(182d) Modeling Tipping Points in the Reduced-Order Stochastic Dynamics of a Population of Interacting Agents

Agent-based models are ubiquitous in complex multiscale systems, especially in domains ranging from biology and epidemiology to the social sciences. In real world applications, it becomes intractable to individually model a large population of interacting agents, and instead reduced continuum partial differential equation (PDEs) approximations are used.

Our agent-based model describes buying and selling actions of individuals/agents in a financial market in the presence of mimesis. For a large population of agents, we can model the probability of finding an agent at some state (the agent density), rather than the state of every single agent, and the corresponding population-level partial differential equation (PDE) can be approximated by a Fokker-Planck (FP) type PDE [1]. Numerical continuation was used on the PDE to find the turning points (TP) that lead to tipping/unstable agent density distributions. The FP deterministic model is not a good approximation close to the TP because the model exhibits rare event transitions (fluctuations that cause instabilities), and a 1D effective Langevin-type stochastic differential equation (SDE) is used in that region [2].

We propose approximating the drift and diffusivity coefficients in these effective SDEs using a novel neural network (NN) algorithm, validated using Kramers-Moyal expansions [3]. In addition, we learn a coarse-grained SDE from agent-based simulations close to the TP on data driven, diffusion map variables using a parameter-dependent neural network. The distribution of escape times is approximated through the surrogate NN-SDE model, and compared to the full agent-based model.

[1] Omurtag, A., & Sirovich, L. (2006). Modeling a large population of traders: Mimesis and stability. Journal of Economic Behavior & Organization, 61(4), 562-576.

[2] Liu, P., Siettos, C. I., Gear, C. W., & Kevrekidis, I. G. (2015). Equation-free model reduction in agent-based computations: Coarse-grained bifurcation and variable-free rare event analysis. Mathematical Modelling of Natural Phenomena, 10(3), 71-90.

[3] B. Øksendal, Stochastic Differential Equations, Sixth. Berlin, Heidelberg: Springer Berlin Heidelberg, 2003.doi: 10.1007/978-3-642-14394-6.