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

1 Introduction

• Existing measures treat bipartite square clustering without phase information or retain phase on triangles, which bipartite networks lack.The paper frames this as a gap in quantifying local structure when networks are both bipartite and complex weighted.Source: §1 · p. 2
• A four-node square is the smallest carrier of gauge-invariant phase information in a bipartite network.Link phases depend on convention, but the phase accumulated around a square is invariant under node-wise phase-reference changes.Source: §1 · pp. 2–3
• The coefficient uses square amplitudes, reduces to classical bipartite clustering when phases vanish, can become negative under destructive interference, and is gauge-invariant.One sparse matrix product computes it for all nodes.Source: §1 · p. 3
• The coefficient factorises into a connectivity-and-magnitude factor and a phase factor, separating structural and phase-dependent effects.This separation makes the coefficient analytically tractable.Source: §1 · p. 3
• For a bipartite Watts–Strogatz ensemble, the mean phase contribution is (sin ∆/∆)^4, independent of node, degree, and topology.Closed-form results also cover clustering, open-path visibility, and phase-factor variance.Source: §1 · p. 3
• Phase disorder leaves hop-count distance unchanged but increases apparent distance by sixteen per cent on networks with identical connectivity.The paper also reports that the phase contribution cancels from the small-world coefficient when a network is compared with its randomized version.Source: §1 · p. 3

2 Related Literature

• Bipartite clustering replaces absent triangles with four-cycles, but existing measures operate on real, non-negative weights [26].These approaches include global four-cycle transitivity and local or weighted square coefficients.Source: §2 · p. 4
• Complex-weighted network research generalizes standard measures and studies balance, spectra, random walks, and spectral clustering [22].Gauge invariance is also connected to structural balance through magnetic Laplacians [32].Source: §2 · p. 4
• Interferometric clustering compares competing path amplitudes and can become negative under destructive interference [23].The paper's coefficient is its bipartite counterpart, replacing a path and closing link with two two-step paths through distinct intermediaries.Source: §2 · p. 4
• The adopted apparent path length accounts for walks of different lengths, while coherent transport studies connect destructive interference with suppressed transfer [23, 37, 38, 39].Small-worldness comparisons can be sensitive to the randomized reference clustering [36].Source: §2 · p. 4
• No existing measure is simultaneously bipartite, phase-sensitive, and gauge-invariant; this work fills that gap with a coefficient built on square flux.The related approaches separately capture square structure or phase-dependent interference.Source: §2 · p. 4

3.1 Complex-Weighted Bipartite Networks

• The network is represented by a biadjacency matrix whose complex edge weights encode strength and phase, with reciprocal traversal represented by conjugate weights.The two node classes connect only across the bipartition, so the shortest closed paths are squares.Source: §3.1 · p. 5

3.2 The Square as the Elementary Unit of Interference

• A square's alternating link-phase sum is gauge-invariant, making the four-node square the smallest carrier of phase information in a bipartite network.The flux determines whether the two two-step routes reinforce or cancel.Source: §3.2 · pp. 5–6, Figure 2 · p. 6

3.3 Clustering Coefficient for Complex-Weighted Bipartite Networks

• The coefficient combines square amplitudes before taking real parts, normalises by possible closing paths, and can be negative when routes cancel.Its topological factor depends on moduli and connectivity, while its phase factor averages plaquette cosines.Source: §3.3 · pp. 6–7
• For unit strengths, the phase factor's attainable negative extreme is constrained by overlapping routes; complete cancellation requires a locally sparse arrangement.Fluxes on squares sharing a node pair obey additive relations, preventing all such squares from having flux π simultaneously.Source: §3.3 · p. 7
• Sparse matrix products compute node-level clustering in O(nk^2) time and memory, instead of O(n^2k) for explicit square enumeration.The method avoids enumerating squares and extends to nodes in V by replacing B with B^H.Source: §3.3 · p. 8

3.4 Open Paths and Coherent Distance

• Open-path visibility measures reinforcement among routes between same-class nodes, while coherent distance increases when competing walks cancel.The apparent-distance construction returns hop count when one walk dominates and phases vanish.Source: §3.4 · p. 8

