Weighted (Co)homology and Weighted Laplacian

Abstract

In this paper, we generalize the combinatorial Laplace operator of Horak and Jost by introducing the $\varphi$-weighted coboundary operator induced by a weight function $\varphi$. Our weight function $\varphi$ is a generalization of Dawson's weighted boundary map. We show that our above-mentioned generalizations include new cases that are not covered by previous literature. Our definition of weighted Laplacian for weighted simplicial complexes is also applicable to weighted/unweighted graphs and digraphs.

24 pages

References in corpus (4)

• TopologyNet: Topology based deep convolutional neural networks for biomolecular property predictions
• Persistent cohomology for data with multicomponent heterogeneous information
• Computational Tools in Weighted Persistent Homology
• Weighted Persistent Homology

Cited by in corpus (7)

• Discrete Morse Theory for Weighted Simplicial Complexes
• Persistent cohomology for data with multicomponent heterogeneous information
• Weighted persistent homology for biomolecular data analysis
• Persistent spectral graph
• Weighted persistent homology for osmolyte molecular aggregation and hydrogen-bonding network analysis
• Towards a Quantitative Theory of Digraph-Based Complexes and its Applications in Brain Network Analysis
• Weighted Fundamental Group