Coarse-grained methods

The dynamics of network and its topology usually associate with multiscale processes spanning from microscopic to mesoscopic, and to macroscopic level. It is known that brute-force simulations are quite expensive and sometimes even become impossible. While phenomenological models, such as mean-field description, may capture certain properties of the system, but often ignore microscopic details and fluctuation effects that may be important near critical points. Therefore a promising way is to develop multiscale theory and approaches, aiming at significantly reducing the degree of freedom while properly preserving the microscopic information of interest. In the present work, we have proposed a strength-based CG (s-CG) method to study critical phenomena of the Potts model on weighted complex networks. By merging nodes with close strengths together, the original network is reduced to a CG network with much smaller size, on which the CG Hamiltonian can be well defined. In particular, we make an error analysis and show that our s-CG approach satisfies the condition of statistical consistency, which demands that the equilibrium probability distribution of the CG model matches that of the microscopic counterpart. Extensive numerical simulations are performed on scale-free networks and random networks, without or with strength correlation, showing that this s-CG approach works very well in reproducing the phase diagrams, fluctuations, and finite-size effects of the microscopic model.