3.5 A Bipartite Watts-Strogatz Ensemble

• The network model independently varies topology through rewiring probability p and phases through half-width ∆, allowing the mean clustering contribution to factorise.Rewiring preserves degrees in U and bipartite structure; the phase-dependent contribution is obtained in closed form, while the connectivity contribution requires numerical evaluation.Source: §3.5 · pp. 8–9

Mean phase factor

• For uniformly distributed link phases, the mean phase factor is (sin ∆/∆)^4, independent of topology, degree, and system size.Its response to weak phase disorder is quadratic, and it vanishes at ∆ = π.Source: §4 · p. 9

Factorisation

• The clustering coefficient factorises in expectation into a connectivity-only topological factor and a phase factor, rather than relying on an approximate decoupling.This lets the coefficient be predicted from the topological factor and a closed-form phase curve; a low coefficient alone cannot distinguish sparse connectivity from weak coherence.Source: §4 · p. 9

Estimator resolution

• The phase-factor estimator’s variance includes covariances from overlapping squares; at complete disorder its standard deviation is exactly m_u^-1/2.Away from complete disorder, shared links raise the variance, so the independent-square estimate 2σ²/m_u can understate uncertainty; at degree 6 it understates the standard deviation by factors 1.67 at ∆ = π/4 and 1.42 at ∆ = π/2.Source: §4 · p. 10
• On the regular lattice, for fixed ∆ < π, Var(Q_u) = 81κ₂/(20k) + O(k^-2), while at ∆ = π it equals 1/m_u.For ∆ < π, the standard deviation decays as k^-1/2 because squares share link phases; rewiring destroys squares and raises the detection threshold.Source: §4 · p. 10

Signed coefficient

• The signed coefficient has mean zero in an incoherent network, whereas an unsigned average can appear positive as the number of squares falls.At nodes with few squares, the variance determines how many realisations are needed to resolve coherence.Source: §4 · pp. 10–11

Open-path visibility

• Open-path visibility approaches q^-1/2 under complete disorder, the shot-noise level from adding q random phasors; unlike squares at a node, the paths in a pair share no links.The topological factor has no simple closed form, and the paper finds no comparable factorisation for global structure.Source: §4 · p. 11

5 Numerical Results

• The numerical study used nU = nV = 200, degree k = 6, 21 rewiring probabilities from 10^-3 to 1, five phase-disorder levels, and 40 realisations per parameter pair.The study also tested system sizes of 100 and 200 nodes per class and degrees 4, 6, and 10 for the clustering factorisation.Source: §5 · p. 11, §5.1 · p. 11

5.1 Topology and Coherence in the Clustering Coefficient

• For ∆ = π/2, phase disorder removes more than four fifths of the local closure supported by connectivity; the measured phase factor follows (sin ∆/∆)^4 across tested sizes, degrees, and rewiring probabilities.The measured clustering coefficient also matches the product of the measured topological factor and predicted phase factor across the parameter plane.Source: §5.1 · p. 11, Figure 3 · p. 12

5.2 Coherence and Global Structure

• At p ≈ 0.09, phase disorder increases apparent distance by 15.7 per cent while hop-count distance changes by less than one per cent.Across the tested networks, phase disorder leaves hop-count distance unchanged, while apparent distance, which sums over all walks, rises through cancellation between competing routes.Source: §5.2 · p. 12, Figure 4 · p. 13

5.3 The Small-World Coefficient and its Reference Network

• With a matched reference, the small-world coefficient is independent of phase disorder; its peak values are 12.37, 12.37, and 12.08 for ∆ = 0, π/4, and π/2.A fixed reference can instead make phase dependence artificial because its clustering denominator tends to zero in expectation under fully disordered phases.Source: §5.3 · pp. 12–13, Figure 4 · p. 13

5.4 Interference between Open Paths

• With connectivity fixed, changing phase disorder leaves the pairs that can interfere unchanged but alters their visibility; rewiring changes which pairs interfere.Pairs with more shared neighbours lose visibility faster and approach lower limiting values q^-1/2 under complete disorder.Source: §5.4 · pp. 13–14, Figure 5 · p. 14

