Modularity spectra, eigen-subspaces, and structure of weighted graphs

The role of the normalized modularity matrix in finding homogeneous cuts will be presented. We also discuss the testability of the structural eigenvalues and that of the subspace spanned by the corresponding eigenvectors of this matrix. In the presence of a spectral gap between the k − 1 largest absolute value eigenvalues and the remainder of the spectrum, this in turn implies the testability of the sum of the inner variances of the k clusters that are obtained by applying the k -means algorithm for the appropriately chosen vertex representatives.