## Equation-Free Analysis for Agent-Based Computation

**Speaker:**Ping Liu

**Series:**Final Public Oral Examinations

**Location:**Lapidus Lounge (E-Quad A210)

**Date/Time:**Monday, December 10, 2012, 2:30 p.m. - 4:00 p.m.

In recent years, individual-based/agent-based modeling has been applied to study a wide range of problems, from engineering problems to scocial/economical phenomena, and to biological and medical problems. Despite the wide application of agent-based models, a challenge which modelers often encounter is the effective nonlinear and stochastic nature of individual dynamics and the inherent complexity of microscopic descriptions; closures that allow us to write macroscopic evolution equations for the coarse-grained dynamics are usually not available or overwhelmingly difficult to derive in closed form. Because these closures are not available, the conventional algorithmic tools that allow us to extract information from continuum models “easier, faster, better” than simulations (e.g. numerical bifurcation theories, optimization, control etc.) simply cannot be applied. This caused a major stumbling block in current complex systems modeling.

As an attempt to alleviate this problem for agent-based modeling, a recently developed computational methodology - the Equation-Free (EF) approach - is applied in this Thesis to study two different agent-based models. The first one is an agent-based financial market model which describes the dynamic behavior of many interacting investors in the presence of mimesis. Three aspects of the EF framework are explored through the investigation of this model, namely, the coarse bifurcation analysis, the coarse rare event analysis and the patch-dynamics scheme. Using appropriately initialized short runs of the microscopic agent-based simulations, coarse-grained bifurcation diagrams of the probability densities of the agents’ preference states are constructed, and the stability of its multiple solution branches are investigated. When the mimetic coupling between agents becomes strong enough, the stable steady state loses its stability at a coarse turning point bifurcation. Close to this turning point, interesting stochastic rare events are observed. A one-dimensional coarse reaction coordinate is first constructed based on physical intuition, based on which an effeciii tive Fokker-Planck (FP) equation is constructed using the EF approach. The mean escape time of the rare event computed using this effective FP shows good agreement with the results from expensive direct agent-based simulations.

Running agent-based simulations in the entire spatial and temporal domain could be prohibitively expensive. Based on the smooth behavior in space and time on a macroscopic scale, a patch-dynamics scheme is designed to perform numerical simulations of the unavailable macroscopic equation on macroscopic time and length scales; it uses appropriately initialized available microscopic agent-based simulations in a number of small patches of the spatial-temporal domain. To reduce the artificially introduced boundary artifacts, buffer regions are used to “shield” the interior of the domain from the boundary artifacts. This scheme is first applied to the continuum PDE approximation of the agent-based model, and the performance and the accuracy of the scheme are analyzed. It is then applied to the agent-based model itself. In the patch-dynamics facilitated agent-based simulations, expensive agent-based simulations are performed in only 20% of the spatial domain and 10% of the temporal domain.

The second agent-based model studied in this Thesis is an animal swarming model. A recently developed data-mining tool - Diffusion Maps (DMAP) - was applied to the simulation data and revealed interesting coarse level features about the underlying dynamics. The first two dominant DMAP coarse variables characterize how the group of animals switches its direction of collective motion, with the two coordinates corresponding to the “left-right” and “up-down” directions of motion respectively. Based on the identified DMAP coarse variables, a reduced stochastic differential equation (SDE) model is successfully constructed using the EF framework. Using the reduced SDE model, the associated coarse rare events are efficiently studied, circumventing expensive long-term agent-based simulations.

Beyond agent-based coarse-graining studies, an interesting project was studied: a reaction network model for a biologically important gene signaling pathway – the extracellular signal-regulated Kinase (ERK) pathway. The ERK protein can use the same domains to interact with phosphatases, which dephosphorylate and deactivate ERK, and with substrates, which connect ERK to its downstream effects. As a consequence, substrates can compete with phosphatases and control the level of ERK phosphorylation. Using a combination of numerical solution of the model and its steady-state bifurcation analysis, a parameter regime which supports large amplitude limit cycles is successfully identified. This suggests that the substrate competing effect can qualitatively change the dynamics of a network that controls ERK activation. On its own, this network can be bistable, but in a larger system, where ERK accelerates the degradation of a substrate competing with a phosphatase, this network can oscillate. Previous studies proposed that oscillatory ERK signaling requires a negative feedback in which active ERK reduces the rate at which it is phosphorylated by upstream kinase. The findings in this study demonstrate that oscillations can also emerge even without such feedback, due to substrate-dependent control of ERK phosphorylation.