Markov Chain Switching Research Articles

Approximate Sampling of Graphs with Near-P-Stable Degree Intervals

The approximate uniform sampling of graph realizations with a given degree sequence is an everyday task in several social science, computer science, engineering etc. projects. One approach is using Markov chains. The best available current result about the well-studied switch Markov chain is that it is rapidly mixing on P-stable degree sequences (see DOI:10.1016/j.ejc.2021.103421). The switch Markov chain does not change any degree sequence. However, there are cases where degree intervals are specified rather than a single degree sequence. (A natural scenario where this problem arises is in hypothesis testing on social networks that are only partially observed.) Rechner, Strowick, and Müller–Hannemann introduced in 2018 the notion of degree interval Markov chain which uses three (separately well studied) local operations (switch, hinge-flip and toggle), and employing on degree sequence realizations where any two sequences under scrutiny have very small coordinate-wise distance. Recently, Amanatidis and Kleer published a beautiful paper (DOI:10.4230/LIPIcs.STACS.2023.7), showing that the degree interval Markov chain is rapidly mixing if the sequences are coming from a system of very thin intervals which are centered not far from a regular degree sequence. In this paper, we substantially extend their result, showing that the degree interval Markov chain is rapidly mixing if the intervals are centered at P-stable degree sequences.

S&P 500 volatility, volatility regimes, and economic uncertainty

AbstractWe assess the relationship between regime‐dependent volatility in S&P 500, economic policy uncertainty, the S&P 500 bull and bear sentiment spread (bb_sp), as well as the Chicago Board Options Exchange's VIX over the period 2000–2018. Our findings from two‐covariate GARCH–MIDAS (GM) methodology, regime switching Markov Chain, and quantile regressions suggest that the association of realized volatility and sentiment varies across high‐ and low‐volatility regimes and depends on investors’ sensitivity toward incidents of market uncertainties under these regimes. The findings suggest that these indicators may not be useful in volatility forecasting, especially under high‐volatility regimes.

Joint nonlinear-drift-driven Wiener process-Markov chain degradation switching model for adaptive online predicting lithium-ion battery remaining useful life

The accurate prediction of the remaining useful life (RUL) of lithium-ion batteries is very important for battery management systems and predictive maintenance. However, lithium-ion batteries have a high degree of internal nonlinearity. There are two switching states during the operation of batteries operating, while the switching time point is also uncertain. In different switching states and random switching times, various unpredictable phenomena, such as capacity recovery or capacity decline could occur, which renders the accurate prediction of RUL challenging. To address this problem, a method for predicting the RUL was proposed in this work based on the nonlinear-drift-driven Wiener process and the Markov chain switching model. First, the nonlinear-drift-driven Wiener process was used to describe the time-varying battery degradation characteristics. The switching model was then applied to predict the future battery working state. Finally, the fuzzy system was employed to integrate the two by combining the battery degradation characteristics. The online update strategy of the model was simulated and validated, resulting in good adaptability and robustness. Two sets of real-case battery data from the National Aeronautics and Space Administration were also included during the validation process. The proposed method was systematically compared to other models in predicting the RUL of the batteries. From the acquired results, it was demonstrated that the proposed method was superior in predicting the RUL of batteries with improved accuracy and safety.

Parallel global edge switching for the uniform sampling of simple graphs with prescribed degrees

The uniform sampling of simple graphs matching a prescribed degree sequence is an important tool in network science, e.g. to construct graph generators or null-models. Here, the Edge Switching Markov Chain (ES-MC) is a common choice. Given an arbitrary simple graph with the required degree sequence, ES-MC carries out a large number of small changes, called edge switches, to eventually obtain a uniform sample. In practice, reasonably short runs efficiently yield approximate uniform samples.In this work, we study the problem of executing edge switches in parallel. We discuss parallelizations of ES-MC, but find that this approach suffers from complex dependencies between edge switches. For this reason, we propose the Global Edge Switching Markov Chain (G-ES-MC), an ES-MC variant with simpler dependencies.We show that G-ES-MC converges to the uniform distribution and design shared-memory parallel algorithms for ES-MC and G-ES-MC. In an empirical evaluation, we provide evidence that G-ES-MC requires not more switches than ES-MC (and often fewer), and demonstrate the efficiency and scalability of our parallel G-ES-MC implementation.

