scholarly journals Detailed Balance $$=$$ Complex Balance $$+$$ Cycle Balance: A Graph-Theoretic Proof for Reaction Networks and Markov Chains

2020 ◽  
Vol 82 (9) ◽  
Author(s):  
Stefan Müller ◽  
Badal Joshi
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Detailed Balance ◽  
Reaction Networks ◽  
Graph Theoretic ◽  
Underlying Graph ◽  
Theoretic Proof ◽  
Reversible Chemical Reaction ◽  
Balance Complex ◽  
Algebraic Proofs

Abstract We further clarify the relation between detailed-balanced and complex-balanced equilibria of reversible chemical reaction networks. Our results hold for arbitrary kinetics and also for boundary equilibria. Detailed balance, complex balance, “formal balance,” and the new notion of “cycle balance” are all defined in terms of the underlying graph. This fact allows elementary graph-theoretic (non-algebraic) proofs of a previous result (detailed balance = complex balance + formal balance), our main result (detailed balance = complex balance + cycle balance), and a corresponding result in the setting of continuous-time Markov chains.

scholarly journals FORMALISMS FOR SPECIFYING MARKOVIAN POPULATION MODELS

2011 ◽  
Vol 22 (04) ◽  
pp. 823-841 ◽  
Author(s):  
THOMAS HENZINGER ◽  
BARBARA JOBSTMANN ◽  
VERENA WOLF
Keyword(s):  
Stochastic Process ◽  
Markov Chains ◽  
Continuous Time ◽  
Population Models ◽  
Ease Of Use ◽  
Stochastic Petri Nets ◽  
Reaction Networks ◽  
Process Algebras ◽  
Continuous Time Markov Chains ◽  
Stochastic Process Algebras

In this survey, we compare several languages for specifying Markovian population models such as queuing networks and chemical reaction networks. All these languages — matrix descriptions, stochastic Petri nets, stoichiometric equations, stochastic process algebras, and guarded command models — describe continuous-time Markov chains, but they differ according to important properties, such as compositionality, expressiveness and succinctness, executability, and ease of use. Moreover, they provide different support for checking the well-formedness of a model and for analyzing a model.

Threshold Functions for Markov Chains: a Graph Theoretic Approach

1993 ◽  
Vol 2 (3) ◽  
pp. 351-362
Author(s):  
James F. Lynch
Keyword(s):  
Markov Chains ◽  
Transition Probabilities ◽  
Theoretic Approach ◽  
Limiting Distributions ◽  
Graph Theoretic ◽  
Threshold Functions ◽  
Theoretic Proof ◽  
Graph Theoretic Approach

A new graph theoretic proof of the convergence of Markov chains with variable transition probabilities and a new algorithm for computing the limiting distributions are presented.

Characterization of the conditional stationary distribution in Markov chains via systems of linear inequalities

2020 ◽  
Vol 52 (4) ◽  
pp. 1249-1283
Author(s):  
Masatoshi Kimura ◽  
Tetsuya Takine
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Continuous Time ◽  
State Transition ◽  
Convex Polytope ◽  
Linear Inequalities ◽  
Systems Of Linear Inequalities ◽  
Free Sets ◽  
Practical Implications

AbstractThis paper considers ergodic, continuous-time Markov chains $\{X(t)\}_{t \in (\!-\infty,\infty)}$ on $\mathbb{Z}^+=\{0,1,\ldots\}$ . For an arbitrarily fixed $N \in \mathbb{Z}^+$ , we study the conditional stationary distribution $\boldsymbol{\pi}(N)$ given the Markov chain being in $\{0,1,\ldots,N\}$ . We first characterize $\boldsymbol{\pi}(N)$ via systems of linear inequalities and identify simplices that contain $\boldsymbol{\pi}(N)$ , by examining the $(N+1) \times (N+1)$ northwest corner block of the infinitesimal generator $\textbf{\textit{Q}}$ and the subset of the first $N+1$ states whose members are directly reachable from at least one state in $\{N+1,N+2,\ldots\}$ . These results are closely related to the augmented truncation approximation (ATA), and we provide some practical implications for the ATA. Next we consider an extension of the above results, using the $(K+1) \times (K+1)$ ( $K > N$ ) northwest corner block of $\textbf{\textit{Q}}$ and the subset of the first $K+1$ states whose members are directly reachable from at least one state in $\{K+1,K+2,\ldots\}$ . Furthermore, we introduce new state transition structures called (K, N)-skip-free sets, using which we obtain the minimum convex polytope that contains $\boldsymbol{\pi}(N)$ .

scholarly journals Trace Machines for Observing Continuous-Time Markov Chains

2006 ◽  
Vol 153 (2) ◽  
pp. 259-277 ◽  
Author(s):  
Verena Wolf ◽  
Christel Baier ◽  
Mila Majster-Cederbaum
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Continuous Time Markov Chains

On quasi-stationary distributions in absorbing continuous-time finite Markov chains

1967 ◽  
Vol 4 (1) ◽  
pp. 192-196 ◽  
Author(s):  
J. N. Darroch ◽  
E. Seneta
Keyword(s):  
Markov Chains ◽  
Discrete Time ◽  
Continuous Time ◽  
Present Note ◽  
Time Parameter ◽  
Stationary Distributions ◽  
Finite State ◽  
Absorbing Markov Chains ◽  
Finite Markov Chains ◽  
Finite State Space

In a recent paper, the authors have discussed the concept of quasi-stationary distributions for absorbing Markov chains having a finite state space, with the further restriction of discrete time. The purpose of the present note is to summarize the analogous results when the time parameter is continuous.

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