# Detecting phase transitions in collective behavior using manifold's curvature

arXiv: https://arxiv.org/abs/1509.07078 · HTML: https://mlsanity.com/paper/1509.07078/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Sep 2015.

## Built on by (most cited first, 3 of 3)

- [Recurrent Neural Networks for Dynamical Systems: Applications to Ordinary Differential Equations, Collective Motion, and Hydrological Modeling](https://mlsanity.com/paper/2202.07022/) (2022)
- [Reconstruction of Fragmented Trajectories of Collective Motion using Hadamard Deep Autoencoders](https://mlsanity.com/paper/2110.10428/) (2021)
- [Fraud Detection Using Optimized Machine Learning Tools Under Imbalance Classes](https://mlsanity.com/paper/2209.01642/) (2022)

## Abstract (arXiv)

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.