A Mass Preserving Numerical Scheme for Kinetic Equations that Model Social Phenomena

TL;DR

The paper tackles the challenge of simulating social-kinetic dynamics described by integro-differential equations with Dirac-delta transition rates while preserving mass. It develops the Mass Preserving Collocation Method (MPCM), a fully deterministic, mass-conserving solver that uses a collocation discretization augmented with a corrective factor to guarantee discrete mass conservation. The authors prove existence/uniqueness for the kinetic system and demonstrate MPCM on models with two and five subsystems, showing strong accuracy and dramatic speedups compared to stochastic Tau-leaping and hybrid ABM methods, with no need for hyperparameter tuning. The method scales to larger networks and provides a practical, reliable tool for analyzing complex social dynamics modeled by kinetic equations.

Abstract

In recent years, kinetic equations have been used to model many social phenomena. A key feature of these models is that transition rate kernels involve Dirac delta functions, which capture sudden, discontinuous state changes. Here, we study kinetic equations with transition rates of the form $$ T(x,y,u) = δ_{φ(x,y) - u}. $$ We establish the global existence and uniqueness of solutions for these systems and introduce a fully deterministic scheme, the \emph{Mass Preserving Collocation Method}, which enables efficient, high fidelity simulation of models with multiple subsystems. We validate the accuracy, efficiency, and consistency of the solver on models with up to five subsystems, and compare its performance against two state-of-the-art agent-based methods: Tau-leaping and hybrid methods. Our scheme resolves subsystem distributions captured by these stochastic approaches while preserving mass numerically, requiring significantly less computational time and resources, and avoiding variability and hyperparameter tuning characteristic of these methods.

Figures (16)

• Figure 1: The transition from agent-based models to kinetic equations involves shifting from tracking individual agents to grouping them into subsystems with assigned microstates. The system is then described using density distributions for each subsystem. The objective is to capture macroscopic behavior while preserving essential microscopic dynamics. This approach bridges discrete agent interactions and continuous statistical representations.
• Figure 2: Illustrations of transition effects based on the first two functions listed in Table \ref{['tab:phitab']}. Each function models the rate at which an agent in subsystem $i$ with microstate $x$ will transition to a microstate $u$ in subsystem $k$. All transition functions depend on $\gamma$. (a) The transition function $\phi_L$, where an agent in subsystem $i$ with microstate $x$ transitions to a lower microstate $x - \gamma xy$, transitioning closer to 0. (b) A depiction of the transition function $\phi_R$, where an agent in subsystem $i$ with microstate $x$ transitions to a higher microstate $x + \gamma(1 - x)(1 - y)$, transitioning closer to 1.
• Figure 3: Illustrations of transition effects based on function $\phi$, which models an agent in subsystem $i$ with microstate $x$ moving to a microstate $u$ in subsystem $k$. For these functions, the direction of the transition depends on threshold $a$. (a) Transition function $\phi_T$: if $x < a$, the agent shifts to the right, and if $x > a$, the agent shifts to the left. In both cases, the agents move toward $a$. (b) Transition function $\phi_A$: if $x < a$, the agent shifts to the left, and if $x > a$, the agent shifts to the right. In both cases, the agents move away from $a$, toward $1$ if $x > a$ and toward $0$ if $x < a$.
• Figure 4: Evolution of density $f_1$ for different transition functions $\phi$ (defined in Table \ref{['tab:phitab']}) and $\gamma = 0.4$; each row corresponds to one of the four representative families of functions. Initial conditions are the same for $\phi_L$, $\phi_R$, and $\phi_A$ to allow for direct comparison of agent movement patterns; the example for $\phi_T$ is otherwise chosen to show a case starting from a bimodal distribution. All simulations are run using MPCM with $100$ collocation points over the time interval $[0,20]$.
• Figure 5: Comparison of the self-convergence metric and time needed to run for system \ref{['EQ:oneparticles']} with different transition rates from Table \ref{['tab:phitab']}. (a) Log-log plot of the self-convergence metric for different numbers of nodes $N$. This plot shows that the difference between simulations with $N$ and $2N$ approaches zero superalgebraically as $N$ grows. (b) Log-log plot of the average runtime needed to run for different initial values and transition rates; runtime and computational cost are observed to be around $O(N^3)$.

Theorems & Definitions (6)

• Theorem 1: Existence and Uniqueness of Solution
• Proposition 1
• proof
• Proposition 2
• proof
• proof