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

The gist: A gauge-invariant clustering coefficient for complex-weighted bipartite networks reduces to classical measures when phases vanish, becomes negative under destructive interference, and factorizes into topological and phase contributions.

Introduction

Structural measures like the clustering coefficient characterize how a network is organized, typically formulated for real weights (pg1). However, a class of quantum and photonic systems utilizes complex edge weights whose phases determine interference between alternative routes (pg3). This work defines 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 (pg3). The authors show that the phase accumulated around a square is the smallest gauge-invariant carrier of structural phase information in a bipartite network (pg3).

Complex-Weighted Bipartite Networks

A bipartite network is defined as a triple G = (U, V, E) where links carry complex weights wuv = wuv e iφuv (pg5). The weight matrix A is Hermitian because links only connect nodes of different classes (pg5). The smallest closed cycle in such networks is a four-node square instead of a triangle (pg3). The flux of a square, denoted as Φs, is the quantity entering a tight-binding Hamiltonian through the Peierls substitution (pg5). This flux is invariant under a change of phase reference because it is invariant under node-wise phase redefinitions (pg5).

Clustering Coefficient Definition and Factorization

The complex square-based clustering coefficient Cu is defined as the product of a topological factor C(0)u and a phase factor Qu, such that Cu = C(0)u Qu (pg7). The topological factor C(0)u depends only on the moduli and connectivity alone and coincides with the classical weighted bipartite clustering coefficient (pg7). The phase factor Qu is a strength-weighted average of the plaquette cosines and depends on the fluxes alone (pg7). This decomposition permits the ensemble of Section 3.5 to be solved (pg7).

Topological and Phase Factors

Theorem 4.1 states that for any bipartite network and any node u lying on at least one square, E[Qu] = χ(1)4, and for φ ∼ U[−∆, ∆], E[Qu] = sin ∆/∆4 (pg9). Theorem 4.2 shows that the clustering coefficient factorizes into a topological contribution and a phase contribution, such that E[Cu] = C(0)u χ(1)4 (pg9). The mean phase contribution is (sin ∆/∆)4, independent of node, degree, and topology (pg7).

Open Paths and Coherent Distance

The open-path visibility I(uu') is defined as Muu' / Nuu', which equals unity when the routes reinforce perfectly and zero when they cancel completely (pg8). The coherent distance between two nodes implies the exponent at which the amplitude reaching one from the other decays, realized through apparent path length (pg8). Phase disorder leaves hop-count distances unchanged but increases the distance experienced by coherent signals (pg8).

Ensemble Analysis and Results

For phases uniformly distributed on [−∆, ∆], the mean phase contribution is (sin ∆/∆)4, independent of node, degree, and topology (pg7). The ensemble is a bipartite Watts–Strogatz model where rewiring modifies only connectivity and phase disorder modifies only complex weights (pg8). The mean phase factor is E[Qu] = χ(1)4, and the variance of the phase factor equals 1/mu, where mu is the number of squares at a node, only at complete phase disorder (pg7).

Conclusion

The clustering coefficient is robust to weak disorder as it contains a topological factor that depends only on connectivity and a phase factor that depends on the fluxes alone (pg8). The topological factor C(0)u depends only on connectivity and retains the structural dependence of the classical Watts–Strogatz model, whereas Qu depends only on the square fluxes and can be analyzed independently (pg8). The phase disorder leaves hop-count distances unchanged but increases the apparent distance by sixteen per cent on networks with identical connectivity (pg8). The small-world coefficient is exactly independent of phase against a matched reference and equal to its classical real-weighted value (pg8). The final result extends the argument from closed motifs to open ones, showing that the topological factor C¯(0) depends on p alone, decreasing (pg8). The phase factor Q¯ depends on ∆ alone and follows (sin ∆/∆)4 (pg8). The product of the topological factor and the phase factor is the clustering coefficient (pg8). The standard deviation decays as k − 1/2 with degree k, rather than as m − 1/2u (pg8). The topological factor falls from 0.53 to 0.02 across the rewiring range, while the mean hop-count distance falls from 18.7 to 3.8 (pg8). The phase disorder leaves the hop-count distance unchanged, since it depends only on the pattern of links, but increases the apparent distance by sixteen per cent on networks with identical connectivity (pg8). The flux disorder therefore does not disconnect the network; it introduces cancellation between alternative routes (pg8). The network remains equally navigable in a combinatorial sense, while becoming measurably harder to traverse coherently (pg8). The scope of our results is set by several assumptions, including phases are drawn independently for each link and are static (pg8). The treatment is moreover static, the propagator describing steady-state amplitude transfer rather than dynamics (pg8). The final result extends the argument from closed motifs to open ones (pg8). The phase factor is measurable on existing hardware, as circuit quantum electrodynamics devices are characterized by both edge strengths and phases (pg8).

How it works

The clustering coefficient Cu is built from the real parts of square amplitudes, normalized by the strength of the three-link paths that could close into squares (pg7). The numerator retains those closing into a square, each weighted by the cosine of its flux (pg7). A node whose alternative routes reinforce contributes positively and one whose routes cancel contributes negatively (pg7). The coefficient admits a factorisation into two parts of distinct character, C(0)u and Qu (pg7).

Topological and Phase Factors

The topological factor C(0)u depends on the moduli and the connectivity alone and coincides with the classical weighted bipartite clustering coefficient (pg7). The phase factor Qu is a strength-weighted average of the plaquette cosines and depends on the fluxes alone (pg7). This decomposition permits the ensemble of Section 3.

1. Bipartite Network Structure Analysis for Quantum/Photonic Systems: The improved system can calculate a gauge-invariant clustering coefficient for complex-weighted bipartite networks, which is defined as a measure that reduces to the classical bipartite coefficient when the phases vanish, becomes negative when alternative routes cancel, allowing researchers to quantify local closure while explicitly incorporating destructive interference effects.

2. Phase Disorder Robustness Assessment: The system can predict the effect of phase noise on local structure by calculating the mean phase contribution as (sin ∆/∆)4, which is independent of node, degree, and topology, providing a predictable metric for how phase disorder increases coherent distance.

3. Coherent Distance Mapping: The improved system can calculate a coherent distance via the apparent path length formula, which the paper defines as log Gxy / log α, enabling it to distinguish between combinatorial reachability and coherent transport by showing that phase disorder makes a pair further apart than a pair of the same hop count whose routes reinforce.

4. Structural Factorization for Tractable Analysis: The system can analytically factorize the clustering coefficient into a topological contribution and a phase contribution, which allows researchers to separates the structural and phase-dependent components and makes the coefficient analytically tractable for studying network organization independently of interference effects.

5. Small-World Coefficient Interpretation: The system can determine that the small-world coefficient is exactly independent of phase disorder and equal to its classical real-weighted value when comparing networks with a matched reference, clarifying that the observed phase dependence in this measure is a consequence of the choice of reference network, not an intrinsic property.

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 (Δ/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.