Random walks on simplicial complexes and harmonics.

Area of Science:

• Mathematics
• Computer Science
• Data Science

Background:

• Random walks are fundamental tools in analyzing complex systems.
• Absorbing states introduce boundaries or termination points in random processes.
• Simplicial complexes provide a higher-dimensional generalization of graphs.

Purpose of the Study:

• To define and analyze random walks with absorbing states on simplicial complexes.
• To explore the relationship between these random walks and the spectra of k-dimensional Laplacians.
• To apply this framework to semi-supervised learning problems, particularly label propagation.

Main Methods:

• Definition of random walks on simplicial complexes with absorbing states.
• Analysis of the k-dimensional Laplacian spectrum for 1 <= k <= d.
• Application to a generalized label propagation algorithm on oriented edges.

Main Results:

• A novel class of random walks on simplicial complexes is introduced.
• Connections are established between these walks and the spectral properties of generalized Laplacians.
• The method is demonstrated to be effective for semi-supervised learning tasks.

Conclusions:

• Random walks on simplicial complexes offer a powerful framework for analyzing complex data structures.
• The spectral properties of Laplacians provide insights into the behavior of these random walks.
• This approach advances semi-supervised learning by generalizing graph-based methods to higher dimensions.