title: Detecting phase transitions in collective behavior using manifold's curvature.
abstract: If a given behavior of a multi-agent system restricts the phase variable to an invariant manifold, then we define a phase transition as a 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 firstly 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. Secondly, 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 Transition Detection (PTD) method using one particle simulation and three real world examples.
authors.affiliation: Department of Mathematics, Clarkson University, Potsdam, NY-13699, United States. email: dineshhk@clarkson.edu.
authors.forename: Kelum
authors.initials: K
authors.lastname: Gajamannage
authors.forename: Erik M
authors.initials: EM
authors.lastname: Bollt
mesh_terms: D000465:Algorithms; D003198:Computer Simulation; D008962:Models, Theoretical; D015203:Reproducibility of Results