Properties of Discrete Laplacians With Application on Brain Networks

Abstract: This thesis investigates three discrete Laplace operators: the graph Laplacian, combinatorial Laplacian, and the more recently introduced persistent Laplacian. We discuss how these operators relate to each other and study their spectral properties. The graph Laplacian is a well-studied operator that plays a central role in spectral graph theory. Its spectrum contains information about the connectivity of the underlying graph and is the foundation for spectral clustering. The combinatorial Laplacian is a natural generalization of the graph Laplacian to simplicial complexes. It encodes the homology of the underlying simplicial complex and provides a way to compute the Betti numbers of the simplicial complex and cluster its simplicies based on the homology. The persistent Laplacian is a bridge between persistent homology and the theory of discrete Laplacians. It extends the combinatorial Laplacian to simplicial pairs and can be used to compute persistent Betti numbers. The last part of this thesis is a brief exploratory analysis of brain network models representing a progression of Parkinson's based on these discrete Laplacians.