A fully-connected graph is one of the simplest model for general networks (Internet, citations, polymer conformations, etc.) with high node connectivity [1,2]. The propagation of information through the network can be described in terms of the vibrational problem for disordered atomic network (nodes-atoms, links-springs). Thus, well-known analytical techniques (e.g., multiple-scattering formalism) from condensed matter physics  can be applied to study the important properties of the networks.

The aim of the project is to investigate the vibrational properties of the fully-connected graph with mass disorder assuming no force-constant disorder. For such a problem, the exact solution can be obtained in terms of the Green's functions  for a particular realization of disorder. The next very challenging step is inperforming the configurational averaging. Recently, we have found the solution for bimodal distribution. How to treat more general continuous mass distribution is not clear. This question is supposed to be studied by the applicant.