Geometric renormalization of weighted networks.

The geometric renormalization technique for complex networks has successfully revealed the multiscale self-similarity of real network topologies and can be applied to generate replicas at different length scales. Here, we extend the geometric renormalization framework to weighted networks, where the intensities of the interactions play a crucial role in their structural organization and function. Our findings demonstrate that the weighted organization of real networks exhibits multiscale self-similarity under a renormalization protocol that selects the connections with the maximum weight across increasingly longer length scales. We present a theory that elucidates this symmetry, and that sustains the selection of the maximum weight as a meaningful procedure. Based on our results, scaled-down replicas of weighted networks can be straightforwardly derived, facilitating the investigation of various size-dependent phenomena in downstream applications. Geometric renormalization reveals hidden network symmetries by scaling them down while retaining key features. Extended to weighted networks, in which link intensities matter, here the authors present empirical evidence and theory to justify selecting links with maximum weights across increasingly longer length scales to reduce resolution, enabling self-similar replicas and study of size-dependent phenomena. [ABSTRACT FROM AUTHOR]