Mean-field dynamics of attractive resource interaction: From uniform to aggregated states

TL;DR

The model generalizes several classical mean-field and opinion-dynamics frameworks and is defined on the standard simplex, where each coordinate evolves according to an interaction rule depending on pairwise preference functions.

Abstract

We introduce and study a nonlinear discrete dynamical system describing the evolution of a resource distribution among interacting agents. The model generalizes several classical mean-field and opinion-dynamics frameworks and is defined on the standard simplex, where each coordinate evolves according to an interaction rule depending on pairwise preference functions. We provide a complete analytical description of the long-term behavior of the system.

Figures (7)

• Figure 1: Trajectories of the dynamic conflict system for the case of three-dimensional space under the action of conflict transformation \ref{['eq:2']}, i.e., the dynamics of change of arbitrary stochastic vectors ${\bf{p}}=(p_{1}; p_{2}; p_{3})$. The vertices of the triangle are identified with the coordinate sets $(1; 0; 0)$, $(0; 1; 0)$, and $(0; 0; 1)$. The limit values of the vectors ${\bf{p}}$ converge to the center of this triangle, meaning the center of the triangle is an attractor. The vertices of the triangle $P_1(1; 0; 0)$, $P_2(0; 1; 0)$, and $P_3(0; 0; 1)$ are unstable repellers. Only the center of the triangle, i.e., the point $\left(\frac{1}{3}; \frac{1}{3}; \frac{1}{3}\right)$, is stable. All other points of the triangle are unstable.
• Figure 2: Trajectories of the dynamic conflict system for the three-dimensional space case under the action of the conflict transformation \ref{['eq:3.2']}. Dynamics of change of arbitrary stochastic vectors ${\bf{p}}=(p_{1}; p_{2}; p_{3})$ with a set of constants: a) ${\bf{c}} = (0.3; 0.4; 0.25)$, ${\bf{p}}^\infty = (0.322; 0.4915; 0.1865)$ and b) ${\bf{c}} = (0.8; 0.1; 0.9)$, ${{\bf{p}}^\infty = (0.4706; 0; 0.5294)}$. In case b), the condition is violated, i.e., ${c_2 < \frac{c_1c_3}{c_1+c_3}}$, and therefore $p_2^\infty = 0$.
• Figure 3: Diagram of the transcritical bifurcation for the component $p_2^\infty$ as a function of the parameter $c_2$. This diagram visualizes the conclusions of Theorem \ref{['thm:bifurcation']} for parameters from Fig. \ref{['ris:3pop']}.b, $c_1=0.8$, $c_3=0.9$. The vertical green dotted line at $c_2=0.1$ indicates the parameter value from the example in Fig. \ref{['ris:3pop']}.b. It shows how the coordinate $p_2^\infty$ changes as the parameter $c_2$ is varied. For $c_2 > c_2^{\text{crit}}$, an exchange of stability occurs: the trivial state $p_2^\infty=0$ loses stability, and the new state with $p_2^\infty > 0$ becomes stable. This behavior is characteristic of a transcritical bifurcation.
• Figure 4: Bifurcation diagram in the parameter space $(c_1, c_2)$ illustrating the conclusions of Theorem \ref{['twobif']} for a system with $n=4$ components and fixed parameters $c_3 = c_4=1$. The diagram shows the plane of control parameters partitioned into four distinct regions by the transcritical bifurcation curves $\Gamma_1: c_1\left(\frac{1}{c_2} + \sum_{j=3}^n \frac{1}{c_j}\right) = n-2$ and $\Gamma_2: c_2\left(\frac{1}{c_1} + \sum_{j=3}^n \frac{1}{c_j}\right) = n-2$.
• Figure 5: Evolution of the vector coordinates from the initial state ${\bf p}^0 = (0.072; 0.183; 0.141; 0.115; 0.030; 0.030; 0.011; 0.1666; 0.116; 0.136)$ to the limit state ${\bf p}^\infty = (0.160; 0.148; 0.135; 0.122; 0.109; 0.096; 0.080; 0.068; 0.054; 0.028)$ under the action of conflict transformation \ref{['eq:2']}. All parameters $c_k$ satisfy the condition $c_k > \Lambda$, $k=\overline{1, 10}$ The limit state is entirely determined by the set of constants $\{c_i\}_{i=1}^{10}$, illustrating the optimization principles from \ref{['thm:max_coord']}--\ref{['thm:simultaneous_opt']}: the coordinate $p_1^t$ with the maximum $c_1=0.71$ converges to the highest value, while $p_{10}^t$ with the minimum $c_{10}=0.61$ converges to the lowest.

Theorems & Definitions (46)

• Remark 1
• Theorem 1
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Lemma 1
• proof