Two-mode geometry controls multiscale organization in bipartite systems

TL;DR

This paper introduces a bipartite-specific Laplacian renormalization method that preserves role separation across scales, revealing multiscale hierarchies and fundamental geometric constraints in bipartite networks.

Contribution

It develops a novel bipartite-aware coarse-graining framework that maintains role differentiation, unlike traditional one-mode projection methods.

Findings

Structural imbalance affects organization across scales.

The framework uncovers nontrivial multiscale hierarchies in empirical networks.

One-mode projection renormalization yields different structures.

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.