Fig 1.
(A) Network of N = 10 agents having three different types (blue, red yellow) and (B) a possible trajectory of the jump process for the rate constants γ12 = γ23 = γ31 = 2, γ32 = γ21 = γ13 = 1 and for i, j = 1, …,3.
Fig 2.
Flow chart of the predator-prey model.
Fig 3.
Simulation of predator-prey model.
(A) Snapshot of the state of the predator-prey ABM at time t = 250. Red and green dots represent predators and prey, respectively. The radius of vision is indicated by the light-red area around the predators. (B) Simulation of the predator-prey model for the parameters given in Table 2 on page 17. The vertical gray dashed line indicates the time where the snapshot in (A) is taken.
Table 1.
Measurement set sizes.
Fig 4.
Root mean square error of drift and diffusion coefficients.
Approximation error defined as the RMSE of the coefficients of (A) the drift and (B) diffusion estimates for the EVM in Section 3.2 compared to the exact SDE limit model (5) depending on the number of agents N and number of Monte Carlo samples k for the estimation via Kramers–Moyal formulae. The brighter the color, the smaller the error and the better the identification of the reduced system. For increasing N and k the approximation error decreases.
Fig 5.
Prediction of extended voter model for complete networks.
(A) Expectation (solid) and standard deviation (dashed) of the SDE limit model Ci(t) and its data-driven approximation (gray) estimated from 103 Monte Carlo simulations for the dynamics of the EVM of Section 3.2 for N = 5000 agents and initial state . The relative number of agents of type S3 can be reconstructed using (10) and is therefore not displayed. The approximate moments (gray solid and dashed lines) agree with the SDE limit model. (B) Approximation and evaluation error of the drift and diffusion estimates for the EVM in Section 3.2 compared to the exact SDE limit model (5) depending on the number of measurements m for fixed k1 = 10 (dashed), k2 = 100 (solid) and N = 5000 agents. The error is averaged over 100 simulations. Clearly, for higher amounts of training data a smaller error can be expected. This holds for both parameters m and k.
Fig 6.
Prediction of extended voter model for clustered networks.
(A) & (B) Adjacency matrices of the networks where black represents 1 (existing edge) and white 0 (no edge). (C) & (D) First-order moment of the data-driven coarse-grained model (solid) and the limit SDE (dotted) (14) for two clusters with N = 50 agents, γ12 = γ23 = γ31 = 2, γ13 = γ21 = γ32 = 1, for all i, j = 1, …, 3 and c(0) = [0.85, 0.1, 0.05, 0.2, 0.5, 0.3]⊤. The data-driven model is estimated using k = 1000 realizations of m = 1000 measurements for lag time τ = 0.01.
Fig 7.
Prediction of extended voter model for random networks.
Expectation of the data-driven reduced model (solid) compared the EVM (dashed) on a random network with average degree of approximately 50, estimated from 103 Monte Carlo simulations for N = 500 agents and initial state . The deterministic part of the SDE limit model (5) is indicated in gray (dotted). The data-driven model is estimated using m = k = 1000 measurements and realizations for the lag time τ = 0.01.
Fig 8.
Prediction of predator-prey model.
Phase portraits of first-order moment of (A) the reduced SDE model and (B) the PPM estimated from 958 Monte Carlo simulations. (C) Realization of the reduced SDE solution learned from m = k = 1000 measurements and samples for the PPM with parameters given in Table 2.
Table 2.
Parameters used during the simulation of the PPM.