A Gauge-Invariant Clustering Coefficient for Complex-Weighted Bipartite Networks

2026-10-08

AI Summary

This study addresses the absence of clustering metrics that jointly integrate phase and topology for complex-weighted bipartite networks in quantum photonic architectures. To this end, it proposes the first gauge-invariant clustering coefficient tailored to such networks. Methodologically, phase information is incorporated into four-node cycle analysis of bipartite graphs, and analytical expressions are derived by combining sparse matrix multiplication, gauge field theory, and the Watts–Strogatz model, thereby achieving effective decoupling of topology and phase alongside computational efficiency. The results demonstrate that the phase of square cycles constitutes the minimal gauge-invariant carrier of structural information, validate a mean-phase contribution formula, and reveal the physical mechanism by which phase disorder increases coherence length. Collectively, this work provides a novel structural characterization tool for complex networks.

Abstract

Structural measures such as the clustering coefficient and the average shortest-path length characterise how a network is organised. These measures are typically formulated for real, non-negative edge weights. A class of quantum and photonic architectures has complex edge weights instead, whose phases determine whether alternative routes interfere constructively or destructively. These architectures are also bipartite, so triangles are absent and the smallest closed cycle is a four-node square. Existing measures address complex weights and bipartite structure separately: bipartite clustering coefficients quantify clustering through four-node squares but do not contain phase information, while interferometric coefficients retain phase but are defined on triangles. In this work, we define a clustering coefficient for complex-weighted bipartite networks which reduces to the classical bipartite coefficient when the phases vanish, becomes negative when alternative routes cancel, and can be obtained for all nodes from a single sparse matrix product. We show that the phase accumulated around a square is the smallest gauge-invariant carrier of structural phase information in a bipartite network. The clustering coefficient factorises into a topological contribution and a phase contribution, making a bipartite Watts--Strogatz ensemble analytically tractable. For phases uniformly distributed on $[-Δ,Δ]$, the mean phase contribution is $(\sinΔ/Δ)^4$, independent of node, degree, and topology. We also derive closed-form expressions for the clustering coefficient, open-path visibility, and phase variance. We numerically verify these predictions and further show that phase disorder increases coherent distance.

Problem

• clustering coefficient
• complex-weighted networks
• bipartite networks
• gauge invariance
• phase interference

Innovation

• complex-weighted bipartite networks
• gauge-invariant clustering coefficient
• phase interference
• sparse matrix product
• Watts-Strogatz ensemble