There is a rich interplay between network topologies and critical phenomena on networks.
The Ising model, a paradigmatic example of dynamical process, in scale-free networks shows a very particular phase diagram. In fact when the second moment of the degree distribution <k^{2}> diverges the critical temperature for the onset of the paramagnetic phase is infinity, (i.e. the system is always ordered!)
Dynamical process on networks might be very diverse: the percolation phase transition, the spreading of diseases on networks, the synchronization phenomena on networks, congestion phase transitions in technological networks and rumor spreading in society. For all these networks the small world behavior, the heterogeneous degree distribution or their spectral properties play a crucial role in determining their phase diagram.
I have worked in the Ising model of networks setting up the theoretical framework to treat the Ising model in annealed networks and extending this work to quantum critical phenomena including the Traverse Ising Model, the Bose-Hubbard model. Moreover I have worked in percolation problems in correlated network ensembles and in hypergraphs, in rumor spreading and in congestion transition using analytically solvable models. Recently I actively work on percolation phenomena on multilayer network including work on multiplex networks and network of networks.