Mesoscopic structures and the Laplacian spectra of random geometric graphs

Abstract

Max Planck Institute for the Physics of Complex Systems,No¨thnitzer Strase 38, Dresden D-01187, Germany(Dated: December 3, 2014)We investigate the Laplacian spectra of random geometric graphs (RGGs). The spectra are foundto consist of both a discrete and a continuous part. The discrete part is a collection of Diracdelta peaks at integer values roughly centered around the mean degree. The peaks are mainlydue to the existence of mesoscopic structures that occur far more abundantly in RGGs than innon-spatial networks. The probability of certain mesoscopic structures is analytically calculated forone-dimensional RGGs and they are shown to produce integer-valued eigenvalues that comprise asignificant fraction of the spectrum, even in the large network limit. A phenomenon reminiscent ofBose-Einstein condensation in the appearance of zero eigenvalues is also found.