Quantifying Emergent Behaviors in Agent-Based Models using Mean Information Gain

TL;DR

This Letter proposes using the Mean Information Gain (MIG) as a metric to quantify emergence in Agent-Based Models and applies it to a multi-agent biased random walk that reproduces Wolfram's four behavioral classes and shows that MIG differentiates these behaviors.

Abstract

Emergent behaviors are a defining feature of complex systems, yet their quantitative characterization remains an open challenge, as traditional classifications rely mainly on visual inspection of spatio-temporal patterns. In this Letter, we propose using the Mean Information Gain (MIG) as a metric to quantify emergence in Agent-Based Models. The MIG is a conditional entropy-based metric that quantifies the lack of information about other elements in a structure given certain known properties. We apply it to a multi-agent biased random walk that reproduces Wolfram's four behavioral classes and show that MIG differentiates these behaviors. This metric reconnects the analysis of emergent behaviors with the classical notions of order, disorder, and entropy, thereby enabling the quantitative classification of regimes as convergent, periodic, complex, and chaotic. This approach overcomes the ambiguity of qualitative inspection near regime boundaries, particularly in large systems, and provides a compact, extensible framework for identifying and comparing emergent behaviors in complex systems.

Figures (4)

• Figure 1: Illustration of agent perception and movement based on the vision and superposition parameters. The green cells define the orthogonal vision range, while the blue cells correspond to the Von Neumann neighborhood. In this example, agent i has orthogonal vision and selects agent j as its target. If superposition is enabled, agent i takes one step in j direction and moves directly onto agent k's cell, as indicated by the orange arrow. If superposition is disabled, agent i moves to the nearest available cell in the direction of agent j, as indicated by the green arrow.
• Figure 2: Emergent phenomena for each system configuration. (a) Convergent regime: Initially oscillates in one direction until it converges in the border. (b) Periodic regime: Quickly converges into oscillating clusters, in some cases they all converge to only one. (c) Complex regime: Present a close-to-chaotic behavior but agents tend to move together. (d) Chaotic regime: Does not present any recognizable pattern.
• Figure 3: MIG per direction, averaged over time and across all repetitions, for each behavioral regime. Convergent: $G_{up}=0.1181 \pm 0.0024$, $G_{down}=0.1182 \pm 0.0025$, $G_{right}=0.1180 \pm 0.0040$, $G_{left}=0.1220 \pm 0.0040$; Periodic: $G_{up}=0.14 \pm 0.04$, $G_{down}=0.14 \pm 0.04$, $G_{right}=0.13 \pm 0.04$, $G_{left}=0.13 \pm 0.04$; Complex: $G_{up}=0.928 \pm 0.005$, $G_{down}=0.928 \pm 0.005$, $G_{right}=0.928 \pm 0.006$, $G_{left}=0.928 \pm 0.006$; Chaotic: $G_{up}=0.9776 \pm 0.0025$, $G_{down}=0.9776 \pm 0.0025$, $G_{right}=0.9775 \pm 0.0025$, $G_{left}=0.9775 \pm 0.0025$.
• Figure 4: Trajectory of agent's average position in the complex (left) and chaotic (right) regimes, from initial (green) to final (orange) position. In the complex regime, the system behaves like a coordinated macro agent performing a two dimensional random walk, exploring the available space. In contrast, the chaotic regime lacks collective coordination and remains localized near its starting point, without a clear spatial trend.