Agent-based models (ABMs) are often high-dimensional and complex, making simulations costly and formal analysis hard. Low-dimensional model reduction is hence of great interest. The systems often show a “tightness”: the complex microdynamics of the O(N) many attributes of the individual agents can be approximated by the stochastic evolution of a small number (that is independent of N) of macroscopic collective variables describing the effective dynamics of the system. Moreover, if the number N of agents is large, one can observe a concentration of measure in the sense that the collective variables follow an almost deterministic and smooth evolution. In simple examples, suitable collective variables and approximate ODEs or SDEs governing the effective dynamics can be guessed or derived easily.

We formalize the efficacy of macroscopic modeling of ABMs by showing concentration of the process’ stochastic law transversal to low-dimensional coordinates of the full state. Further, we harness this property to numerically compute corresponding variables for systems where their analytical derivation is out of reach.

Overview EF4-8Figure 1: Concentration effect. The set of transition densities is thightly concentrated around a (low-dimensional) mainfold. Using manifold learning techniques, we can derive collective variables for the ABM. 

Figure 2: Approximation by mean-field equation (MFE). Mean share c(j,i) and standard deviation of opinion i in cluster j for N=10000 agents in a stochastic block model (SBM) with two clusters [Lücke et al., 2023]. 

For an efficient implementation of the discrete-state spreading processes on networks of interacting agents see the software SPoNet.

Publications

2024
Learning interpretable collective variables for spreading processes on networks Physical Review E, 109(2), p. L022301, 2024 Marvin Lücke, Stefanie Winkelmann, Jobst Heitzig, Nora Molkenthin, Péter Koltai BibTeX
arXiv
DOI
Concentration Effects and Collective Variables in Agent-Based Systems
Understanding Memory Mechanisms in Socio-Technical Systems: the Case of an Agent-based Mobility Model Advances in Complex Systems, 2024 (accepted for publication) Gesine Steudle, Stefanie Winkelmann, Steffen Fürst, Sarah Wolf BibTeX
DOI
Concentration Effects and Collective Variables in Agent-Based Systems
2023
Large population limits of Markov processes on random networks Stochastic Processes and their Applications, Vol.166, 2023 Marvin Lücke, Jobst Heitzig, Péter Koltai, Nora Molkethin, Stefanie Winkelmann BibTeX
DOI
arXiv
Concentration Effects and Collective Variables in Agent-Based Systems
2022
Discovering collective variable dynamics of agent-based models 25th International Symposium on Mathematical Theory of Networks and Systems MTNS 2022, 2022 Marvin Lücke, Peter Koltai, Stefanie Winkelmann, Nora Molkethin, Jobst Heitzig BibTeX
DOI
Concentration Effects and Collective Variables in Agent-Based Systems
tgEDMD: Approximation of the Kolmogorov Operator in Tensor Train Format Journal of Nonlinear Science, Vol.32, 2022 Marvin Lücke, Feliks Nüske BibTeX
DOI
Concentration Effects and Collective Variables in Agent-Based Systems