Lie Algebra Solution of Population Models Based on Time-Inhomogeneous Markov Chains

House, Thomas

doi:10.1017/s0021900200009219

Cited by 5 publications

(6 citation statements)

References 12 publications

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“…The method has recently been applied even to the birth-death process. 24 However, in Ref. 24, an infinite-dimensional matrix have been used; on the contrary, we use the creation and annihilation operators in the present paper, and these notations have a kind of flexibility and availability and enable us to obtain the Karlin-McGregor-like formula, as shown later.…”

Section: Brief Review Of Lie Algebraic Methods For Time-inhomogenomentioning

confidence: 99%

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Karlin–McGregor-like formula in a simple time-inhomogeneous birth–death process

Ohkubo¹

2014

J. Phys. A: Math. Theor.

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A novel approach is employed and developed to derive transition probabilities for a simple time-inhomogeneous birth-death process. Algebraic probability theory and Lie algebraic treatments make it easy to treat the timeinhomogeneous cases. As a result, an expression based on the Charlier polynomials is obtained, which can be considered as an extension of a famous Karlin-KcGregor representation for a time-homogeneous birth-death process. I. INTRODUCTIONBirth-death processes have been widely used in various contexts including physics, biology, and social sciences 1-3 ; it is a continuous-time Markov chain with discrete states on non-negative integers. Not only for their applicability to modeling various phenomena, but also for their rich mathematical structures, the birth-death processes have been studied well. For example, the birth-death processes have discrete states, so that a treatment based on generating functions are useful 1 ; using the generating function approach, various quantities, including transition probabilities, can be derived. However, it has been shown that orthogonal polynomials give beautiful representations for the transition probabilities. [4][5][6][7][8][9] The expression for the transition probabilities, so-called Karlin-McGregor spectral representation, is based on a sequence of orthogonal polynomials and a spectral measure. Because the orthogonal polynomials have deep relationship with continued fractions, it would be natural to consider that the birth-death process could be dealt with by using the continued fractions. Actually, numerical algorithms based on the continued fractions have been proposed (for this topic, for example, see the recent paper by Crawford and Suchard. 10 ) However, most of the above discussions are basically for time-homogeneous cases, in which rate constants for the birth-death processes are time-independent. In contrast, studies for time-inhomogeneous cases are not enough. While the generating function approach have been applied to time-inhomogeneous birth-death processes, 3,11 it has not been known even whether transition probabilities for the time-inhomogeneous birth-death processes can be described in terms of the orthogonal polynomials or not. The time-inhomogeneous cases are sometimes important in mathematical modeling of external influences. In addition, in a practical sense, concise expressions for the transition probabilities are demanded; for example, in time-series data analysis for bioinformatics, rapid evaluation of the transition probabilities is needed. While we can use various Monte Carlo simulations in order to deal with the time-inhomogeneous cases, it is important to try to find concise expressions and easy calculations for the transition probabilities.In the present paper, we show that it is possible to describe transition probabilities in terms of orthogonal polynomials at least in a simple time-inhomogeneous birth-death process. The birth-death process has only a state-independent birth rate and linearly-state-dependent death rate. For a time...

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Section: Brief Review Of Lie Algebraic Methods For Time-inhomogenomentioning

confidence: 99%

“…In Ref. 24, a similar discussion has been performed using an infinite-dimensional matrix (not the creation and annihilation operators as in the present paper), and the transition probability for the simplest case of the initial state (n = 0) is derived. In addition, as pointed out in Ref.…”

Section: B Expression 1: Finite Summation Expression Based On Abstramentioning

confidence: 96%

Karlin–McGregor-like formula in a simple time-inhomogeneous birth–death process

Ohkubo¹

2014

J. Phys. A: Math. Theor.

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“…While the application of Wei-Norman method in physical science has a long history [21,25], its application in some simple biological population dynamics, such as SIR and SIS, appears only recently (see e.g. [24,[26][27][28]) due to lack of symmetry in such systems (Hence, it would be much more difficult to construct an appropriate Lie algebra with a low dimension). It is hoped that the approach offered in this study could shed some light on the analytical solution of more complicated (and realistic) social dynamics models.…”

Section: Discussionmentioning

confidence: 99%

“…where H(t) = Q(t) T , and p(t) is a column probability vector with component p i (t) describing the probability of finding the system in state i at time t. Using the 'ket' notation |·⟩ [23,24], the probability vector can alternatively be recast as…”

Section: The Wei-norman Methodsmentioning

confidence: 99%

Lie algebraic discussion for affinity based information diffusion in social networks

Shang

2017

Open Physics

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Abstract:In this paper we develop a dynamical information diffusion model which features the affinity of people with information disseminated in social networks. Four types of agents, i.e., susceptible, informed, known, and refractory ones, are involved in the system, and the affinity mechanism composing of an affinity threshold which represents the fitness of information to be propagated is incorporated. The model can be generally described by a timeinhomogeneous Markov chain, which is governed by its master (Kolmogorov) equation. Based on the Wei-Norman method, we derive analytical solutions of the model by constructing a low-dimensional Lie algebra. Numerical examples are provided to illustrate the obtained theoretical results. This study provides useful insights into the closed-form solutions of complex social dynamics models through the Lie algebra method.

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“…In Ref. [23], chemical reaction systems has been treated via the Wei-Norman method, but a kind of infinite-matrix formulation has been used. The infinite-matrix formulation is a little difficult to treat, and then the Doi-Peliti formulation would be more suitable for the Wei-Norman method.…”

Section: Application Of the Wei-norman Methodsmentioning

confidence: 99%

Lie Algebraic Discussions for Time-Inhomogeneous Linear Birth–Death Processes with Immigration

Ohkubo

2014

J Stat Phys

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Analytical solutions for time-inhomogeneous linear birth-death processes with immigration are derived. While time-inhomogeneous linear birth-death processes without immigration have been studied by using a generating function approach, the processes with immigration are here analyzed by Lie algebraic discussions. As a result, a restriction for time-inhomogeneity of the birth-death process is understood from the viewpoint of the finiteness of the dimensionality of the Lie algebra.

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Lie Algebra Solution of Population Models Based on Time-Inhomogeneous Markov Chains

Cited by 5 publications

References 12 publications

Karlin–McGregor-like formula in a simple time-inhomogeneous birth–death process

Karlin–McGregor-like formula in a simple time-inhomogeneous birth–death process

Lie algebraic discussion for affinity based information diffusion in social networks

Lie Algebraic Discussions for Time-Inhomogeneous Linear Birth–Death Processes with Immigration

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