Graph theory based approach to identify phase transitions in condensed matter

This repository contains the implementation of key algorithms, computed results, and Python scripts for visualizing all figures from the article "Graph theory based approach to identify phase transitions in condensed matter".

The related article is published in Physical Review B (PRB) and can be accessed at https://journals.aps.org/prb/abstract/10.1103/PhysRevB.111.054116.

Molecular Dynamics (MD) Simulation

See "MD_simulation" directory for the LAMMPS scripts.

Graph Construction

Fully connected and node-labelled graphs with edge weights representing the spatial distances between particles.

Algorithm Implementation

This article only outlines the core idea of the algorithm and does not delve into every detail. In the implementation, we use the trick of extending $A$ and $X$ to square matrices and expressing the objective function as taking the strict trace of a square matrix, which can be found in Algorithm.pdf in this repository.

Please note that its content is consistent with Algorithm.md, but since GitHub does not support rendering LaTeX formulas in markdown files, the markdown version may not display correctly.

Implementation of Cut Norm (All Languages)

The Implementation of Cut Distance (for node-labelled graphs) in pure C, C++, Java, Python, Fortran, Matlab, and Mathematica (including both imperative and functional programming).

See repo https://github.com/anwanguow/cut_dist_all_languages.

Computation and Testing of Cut Distance (Python)

See "cut_dist" directory.

Figures

The generation of Fig.6 requires the complete trajectory data from the KA system simulation. This data has been uploaded to Baidu Netdisk at the following link: https://pan.baidu.com/s/13V_jR030NeefVrGKnXca4Q and the access code is: f9ud. For readers who do not have access to Baidu Netdisk, please contact me directly if you need this data. My email is [email protected].

All python scripts in "Plot" directory reproduces the computed figures in the article, including Fig.2b), Fig.2c), Fig.3a), Fig.3b), Fig.4a), Fig.4b), Fig.5b), Fig.5c), Fig.8a) and Fig.8b). Besides, the generated figures are saved in "Figure" directory, demonstrated as follow:

Fig.2b)

Fig.2c)

Fig.3a)

Fig.3b)

Fig.4a)

Fig.4b)

Fig.5b)

Fig.5c)

Fig.8a)

Fig.8b)

Reference

Please consider adding the following citation if you use this work in your research.

@article{PhysRevB.111.054116,
  title = {Graph theory based approach to identify phase transitions in condensed matter},
  author = {Wang, An and Sosso, Gabriele C.},
  journal = {Phys. Rev. B},
  volume = {111},
  issue = {5},
  pages = {054116},
  numpages = {10},
  year = {2025},
  month = {Feb},
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevB.111.054116},
  url = {https://link.aps.org/doi/10.1103/PhysRevB.111.054116}
}

Contact:

An Wang: [email protected]