6 Discussion

• The coefficient addresses complex-weighted bipartite networks by using square fluxes, the smallest gauge-invariant phase carrier in a bipartite network.It reduces to the classical bipartite coefficient when phases vanish, can be negative under destructive interference, and is computable for all nodes from one sparse matrix product.Source: §6 · p. 14
• The coefficient factorises into connectivity and phase contributions; for uniform phases on [−∆, ∆], the mean phase factor is (sin ∆/∆)^4, independent of degree and topology.The discussion also gives closed-form results for the clustering coefficient, open-path visibility, and phase variance.Source: §6 · pp. 14–15
• Away from complete phase disorder, variance is dominated by pairs of squares sharing a link, while the phase-factor standard deviation decays as k^−1/2.At complete disorder, the variance is 1/mu, where mu is the number of squares at a node.Source: §6 · p. 15
• Across the rewiring range, the topological factor falls from 0.53 to 0.02 and mean hop-count distance from 18.7 to 3.8.On identical connectivity, phase disorder leaves hop-count distance unchanged but increases apparent distance by 16 percent.Source: §6 · p. 15
• Combinatorial distance depends only on links, whereas coherent distance reflects route interference; phase disorder changes the latter without disconnecting the network.The discussion relates this distinction to photonic meshes, whose coupling map may remain unchanged while effective propagation becomes less coherent.Source: §6 · p. 15
• The small-world coefficient’s phase factor cancels against a matched reference, so residual phase dependence reflects reference choice; zero-mean reference clustering at full disorder can cause multi-order distortions.The passage cautions that this normalisation can produce distortions spanning several orders of magnitude.Source: §6 · p. 15
• The analysis assumes independent, static link phases independent of strengths; correlations require recomputing covariances, while the mean factor remains valid if each square’s links are independent.The treatment describes steady-state, single-particle or wave-like amplitudes, not dynamics or entangled multipartite states.Source: §6 · p. 15
• Future work could place disorder directly on square fluxes, study longer-cycle fluxes, and test the square-flux interference prediction using measured circuit-QED edge strengths and phases.The passage connects flux disorder to random-flux localisation models [41].Source: §6 · p. 15

Gauge invariance

• Node-wise phase-reference changes cancel around a square, leaving its flux invariant; the proof also uses the invariance of moduli under phase changes.This establishes why the square flux, rather than an individual link phase, carries structural phase information.Source: A Proofs · p. 17

Coefficient properties

• The coefficient is bounded by [−1, 1], equals one when all phases vanish, and reduces to the classical bipartite coefficient for unit strengths.Its numerator sums real square-amplitude contributions, while the denominator normalises by possible three-link paths; the appendix also relates its extremes to open-path visibility.Source: A Proofs · pp. 17–18

Mean phase factor

• For independent link phases, the mean phase factor is |χ(1)|^4; uniform phases on [−∆, ∆] give χ(1) = sin ∆/∆.The derivation uses independence of the four links within each square and linearity of expectation over squares, not independence between squares.Source: A Proofs · p. 18

Square covariances

• Two squares can share a phase difference, share one link, or share no link; in the last case their fluxes are independent.At uniform phases, fluxes are pairwise independent, but three squares on a common node pair obey an additive flux relation.Source: A Proofs · p. 18
• The appendix notes that pairwise independence at uniform phases does not extend to triples: for three squares on one node pair, one flux is the sum of the other two.Source: A Proofs · p. 19

Variance

• At complete disorder, Var(Qu) = 1/mu; away from that limit, shared-link square pairs contribute covariance and determine the variance correction.The variance is 1/mu only when ∆ = π for uniform phases; the derivation expresses it using χ(2) and the first-harmonic factor.Source: A Proofs · p. 19

Lattice variance

• On the lattice, nU = nV ≥ 2k prevents wraparound, and partners at 1 ≤ |d| ≤ k−1 share q = k−|d| neighbors; pairs through a common link are counted twice, while type (b) pairs once.Source: A Proofs · p. 19

Open-path visibility

• For unit strengths, the open-path visibility depends on the shared-neighbour count q and has mean-square value q^−1 + (1 − q^−1)r^2, with r = sinc^2 ∆ for uniform phases.The derivation treats the q link-phase differences as independent and identically distributed.Source: A Proofs · p. 20

