Detecting phase transitions in collective behavior using manifold's curvature

TL;DR

This paper introduces a method to detect phase transitions in multi-agent systems by analyzing manifold curvature and singular value ratios, validated through simulations and real-world data.

Contribution

It proposes a novel approach linking curvature and singular value ratios to identify phase transitions and split manifolds into distinct sub-manifolds.

Findings

Effective detection of phase transitions in simulated and real data

Manifold splitting corresponds to physical changes in behavior

Method extends to higher-dimensional manifolds using shape operator

Abstract

If a given behavior of a multi-agent system restricts the phase variable to a invariant manifold, then we define a phase transition as change of physical characteristics such as speed, coordination, and structure. We define such a phase transition as splitting an underlying manifold into two sub-manifolds with distinct dimensionalities around the singularity where the phase transition physically exists. Here, we propose a method of detecting phase transitions and splitting the manifold into phase transitions free sub-manifolds. Therein, we utilize a relationship between curvature and singular value ratio of points sampled in a curve, and then extend the assertion into higher-dimensions using the shape operator. Then we attest that the same phase transition can also be approximated by singular value ratios computed locally over the data in a neighborhood on the manifold. We validate the phase transitions detection method using one particle simulation and three real world examples.