Detecting phase transitions in collective behavior using manifold's curvature
This work addresses the challenge of identifying behavioral shifts in collective systems, offering a novel mathematical framework that could be applied in fields like robotics or biology, though it appears incremental as it builds on existing manifold theory.
The paper tackles the problem of detecting phase transitions in multi-agent systems by analyzing changes in manifold curvature, proposing a method that uses singular value ratios and the shape operator to identify transitions where physical characteristics like speed and coordination change. They validate their approach with one simulation and three real-world examples.
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.