  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.
