Geometric Renormalization Reveals the Self-Similarity of Weighted Networks

The geometric renormalization (GR) group of complex networks based on hidden metric space provides a powerful framework for studying the self-similarity of networks. Recent studies have shown that this framework can significantly reduce the size and complexity of the initial system. In this sense, the smaller-scale replica can be used as an alternative or guidance to the original large-scale network. In this article, we extend the GR framework to the weighted network and prove that this framework can sustain the self-similarity of synthetic weighted networks and real-world weighted networks. Furthermore, we assign the corresponding weights to all edges of reconstructed human connectomes at five different resolutions, and the results show that topological features of these networks exhibit self-similar behaviors. Remarkably, our results also suggest that the GR transform group can generate a series of low-resolution replica networks that are similar to the initial highest-resolution human connectome networks, which greatly promotes the network science, neuroscience, and physics understanding of brain mechanisms, and is of great significance to the research of brain science. Finally, a typical spin-like model is used to further verify the rationality of this framework.