Graph Laplacians, Riemannian Manifolds and their Machine-Learning
This work provides tools for analyzing graph properties using machine learning, which could benefit researchers in combinatorics, geometry, and theoretical physics, though it appears incremental as it applies existing methods to new data.
The researchers applied machine learning techniques to analyze 8,000 finite graphs from the Wolfram database, finding that neural classifiers, regressors, and networks can efficiently and accurately perform tasks such as recognizing graph Ricci-flatness, predicting spectral gaps, and detecting Hamiltonian cycles.
Graph Laplacians as well as related spectral inequalities and (co-)homology provide a foray into discrete analogues of Riemannian manifolds, providing a rich interplay between combinatorics, geometry and theoretical physics. We apply some of the latest techniques in data science such as supervised and unsupervised machine-learning and topological data analysis to the Wolfram database of some 8000 finite graphs in light of studying these correspondences. Encouragingly, we find that neural classifiers, regressors and networks can perform, with high efficiently and accuracy, a multitude of tasks ranging from recognizing graph Ricci-flatness, to predicting the spectral gap, to detecting the presence of Hamiltonian cycles, etc.