Spectra and representations of some special weighted graphs are investigated with weight matrices consisting of homogeneous blocks. It is proved that a random perturbation of the weight matrix or that of the weighted Laplacian with a ''Wigner-noise'' will not have an effect on the order of the protruding eigenvalues and the representatives of the vertices will unveil the underlying block-structure. Such random graphs adequately describe some biological and social networks, the vertices of which belong either to loosely connected strata or to clusters with homogeneous edge-densities between any two of them, like the structure guaranteed by the Regularity Lemma of Szemeredi.
['Marianna Bolla']
Noisy random graphs and their Laplacians