Rapid Mixing of the Switch Markov Chain for 2-Class Joint Degree Matrices

Rapid Mixing of the Switch Markov Chain for 2-Class Joint Degree Matrices

The mixing time of switch Markov chains: A unified approach

Since 1997 a considerable effort has been spent to study the mixing time of switch Markov chains on the realizations of graphic degree sequences of simple graphs. Several results were proved on rapidly mixing Markov chains on unconstrained, bipartite, and directed sequences, using different mechanisms.The aim of this paper is to unify these approaches. We will illustrate the strength of the unified method by showing that on any P-stable family of unconstrained/bipartite/directed degree sequences the switch Markov chain is rapidly mixing. This is a common generalization of every known result that shows the rapid mixing nature of the switch Markov chain on a region of degree sequences. Among the applications of this general result is an almost uniform sampler for power-law and heavy-tailed degree sequences. Another application shows that the switch Markov chain on the degree sequence of an Erdős-Rényi random graph G(n,p) is asymptotically almost surely rapidly mixing if p is bounded away from 0 and 1 by at least 5lognn−1.

Online to offline social interaction on gaming motivations

PurposeThis research examined the social interactions of online game players based on the proposed motivation model in order to understand the transitions of motivation of online game. The authors also separated samples into four categories to compare the difference of different type of online game players.Design/methodology/approachThis study proposed a motivation model for online game player based on existence–relatedness–growth theory. The authors also analyze the transitions of motivations via first-order and second-order Markov chain switching model to obtain the journey of online to offline socialization.FindingsTeamwork–socialization players preferred to make friends in their online gaming network to socialize. Competition–socialization players were mostly students who played games to compete and socialize and may share experience in online or offline activities. Teamwork–mechanics players purely derived pleasure from gaming and were not motivated by other factors in their gaming activities. Competition–mechanics players may already have friends with other gamers in real life.Research limitations/implicationsMore samples can be added to generate more generalizable findings and the proposed motivation model can be extended by other motivations related to online gaming behavior. The authors proposed a motivation model for online to offline socialization and separated online game players into four categories: teamwork–socialization, competition–socialization, teamwork–mechanics and competition–mechanics. The category of teamwork–socialization may contribute to online to offline socialization area. The category of competition–mechanics may add value to the area of traditional offline socialization. The categories of competition–socialization and teamwork–mechanics may help extant literature understand critical stimulus for online gaming behavior.Practical implicationsThe authors’ findings can help online gaming industry understand the motivation journey of players through transition. Different types of online games may have various online game player's journey that can assist companies in improving the quality of online games. Online game companies can also offer official community to players for further interaction and experience exchange or the platform for offline activities in the physical environment.Originality/valueThis research proposed a novel motivation model to examine online to offline socializing behavior for online game research. The motivations in model were interconnected via the support of literature. The authors also integrated motivations by Markov chain switching model to obtain the transitions of motivational status. It is also the first attempt to analyze first-order and second-order Markov chain switching model for analysis. The authors’ research examined the interconnected relationships among motivations in addition to the influential factors to online gaming behavior from previous research. The results may contribute to extend the understanding of online to offline socialization in online gaming literature.

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

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.

Mixing time of the switch Markov chain and stable degree sequences

The switch chain is a well-studied Markov chain which can be used to sample approximately uniformly from the set Ω(d) of all graphs with a given degree sequence d. Polynomial mixing time (rapid mixing) has been established for the switch chain under various conditions on the degree sequences. Amanatidis and Kleer introduced the notion of strongly stable families of degree sequences, and proved that the switch chain is rapidly mixing for any degree sequence from a strongly stable family. Using a different approach, Erdős et al. recently extended this result to the (possibly larger) class of P-stable degree sequences, introduced by Jerrum and Sinclair in 1990. We define a new notion of stability for a given degree sequence, namely k-stability, and prove that if a degree sequence d is 8-stable then the switch chain on Ω(d) is rapidly mixing. We also provide sufficient conditions for P-stability, strong stability and 8-stability. Using these sufficient conditions, we give the first proof of P-stability for various families of heavy-tailed degree sequences, including power-law degree sequences, and show that the switch chain is rapidly mixing for these families.We further extend these notions and results to directed degree sequences.

