The Laplacian spectra of small-world networks

Spectral properties of the Laplacian operator are studied numerically on the Watts-Strogatz-type small-world networks. The spectra on small-world networks are shown to have the characteristics that (i) the gap between the lowest frequency mode (Goldstone mode) and the second lowest frequency mode is small in contrast to that of random networks and (ii) the average participation ratio is as small as that of random networks. We also find that the highest frequency is essentially determined by the largest coordination degree.