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.

翻译：我们在单纯复形上定义了加权组合拉普拉斯算子，并研究了其谱性质。我们特别关注其接近零的特征值及对应的特征向量，并证明它们能够检测给定复形中的几乎n维空洞。单纯形上的实值权重允许基于梯度下降的优化，进而为移动机器人团队的传感器网络提供了一种高效的动态覆盖修复算法。