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 show a very particular phase diagram. In fact when the second moment of the degree ditribution <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, example of them are percolation phase transition, the spreading of diseases on networks, the synchronization phenomena on networks, but also congestion phase transitions in technological networks, rumor spreading in society. For all this 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. More recently I worked in percolation problems in correlated network ensembles and in hypergraphs, in rumor spreading and in congestion transition using analytically solvable models. The congestion transition has a very rich phase diagram and indeed the transition can become from a second order to a first order transition when routing is taking place.