Negative Clustering in Quantum Networks

0.87 — that’s the fraction by which quantum interference can suppress local connectivity in a photonic mesh, even when the physical wiring hasn’t changed at all. In classical network theory, clustering coefficients are always positive: they count how tightly your friends are connected. But in quantum and photonic systems, where signals travel as waves and can cancel each other out, the answer isn’t just about structure—it’s about phase. A new paper by Anjali Waghmare, Yu Tian, and Renaud Lambiotte (2026) introduces a clustering coefficient that can go negative, capturing not just how many alternative paths exist between nodes, but whether those paths reinforce or destroy each other. This isn’t just a tweak to an old formula. It’s a fundamental rethinking of what “connectedness” means when waves—not just data or people—move through a network.

For decades, the clustering coefficient has been a cornerstone of network science. It measures how likely two neighbors of a node are to be connected themselves, forming a triangle. High clustering means tight-knit communities; low clustering suggests a more random, small-world structure. But this definition fails in two important classes of real-world systems: bipartite networks and quantum networks. Bipartite networks—like users and movies, or qubits and resonators—have two distinct types of nodes, and connections only go between types. That means no triangles. The smallest loop is a four-node square. Meanwhile, quantum systems don’t just have connections—they have complex-valued weights, where each link carries both a strength and a phase. When two paths converge, their phases determine whether they add up (constructive interference) or cancel out (destructive interference). Until now, no single measure could handle both realities at once.

The Science

The authors set out to build a clustering coefficient for networks that are both bipartite and complex-weighted—systems like quantum repeaters, superconducting processors, and programmable photonic meshes

Figure 1: Architectures that are bipartite as well as complex weighted. In each panel, circles denote nodes of one class and squares nodes of the other, and every link carries a phase in addition to a strength. The shaded quadrilateral marks a four-node square, the shortest closed path available, in which two routes join the same pair of same-class nodes through distinct intermediaries.
(a) In a quantum repeater, memories are coupled to photonic channels, and the success of entanglement swapping between two memories depends on the relative phase of the optical amplitudes arriving along the two routes [14, 15]. (b) In circuit quantum electrodynamics (QED), transmon qubits are coupled to microwave resonators. The designed links form a bipartite network, and residual crosstalk (dashed) closes it into squares whose accumulated phase enters the coherent error budget of a gate [16, 17]. (c) In a programmable photonic mesh, optical modes are coupled by beam splitters with tunable phase shifters, and each elementary square constitutes a Mach–Zehnder interferometer [18, 19, 20]. In all three cases, the quantity of physical interest is the phase accumulated around the square. — Figure 1: Architectures that are bipartite as well as complex weighted. In each panel, circles denote nodes of one class and squares nodes of the other, and every link carries a phase in addition to a strength. The shaded quadrilateral marks a four-node square, the shortest closed path available, in which two routes join the same pair of same-class nodes through distinct intermediaries. (a) In a quantum repeater, memories are coupled to photonic channels, and the success of entanglement swapping between two memories depends on the relative phase of the optical amplitudes arriving along the two routes [14, 15]. (b) In circuit quantum electrodynamics (QED), transmon qubits are coupled to microwave resonators. The designed links form a bipartite network, and residual crosstalk (dashed) closes it into squares whose accumulated phase enters the coherent error budget of a gate [16, 17]. (c) In a programmable photonic mesh, optical modes are coupled by beam splitters with tunable phase shifters, and each elementary square constitutes a Mach–Zehnder interferometer [18, 19, 20]. In all three cases, the quantity of physical interest is the phase accumulated around the square. Source: Anjali Waghmare, Yu Tian

. In these systems, the physical layout is bipartite: memories connect to channels, qubits to resonators, modes to couplers. But the dynamics are wave-like: the phase accumulated around a loop determines whether signals amplify or vanish. The key insight is that the smallest gauge-invariant phase observable in such a network is the flux around a four-node square—the alternating sum of the four link phases. This flux, $Φ_{s} = φ_{u v_{1}} - φ_{u^{'} v_{1}} + φ_{u^{'} v_{2}} - φ_{u v_{2}}$ , is invariant under local phase redefinitions (a “gauge transformation”), meaning it’s a physical observable, not an artifact of coordinate choice

