Extending Transition Path Theory: Periodically Driven and Finite-Time Dynamics

Helfmann, Luzie; Borrell, Enric Ribera; Schütte, Christof; Koltai, Péter

doi:10.1007/s00332-020-09652-7

Cited by 28 publications

(40 citation statements)

References 48 publications

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“…We reduce the temperature in time, akin to the process in simulated annealing. More precisely, we consider the overdamped Langevin equation in

R^{2}

d Y_{t} = - \nabla V false(Y_{t} false) d t + \sqrt{2 β {(t)}^{- 1}} d W_{t}

with a triple well potential V with 2 deep wells at ( − 1, 0), (1,0) and a shallow well at

(0, \frac{3}{2})

as in [22].

W_{t}

denotes standard Brownian motion and

β (t)

is the varying inverse of the temperature / the coldness.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…[2] This approach was recently extended to non-autonomous dynamics for finite-time and periodic systems. [22] Generalizing furthermore to time-dependent target sets it may be useful to think of committor functions on space-time. Indeed the Koopman operator (39) applied to an indicator function of some set G ⊂ can then be interpreted as such a generalized committor function, that is, the probability to hit space-time set A = G × {t} before B = ∖G × {t} (see sketch 3 of Figure 2):…”

Section: Connections To Committor Functionsmentioning

confidence: 99%

“…[ 2 ] This approach was recently extended to non‐autonomous dynamics for finite‐time and periodic systems. [ 22 ]…”

Section: The Augmented Jump Chainmentioning

confidence: 99%

See 2 more Smart Citations

The Augmented Jump Chain

Sikorski

Weber

Schütte

2021

Advcd Theory and Sims

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Modern methods of simulating molecular systems are based on the mathematical theory of Markov operators with a focus on autonomous equilibrated systems. However, non‐autonomous physical systems or non‐autonomous simulation processes are becoming more and more important. A representation of non‐autonomous Markov jump processes is presented as autonomous Markov chains on space‐time. Augmenting the spatial information of the embedded Markov chain by the temporal information of the associated jump times, the so‐called augmented jump chain is derived. The augmented jump chain inherits the sparseness of the infinitesimal generator of the original process and therefore provides a useful tool for studying time‐dependent dynamics even in high dimensions. Furthermore, possible generalizations and applications to the computation of committor functions and coherent sets in the non‐autonomous setting are discussed. After deriving the theoretical foundations, the concepts with a proof‐of‐concept Galerkin discretization of the transfer operator of the augmented jump chain applied to simple examples are illustrated.

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“…We reduce the temperature in time, akin to the process in simulated annealing. More precisely, we consider the overdamped Langevin equation in

R^{2}

d Y_{t} = - \nabla V false(Y_{t} false) d t + \sqrt{2 β {(t)}^{- 1}} d W_{t}

with a triple well potential V with 2 deep wells at ( − 1, 0), (1,0) and a shallow well at

(0, \frac{3}{2})

as in [22].

W_{t}

denotes standard Brownian motion and

β (t)

is the varying inverse of the temperature / the coldness.…”

Section: Numerical Examplesmentioning

confidence: 99%

Section: Connections To Committor Functionsmentioning

confidence: 99%

See 1 more Smart Citation

The Augmented Jump Chain

Sikorski

Weber

Schütte

2021

Advcd Theory and Sims

Self Cite

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show abstract

“…Here, i denotes the imaginary unit. In our case we compare the level of synchronization for different edge probabilities W 12 between the two blocks and θ j t is the angle in the plane of "number of agents with green opinion" vs "number of agents with green behaviour" in block j at time t as measured from the center point (20,20). By placing every oscillator according to its phase on the unit circle, the order parameter measures the distance of the average of the positions on the unit circle from the origin.…”

Section: An Oscillating Bivariate Complex Contagion Modelmentioning

confidence: 99%

Statistical Analysis of Tipping Pathways in Agent-Based Models

Helfmann¹,

Heitzig²,

Koltai³

et al. 2021

Preprint

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Agent-based models are a natural choice for modeling complex social systems. In such models simple stochastic interaction rules for a large population of individuals can lead to emergent dynamics on the macroscopic scale, for instance a sudden shift of majority opinion or behavior. Here, we are concerned with studying noise-induced tipping between relevant subsets of the agent state space representing characteristic configurations. Due to the large number of interacting individuals, agent-based models are high-dimensional, though usually a lower-dimensional structure of the emerging collective behaviour exists. We therefore apply Diffusion Maps, a non-linear dimension reduction technique, to reveal the intrinsic low-dimensional structure. We characterize the tipping behaviour by means of Transition Path Theory, which helps gaining a statistical understanding of the tipping paths such as their distribution, flux and rate. By systematically studying two agent-based models that exhibit a multitude of tipping pathways and cascading effects, we illustrate the practicability of our approach.

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“…But as mentioned in the introduction, agent-based models are often not stationary. Therefore at next it would be important to also consider tipping in non-stationary ABMs, e.g., ABMs influenced by some external parameter variations [22].…”

mentioning

confidence: 99%

Statistical analysis of tipping pathways in agent-based models

Helfmann

Heitzig

Koltai

et al. 2021

Eur. Phys. J. Spec. Top.

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Agent-based models are a natural choice for modeling complex social systems. In such models simple stochastic interaction rules for a large population of individuals on the microscopic scale can lead to emergent dynamics on the macroscopic scale, for instance a sudden shift of majority opinion or behavior. Here we are introducing a methodology for studying noise-induced tipping between relevant subsets of the agent state space representing characteristic configurations. Due to a large number of interacting individuals, agent-based models are high-dimensional, though usually a lower-dimensional structure of the emerging collective behaviour exists. We therefore apply Diffusion Maps, a non-linear dimension reduction technique, to reveal the intrinsic low-dimensional structure. We characterize the tipping behaviour by means of Transition Path Theory, which helps gaining a statistical understanding of the tipping paths such as their distribution, flux and rate. By systematically studying two agent-based models that exhibit a multitude of tipping pathways and cascading effects, we illustrate the practicability of our approach.

show abstract

Extending Transition Path Theory: Periodically Driven and Finite-Time Dynamics

Cited by 28 publications

References 48 publications

The Augmented Jump Chain

The Augmented Jump Chain

Statistical Analysis of Tipping Pathways in Agent-Based Models

Statistical analysis of tipping pathways in agent-based models

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