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We define the weighted combinatorial Laplacian operators on a simplicialcomplex and investigate their spectral properties. Eigenvalues close to zeroand the corresponding eigenvectors of them are especially of our interest, andwe 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 thesensor network of a mobile robot team.

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