B Relation to existing measures

• With phases set to zero, the coefficient becomes a bipartite analogue of Zhang and Horvath’s weighted coefficient; for binary weights it becomes the local four-cycle to three-path ratio of Robins and Alexander.Lind et al. [26] and Zhang et al. use different potential-square counting conventions and are monotonically related on the considered ensembles.Source: B Relation to existing measures · p. 20
• Unlike the triangle-based interferometric coefficient, the bipartite coefficient compares two two-step routes through distinct intermediaries, weighting their interference by route amplitudes.The square amplitude is the product of the two route amplitudes, and its real part contains their phase difference.Source: B Relation to existing measures · p. 20
• Summing square amplitudes for fixed node pairs gives the squared open-path visibility, so the clustering coefficient is a degree-normalised aggregate of those squared visibilities.Source: B Relation to existing measures · p. 20

C.1 Generation and cost

• Rewiring removes and replaces all selected links at a node together, preserving degrees in U exactly and avoiding collision bias from link-by-link rewiring.The seed lattice costs O(nk) to construct.Source: §C.1 · p. 20
• The sparse-product evaluation of clustering costs O(nk^2) in time and memory, compared with O(n^2k) for explicit square enumeration.The overlap counts used in the variance theorem also cost O(nk^2).Source: §C.1 · p. 20
• Hop-count distances cost O(n^2k), while the dense inverse used for apparent distance costs O(n^3) and limits reported system sizes.The passage suggests sampled linear solves as an alternative for larger networks; Figure 6 compares simulated phase-factor fluctuations with the prediction and independent-squares approximation.Source: §C.1 · p. 20, Figure 6 · p. 21

• 3 × 10^-14: matrix identities (9)–(11) agree with direct enumeration on random bipartite networks with heterogeneous strengths and random phases.Source: §C.2 · p. 21
• 10^-10: setting all phases to zero reproduces the directly enumerated classical bipartite clustering coefficient, with Qu = 1 at every node.Source: §C.2 · p. 21
• 10^-16: random phase-reference redefinitions leave Cu, C(0), Qu, and I(2) unchanged, confirming gauge invariance.Source: §C.2 · p. 21
• Within statistical error, the measured mean phase factor matches (sin ∆/∆)^4 for every tested network size, degree, and rewiring probability.Source: §C.2 · p. 21

C.3 Resolution

• 4 × 10^6 samples per value: κ1 and κ2 agree with direct phase sampling within error for three overlapping-pair types; the theorem's counts agree exactly with direct square-pair enumeration.The checks cover ∆∈{π/4, π/2, 2, π}, lattice k from 3 to 15, and rewired networks at p∈{0.1, 0.5, 1}.Source: §C.3 · p. 21
• Up to a factor of 3.4: the independence estimate 2σ2/mu underestimates the measured standard deviation of Qu on the regular lattice, while Corollary 4.4 agrees across ∆ and nearly three orders of magnitude in mu.Source: §C.3 · p. 21
• 2.4 at p = 0.1 and ∆ = π/4, versus 1.1 at p = 1: the independence estimate underestimates the predicted variance on rewired networks.Across 4000 phase realisations, measured variance matches the prediction within sampling error, with no trend in p.Source: §C.3 · p. 21

C.4 Attainable range

• −1, −1/2, and −1/3 are minima for K2,2, K3,3, and K4,4; the ring lattice reaches −1, −0.625, and −0.35 at degrees 3, 4, and 6.Random bipartite networks also match the predicted minimum to four decimal places; the degree-6 ensemble coefficient cannot fall below approximately −0.35 for any phase choice.Source: §C.4 · p. 22

C.5 Heterogeneous strengths

• 0.011: for strengths drawn uniformly from (0.5, 1), the largest deviation of the measured phase factor from sinc4 ∆ is 0.011; the formula is exact for unit strengths.With varying strengths, the weighting introduces a second-order correction in their dispersion.Source: §C.5 · p. 22

• Standard errors remain below two per cent of the mean for all clustering, distance, and interference quantities across forty realizations per parameter pair.At complete phase disorder, the expectation of ¯C vanishes, its relative error is unbounded, and the reported value is not interpreted.Source: §C.6 · p. 22