Multifractal analysis of complex networks reconstructed with the geometric information from the embedded hyperbolic space

In recent years, multifractal analysis of complex networks primarily focuses on the topological scale, where the distances between nodes are characterized through their topological shortest-path lengths. In this study, we integrate geometric information into the multifractal analysis framework of networks, enabling the distances between nodes to be expressed through geometric information. We utilize these geometric information to assign weights to each edge of the original network, thereby reconstructing the network in a way that simultaneously captures both topological and geometric information. We analyze changes in the multifractal spectrum of these reconstructed networks using the sandbox algorithm for multifractal analysis of weighted networks. This approach not only enriches our understanding of the network structures but also provides new insights into the intrinsic mechanisms of complex systems, specifically revealing that the synergistic interplay between network topology and geometric weight assignments critically regulates the emergence of multiscale complexity. By combining topological and geometric information, we can more comprehensively reveal the multifractal structure and heterogeneity of networks, particularly the relationships between hub nodes and non-hub nodes and their impact on the overall network characteristics. We conduct experimental analyses on both model networks, computational mesh networks and real-world networks, and find that the introduction of geometric information has varying degrees of influence on their generalized fractal dimensions.