# Weighted Combinatorial Laplacian and its Application to Coverage Repair in Sensor Networks

arXiv: https://arxiv.org/abs/2312.04825 · HTML: https://mlsanity.com/paper/2312.04825/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Dec 2023.

## Built on by (most cited first, 2 of 2)

- [Topology-Preserving Neural Operator Learning via Hodge Decomposition](https://mlsanity.com/paper/2605.13834/) (2026)
- [A differentiability framework for zigzag persistent homology via linear interpolation](https://mlsanity.com/paper/2609.39242/) (2026)

## Abstract (arXiv)

We define the weighted combinatorial Laplacian operators on a simplicial complex and investigate their spectral properties. Eigenvalues close to zero and the corresponding eigenvectors of them are especially of our interest, and we show that they can detect almost n-dimensional holes in the given complex. Real-valued weights on simplices allow gradient descent based optimization, which in turn gives an efficient dynamic coverage repair algorithm for the sensor network of a mobile robot team. Using the theory of relative homology, we also extend the problem of dynamic coverage repair to environments with obstacles.