Statistical physics is a method of describing our many-body world in terms of probability, and network science is part of it. One advantage network science has over the conventional statistical mechanics is that it deals with quenched disorder system relatively easily. This thesis, with the help of scaling analysis, focuses on a nonequilibrium process called generalized epidemic process (GEP), which is a epidemic spreading model with synergy effect, on several structures with different types of quenched disorder: Poisson random, modular and scale-free networks.
Chapter 2 discusses GEP on mean-field level, which means it is implemented on Poisson random network. A self-contained argument of the universality classes of GEP is proposed analytically and veri- fied numerically by extended-finite-size scaling theory. The results show that GEP with large cooperative effect exhibits a discontinuous phase transition at percolation threshold, while small cooperative effect results in continuous phase transition that belongs to the universality class of bond-percolation.
Chapter 3 discusses the universality class of the cooperative process as modular structure is intro- duced by a rewiring parameter. Community structure is expected to change the critical properties of GEP since cooperative effect is amplified in clustered network. However, by mapping modular network to clique-based random network, we analytically show that cooperation only affects local infection and is helpless in infecting other neighboring communities. Therefore, modularity doesn’t change the universal- ity class of GEP but it only shifts the epidemic threshold and tricritical point, which are also numerically verified through finite-size scaling forms and bimodality coefficient respectively.
Chapter 4 focuses on the role of hubs in GEP since hubs influence many processes in nontrivial way as in percolation. Therefore, GEP is implemented on uncorrelated scale-free network with varying degree exponent, $\alpha$, and it is analytically and numerically discussed with the help of self-consistency equation and scaling analysis respectively. The results show that the process exhibits mixed-order transition regardless of the heterogeneity of the structure, owing to the locally tree-likeness of the structure. In addition, the result indicate that as long as cooperative effect is not null, the outbreak size of process always converges to a nonzero value as infection probability goes to 0 on structures with dominant hubs, 2 < $\alpha$ < 3.