This project aims to develop new ways to model complex network evolution, and will build on synergies between research at USP-Sao Carlos on network evolution and work at York on the spectral characterisation of graphs. In the first year of the project we will study how characterisations of graphs based on the eigenvalues and eigenvectors of the node-based Laplacian matrix evolve with times, and how to model this evolution process. In the second year we will develop newmodels of network evolution based around the hitherto unexplored edge-based Laplacian. The project will involve 10-14 day visits academic staff, and exchanges of PhD students. Each site will perform three academic visits and one PhD exhange. During the academic visits, we plan to organise exchange seminars and short research oriented lecture courses. The research will potentially have a transformative role in how complex network evolution is modelled. We plan to continue the work through applications to our national funding bodies, network activities under WUN, and funding for exhange visits under the Royal Society.