Switch-Based Markov Chains for Sampling Hamiltonian Cycles in Dense Graphs

We consider the irreducibility of switch-based Markov chains for the approximate uniform sampling of Hamiltonian cycles in a given undirected dense graph on $n$ vertices. As our main result, we show that every pair of Hamiltonian cycles in a graph with minimum degree at least $n/2+7$ can be transformed into each other by switch operations of size at most 10, implying that the switch Markov chain using switches of size at most 10 is irreducible. As a proof of concept, we also show that this Markov chain is rapidly mixing on dense monotone graphs.

Variational Bayesian identification for bilinear state space models with Markov‐switching time delays

SummaryThis article studies the parameter identification problem for bilinear state space models with time‐varying time delays. Considering the correlation of time delays, the Markov chain switching mechanism is adopted to model the delay sequence. Based on the observer canonical form, the bilinear state space model is transformed into a regressive form. A bilinear state observer is designed to estimate the state variables. Under the variational Bayesian scheme, the system parameters, discrete delays, and the Markov transition probabilities are identified by using the measurement data. A numerical example and a continuous stirred tank reactor simulation are employed to validate the effectiveness of the proposed algorithm.

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

The switch Markov chain has been extensively studied as the most natural Markov chain Monte Carlo approach for sampling graphs with prescribed degree sequences. We show that the switch chain for sampling simple undirected graphs with a given degree sequence is rapidly mixing when the degree sequence is so‐called strongly stable. Strong stability is satisfied by all degree sequences for which the switch chain was known to be rapidly mixing based on Sinclair's multicommodity flow method up until a recent manuscript of Erdős and coworkers in 2019. Our approach relies on an embedding argument, involving a Markov chain defined by Jerrum and Sinclair in 1990. This results in a much shorter proof that unifies (almost) all the rapid mixing results for the switch chain in the literature, and extends them up to sharp characterizations of P‐stable degree sequences. In particular, our work resolves an open problem posed by Greenhill and Sfragara in 2017.

Quasimonotone graphs

For any class C of bipartite graphs, we define quasi-C to be the class of all graphs G such that every bipartition of G belongs to C. This definition is motivated by a generalisation of the switch Markov chain on perfect matchings from bipartite graphs to nonbipartite graphs. The monotone graphs, also known as bipartite permutation graphs and proper interval bigraphs, are such a class of bipartite graphs. We investigate the structure of quasi-monotone graphs and hence construct a polynomial time recognition algorithm for graphs in this class.

Chaotic wind power time series prediction via switching data-driven modes

To schedule wind power efficiently and to mitigate the adverse effects caused by wind's intermittency and variability, an advanced wind power prediction model is proposed in this paper. This model is a combined model via switching different data-driven chaotic time series models. First, inputs of this model come from the reconstructed data based on the chaotic characteristics of wind power time series. Second, three different data mining algorithms are used to construct wind power prediction models individually. To obtain a regime for switching optimal models, a Markov chain is trained. Then, weights of different data-driven modes are calculated by the Markov chain switching regime, and used in the final combined model for wind power prediction. The industrial data from actual wind farms is studied. Results of the proposed model are compared with that of non-reconstructed input data, traditional data-driven models and two typical combined models. These results validate the superiority of proposed model on improving wind power prediction accuracy.

Counting Perfect Matchings and the Switch Chain

We examine the problem of exactly or approximately counting all perfect matchings in hereditary classes of nonbipartite graphs. In particular, we consider the switch Markov chain of Diaconis, Graham, and Holmes. We determine the largest hereditary class for which the chain is ergodic, and define a large new hereditary class of graphs for which it is rapidly mixing. We go on to show that the chain has exponential mixing time for a slightly larger class. We also examine the question of ergodicity of the switch chain in an arbitrary graph. Finally, we give exact counting algorithms for three classes.

