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