐS ◽  
ISTVÁN MIKLÓS ◽  
ZOLTÁN TOROCZKAI
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Directed Graphs ◽  
Degree Sequence ◽  
Degree Sequences ◽  
Network Modelling ◽  
Split Graph ◽  
Simple Graphs ◽  
Large Classes ◽  
Special Degree

In network modelling of complex systems one is often required to sample random realizations of networks that obey a given set of constraints, usually in the form of graph measures. A much studied class of problems targets uniform sampling of simple graphs with given degree sequence or also with given degree correlations expressed in the form of a Joint Degree Matrix. One approach is to use Markov chains based on edge switches (swaps) that preserve the constraints, are irreducible (ergodic) and fast mixing. In 1999, Kannan, Tetali and Vempala (KTV) proposed a simple swap Markov chain for sampling graphs with given degree sequence, and conjectured that it mixes rapidly (in polynomial time) for arbitrary degree sequences. Although the conjecture is still open, it has been proved for special degree sequences, in particular for those of undirected and directed regular simple graphs, half-regular bipartite graphs, and graphs with certain bounded maximum degrees. Here we prove the fast mixing KTV conjecture for novel, exponentially large classes of irregular degree sequences. Our method is based on a canonical decomposition of degree sequences into split graph degree sequences, a structural theorem for the space of graph realizations and on a factorization theorem for Markov chains. After introducing bipartite ‘splitted’ degree sequences, we also generalize the canonical split graph decomposition for bipartite and directed graphs.

scholarly journals A Polynomial Bound on the Mixing Time of a Markov Chain for Sampling Regular Directed Graphs

10.37236/721 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Catherine Greenhill
Keyword(s):  
Markov Chain ◽  
Directed Graphs ◽  
Mixing Time ◽  
Degree Sequence ◽  
Negative Eigenvalue ◽  
Multicommodity Flow ◽  
Degree Sequences ◽  
Regular Graphs ◽  
Significant Extension ◽  
Second Largest Eigenvalue

The switch chain is a well-known Markov chain for sampling directed graphs with a given degree sequence. While not ergodic in general, we show that it is ergodic for regular degree sequences. We then prove that the switch chain is rapidly mixing for regular directed graphs of degree $d$, where $d$ is any positive integer-valued function of the number of vertices. We bound the mixing time by bounding the eigenvalues of the chain. A new result is presented and applied to bound the smallest (most negative) eigenvalue. This result is a modification of a lemma by Diaconis and Stroock [Annals of Applied Probability 1991], and by using it we avoid working with a lazy chain. A multicommodity flow argument is used to bound the second-largest eigenvalue of the chain. This argument is based on the analysis of a related Markov chain for undirected regular graphs by Cooper, Dyer and Greenhill [Combinatorics, Probability and Computing 2007], but with significant extension required.

scholarly journals Rapid mixing of the switch Markov chain for strongly stable degree sequences

2020 ◽  
Vol 57 (3) ◽  
pp. 637-657
Author(s):  
Georgios Amanatidis ◽  
Pieter Kleer
Keyword(s):  
Markov Chain ◽  
Degree Sequences ◽  
Rapid Mixing ◽  
Stable Degree

Sparse Code Multiple Access Decoding Based on a Monte Carlo Markov Chain Method

2016 ◽  
Vol 23 (5) ◽  
pp. 639-643 ◽  
Author(s):  
Jienan Chen ◽  
Zhenbing Zhang ◽  
Shuaining He ◽  
Jianhao Hu ◽  
Gerald E. Sobelman
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Multiple Access ◽  
Monte Carlo Markov Chain ◽  
Sparse Code ◽  
Markov Chain Method

A Bayesian Monte Carlo Markov Chain Method for Loss Models and Risk Measure Assessments

2009 ◽  
Vol 12 (03) ◽  
pp. 529-543
Author(s):  
Ling Hu ◽  
Yating Yang
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Normal Distribution ◽  
Risk Measure ◽  
Markov Chain Method ◽  
Flood Loss ◽  
Bayesian Monte Carlo ◽  
Loss Models ◽  
Log Normal ◽  
Log Normal Distribution

Natural disasters are also known as catastrophes with low frequency but high damages. Typhoons and floods are the major catastrophes which lead to gargantuan losses in Asia. Once a disaster occurs, a broad region will be affected and this will result in huge social loss. If issuers or governments use the wrong loss models or risk measure indexes to price the related insurance products, they will get an inaccurate price and thus be insolvent to the claims. Previous researches often use a Log-Normal distribution to model a catastrophic loss. This is not appropriate since the characteristics of a loss distribution have some empirical facts, including the positive skewness and the heavy-tailed properties. Recently, some studies (McNeil and Frey, 2000; Rootzen and Tajvidi, 2000; Thuring et al., 2008) also point out that using Log-Normal distribution to model a characteristic loss is not suitable. Therefore, we build a typhoon and flood loss model with higher order moments and estimate the parameters through a Bayesian Monte Carlo Markov Chain method. According to the Kolmogorov-Smirnov test, we find that the Pareto distribution is more adaptive for modeling the loss of typhoon and flood. Further, we evaluate different kinds of risk measure indexes through simulating and numerical analysis. It gives the beacon to issuers or governments when they want to issue the insurance products about typhoon and flood loss.

scholarly journals Markov Chain Investigation of Discretization Schemes and Computational Cost Reduction in Modeling Photon Multiple Scattering

2018 ◽  
Vol 8 (11) ◽  
pp. 2288 ◽  
Author(s):  
Shangze Yang ◽  
Di Xiao ◽  
Xuesong Li ◽  
Zhen Ma
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Multiple Scattering ◽  
Cost Reduction ◽  
Computational Cost ◽  
Detailed Comparison ◽  
Iterative Approach ◽  
Computational Costs ◽  
Markov Chain Method ◽  
Discretization Schemes

Establishing fast and reversible photon multiple scattering algorithms remains a modeling challenge for optical diagnostics and noise reduction purposes, especially when the scattering happens within the intermediate scattering regime. Previous work has proposed and verified a Markov chain approach for modeling photon multiple scattering phenomena through turbid slabs. The fidelity of the Markov chain method has been verified through detailed comparison with Monte Carlo models. However, further improvement to the Markov chain method is still required to improve its performance in studying multiple scattering. The present research discussed the efficacy of non-uniform discretization schemes and analyzed errors induced by different schemes. The current work also proposed an iterative approach as an alternative to directly carrying out matrix inversion manipulations, which would significantly reduce the computational costs. The benefits of utilizing non-uniform discretization schemes and the iterative approach were confirmed and verified by comparing the results to a Monte Carlo simulation.

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