Renormalization of complex networks with partition functions

While renormalization groups are fundamental in physics, renormalization of complex networks remains vague in its conceptual definition and methodology. Here, we propose a novel strategy to renormalize complex networks. Rather than resorting to handling the bare structure of a network, we overlay it with a readily renormalizable physical model, which reflects real-world scenarios with a broad generality. From the renormalization of the overlying system, we extract a rigorous and simple renormalization group transformation of arbitrary networks. In this way, we obtain a transparent, model-dependent physical meaning of the network renormalization, which in our case is a scale transformation preserving the transition dynamics of low-density particles. We define the strength of a node in accordance with the physical model and trace the change of its distribution under our renormalization process. This analysis demonstrates that the strength distributions of scale-free networks remain scale-invariant, whereas those of homogeneous random networks do not.