Two-mode geometry controls multiscale organization in bipartite systems

Abstract

Many complex systems are organized around complementary roles and naturally described as bipartite networks. Unveiling their multiscale structure presents a fundamental challenge because coarse-graining procedures must preserve role separation, whereas standard approaches collapse it via one-mode projections. Here we introduce a Laplacian-based renormalization framework that operates directly on the bipartite architecture, enabling scale transformations while retaining role differentiation. Using controlled bipartite ensembles at criticality, we show that structural imbalance systematically reshapes organization across scales while leaving scaling properties invariant, revealing a separation between universality and geometry. Applying the coarse-graining framework to empirical bipartite networks, we uncover nontrivial multiscale hierarchies for both roles. In contrast, renormalization performed after one-mode projection -- which truncates diffusion paths to nearest neighbors -- yields qualitatively different structures. Our results identify two-mode geometry as a fundamental constraint for revealing multiscale organization in systems with role separation.