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

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.