Estimation and Forecasting of Sovereign Credit Rating Migration Based on Regime Switching Markov Chain

Our research aims to develop the regime switching Markov chain (RSMC), a discrete time Markov chain whose underlying regime is depending on a hidden Markov model, which express the dynamics of sovereign credit rating migration. Estimated based on a version of the Expectation-Maximization algorithm, the regime in RSMC indicates either economic expansion or contraction. Then, we apply RSMC to the monthly time series of the sovereign credit rating of 41 nations from January 1994 to December 2018. At first, we confirm that the estimation of RSMC is superior to a homogeneous Markov chain. It implies that the credit rating dynamics are subject to the underlying economic condition. Secondly, we observe that the second tier and non-investment credit ratings in economic contractions are likely to be downgraded. We also detect the continental clustering of economic contractions for the Asian currency and European sovereign debt crises. Lastly, we discover that the forecasting performance of RSMC is superior to that of the benchmark, especially for the second tier and non-investment credit ratings. In conclusion, we claim that RSMC can improve the management of sovereign credit risk exposures.

Load analyzing of nonstationary load history of engineering vehicles by switching Markov chain

Engineering vehicles are widely used under various harsh working conditions. For many components in them, service loadings they suffered are usually random and nonstationary due to their remarkable characteristic called cyclic operation. To deal with that, section method can be applied. However, this method will neglect those transition cycles caused by switching load section, which can contribute a lot to fatigue. In order to consider those transition cycles, this paper applied the model called “Switching Markov Chain of Turning Points” (SMCTP). Then the expected rain-flow matrix is compared with the overall rain-flow matrix conducted by section method. The comparison result shows that SMCTP can perform well in processing nonstationary loadings. As a result, the Switching Markov Chain method (SMC) was proved to be effective in stochastically characterizing the nonstationary switching loadings of engineering vehicles.

Switch chain mixing times and triangle counts in simple random graphs with given degrees

Sampling uniform simple graphs with power-law degree distributions with degree exponent τ∈(2,3) is a non-trivial problem. Firstly, we propose a method to sample uniform simple graphs that uses a constrained version of the configuration model together with a Markov Chain switching method. We test the convergence of this algorithm numerically in the context of the presence of small subgraphs and we estimate the mixing time to be at most O(n log2 n)⁠. Secondly, we compare the number of triangles in uniform random graphs with the number of triangles in the erased configuration model where double edges and self-loops of the configuration model are removed. Using simulations and heuristic arguments, we conjecture that the number of triangles in the erased configuration model is larger than the number of triangles in the uniform random graph, provided that the graph is sufficiently large. Lastly, we argue that certain switch-chain-based proof methods can not be used in the regime τ∈(2,3) due to the possibility of creating many triangles with a single switch.

The switch Markov chain for sampling irregular graphs and digraphs

The problem of efficiently sampling from a set of (undirected, or directed) graphs with a given degree sequence has many applications. One approach to this problem uses a simple Markov chain, which we call the switch chain, to perform the sampling. The switch chain is known to be rapidly mixing for regular degree sequences, both in the undirected and directed setting.We prove that the switch chain for undirected graphs is rapidly mixing for any degree sequence with minimum degree at least 1 and with maximum degree dmax which satisfies 3≤dmax≤13M, where M is the sum of the degrees. The mixing time bound obtained is only a factor n larger than that established in the regular case, where n is the number of vertices. Our result covers a wide range of degree sequences, including power-law density-bounded graphs with parameter γ>5/2 and sufficiently many edges.For directed degree sequences such that the switch chain is irreducible, we prove that the switch chain is rapidly mixing when all in-degrees and out-degrees are positive and bounded above by 14m, where m is the number of arcs, and not all in-degrees and out-degrees equal 1. The mixing time bound obtained in the directed case is an order of m2 larger than that established in the regular case.

Do Brands Compete or Coexist? Evidence from the Cola Wars

Marketers believe that competition and brand loyalty create a tug-of-war: consumers like competition since competing firms must innovate, improve quality, and lower prices, leading to increased consumer welfare. Brands prefer to minimize competition, as innovation requires costly development while competing on price reduces profitability. Rather, brands quasi-compete, using differentiated offerings to compete yet sustain profitability. We examine marketing’s dominant logic of the service-centered model of exchange (Vargo and Lusch 2004) by looking at brand loyal household purchasing patterns between two substantive brands. With the Cola Wars as our context, we use Markov chain switching probabilities of panel data for two product categories. Across three levels of the branded house (parent brand, category brand, and variety brand), we find that at all three levels between 90% and 95% of households that are brand loyal in one quarter continue to be loyal at each level in subsequent quarters. Despite the commonly held belief that brands compete, we find that brands coexist, competing at the consumption fringes (non-heavy users) instead.