Metastable States in a Model of Cancer Initiation

Ashcroft, Peter

doi:10.1007/978-3-319-41213-9_5

Cited by 4 publications

(6 citation statements)

References 50 publications

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“…where τn is the mean first passage time to reach nu given one starts at n. In our case it is possible to find the values of τn exactly, and we do so in line with methods shown in [26,45,54]. Consider again the mean-field microscopic transitions previously seen in Fig.…”

Section: Exact Determination Of First Passage Timesmentioning

confidence: 62%

“…In the common case where each agent is initially assigned a decision at random with probability p0 the number of initially right deciding agents is drawn from a binomial distribution, i.e., Q(n) = Bin(n; N, p0). The steady-state distribution reached as t → ∞, i.e., Ps(n) = P(n, t → ∞), is widely known for birth-death processes with general rates and is given by [26,38,39,45],…”

Section: Master Equation and Analytical Time-dependent Solutionmentioning

confidence: 99%

“…which has the intuitive interpretation that the probability of reaching mu at time t + ∆t is the probability of hopping to i + 1 and reaching mu from there, plus the probability of hopping to i − 1 and reaching mu from there, plus the probability of not hopping and reaching mu from i [45]. If we subtract ψi(t) from both sides of Eq.…”

Section: Exact Determination Of First Passage Timesmentioning

confidence: 99%

“…noting the curious independence of τi on the choice of ∆t. One can perform similar calculations for the higher order moments, although we do not consider them here (see [45] for more details). We have two boundary conditions on this recursion relation.…”

Section: Exact Determination Of First Passage Timesmentioning

confidence: 99%

“…absorbing boundaries (see [26,45]). As stated in the main text, we calculate the 'equilibrium values' of n based on the extrema of the steady-state probability distribution.…”

Section: E Derivation Of φ Rmentioning

confidence: 99%

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Non-equilibrium time-dependent solution to discrete choice with social interactions

Holehouse,

Pollitt

2021

Preprint

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We solve the binary decision model of Brock and Durlauf [1] in time using methods often employed in studies of stochastic complex systems. This solution is valid when not at equilibrium and can be used to exemplify path-dependent behaviours of their model. The solution is computationally fast and is indistinguishable from Monte Carlo simulation. Lock-in effects are observed in some regions of the model's parameter space, and we calculate the time scale of the lock-ins. Curiously, we find that although altruistic agents coalesce more strongly on a particular decision, increasing their utility in the short-term, they are also more prone to being stuck in a non-optimal decision lock-in as compared to selfish agents. Finally, we construct a likelihood function that can be used on non-equilibrium data for model calibration. Even with a well-defined likelihood function, model calibration is difficult unless one has access to data representative of the underlying model.

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Section: Exact Determination Of First Passage Timesmentioning

confidence: 62%

Section: Master Equation and Analytical Time-dependent Solutionmentioning

confidence: 99%

Section: Exact Determination Of First Passage Timesmentioning

confidence: 99%

Section: Exact Determination Of First Passage Timesmentioning

confidence: 99%

“…absorbing boundaries (see [26,45]). As stated in the main text, we calculate the 'equilibrium values' of n based on the extrema of the steady-state probability distribution.…”

Section: E Derivation Of φ Rmentioning

confidence: 99%

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Non-equilibrium time-dependent solution to discrete choice with social interactions

Holehouse,

Pollitt

2021

Preprint

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Non-equilibrium time-dependent solution to discrete choice with social interactions

Holehouse

2022

PLoS ONE

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We solve the binary decision model of Brock and Durlauf (2001) in time using a method reliant on the resolvent of the master operator of the stochastic process. Our solution is valid when not at equilibrium and can be used to exemplify path-dependent behaviours of the binary decision model. The solution is computationally fast and is indistinguishable from Monte Carlo simulation. Well-known metastable effects are observed in regions of the model’s parameter space where agent rationality is above a critical value, and we calculate the time scale at which equilibrium is reached using a highly accurate method based on first passage time theory. In addition to considering selfish agents, who only care to maximise their own utility, we consider altruistic agents who make decisions on the basis of maximising global utility. Curiously, we find that although altruistic agents coalesce more strongly on a particular decision, thereby increasing their utility in the short-term, they are also more prone to being subject to non-optimal metastable regimes as compared to selfish agents. The method used for this solution can be easily extended to other binary decision models, including Kirman’s model of ant recruitment Kirman (1993), and under reinterpretation also provides a time-dependent solution to the mean-field Ising model. Finally, we use our time-dependent solution to construct a likelihood function that can be used on non-equilibrium data for model calibration. This is a rare finding, since often calibration in economic agent based models must be done without an explicit likelihood function. From simulated data, we show that even with a well-defined likelihood function, model calibration is difficult unless one has access to data representative of the underlying model.

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Exact time-dependent dynamics of discrete binary choice models

Holehouse¹,

Moran²

2022

SSRN Journal

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Metastable States in a Model of Cancer Initiation

Cited by 4 publications

References 50 publications

Non-equilibrium time-dependent solution to discrete choice with social interactions

Non-equilibrium time-dependent solution to discrete choice with social interactions

Non-equilibrium time-dependent solution to discrete choice with social interactions

Exact time-dependent dynamics of discrete binary choice models

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