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

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.