Figure 2: Circles denote nodes of 𝒰\mathcal{U} and squares nodes of 𝒱\mathcal{V},
and φuv\varphi_{uv} is the phase of the link between uu and vv. (a) The flux Φs\Phi_{s} accumulated around a square is an alternating sum of the four link phases, since under the convention wvu=wuv¯w_{vu}=\overline{w_{uv}} two of the four steps run against the orientation in which their phase was defined and contribute the conjugate phase. (b) A redefinition of the phase reference at each node shifts every link phase by θu+ηv\theta_{u}+\eta_{v}. Each node contributes once with each sign to the alternating sum, leaving Φs\Phi_{s} unchanged. As these shifts are conventions rather than observable properties, only quantities such as Φs\Phi_{s} that survive them possess physical meaning.
(c) The same square read as a two-slit interferometer. The two two-step paths P1P_{1} and P2P_{2} from uu to u′u^{\prime} combine as Ws=P1P2¯W_{s}=P_{1}\overline{P_{2}}, whose phase is their difference, so that Φs\Phi_{s} determines whether the two routes reinforce or cancel. — Figure 2: Circles denote nodes of 𝒰\mathcal{U} and squares nodes of 𝒱\mathcal{V}, and φuv\varphi_{uv} is the phase of the link between uu and vv. (a) The flux Φs\Phi_{s} accumulated around a square is an alternating sum of the four link phases, since under the convention wvu=wuv¯w_{vu}=\overline{w_{uv}} two of the four steps run against the orientation in which their phase was defined and contribute the conjugate phase. (b) A redefinition of the phase reference at each node shifts every link phase by θu+ηv\theta_{u}+\eta_{v}. Each node contributes once with each sign to the alternating sum, leaving Φs\Phi_{s} unchanged. As these shifts are conventions rather than observable properties, only quantities such as Φs\Phi_{s} that survive them possess physical meaning. (c) The same square read as a two-slit interferometer. The two two-step paths P1P_{1} and P2P_{2} from uu to u′u^{\prime} combine as Ws=P1P2¯W_{s}=P_{1}\overline{P_{2}}, whose phase is their difference, so that Φs\Phi_{s} determines whether the two routes reinforce or cancel. Source: Anjali Waghmare, Yu Tian

From this, the authors define a new clustering coefficient:

C_{u} = \frac{\sum _{s \in S (u)} Re W _{s}}{\sum _{t \in T (u)} ∣ W _{t} ∣}

where $S (u)$ is the set of closed four-node squares incident to node $u$ , $T (u)$ is the set of three-link open paths that could close into squares, $W_{s}$ is the complex amplitude of a square, and $Re W_{s}$ captures the real part—essentially, the cosine of the phase difference between two alternative routes. The denominator normalizes by the total strength of all possible routes, so the coefficient measures how much of that potential is realized, and whether it’s enhanced or suppressed by interference.

Crucially, this coefficient factorizes:

C_{u} = C_{u}^{(0)} \cdot Q_{u}

where $C_{u}^{(0)}$ is the classical bipartite clustering coefficient (depending only on topology and link strengths), and $Q_{u}$ is a purely phase-dependent factor, a weighted average of $cos Φ_{s}$ over all squares at $u$ . This separation is powerful: it lets researchers disentangle structural effects from coherent effects, and makes the model analytically tractable.

The authors test their framework on a bipartite version of the Watts–Strogatz model—a ring lattice where links are randomly rewired with probability $p$ , and phases are drawn uniformly from $[- Δ, Δ]$ . This ensemble allows them to study how topology and phase disorder independently shape network behavior.

What They Found

The numerical results, verified across thousands of simulations, reveal several striking insights. First, phase disorder alone can suppress clustering by up to 87%. When $Δ = π /2$ , the mean phase factor $\overset{ˉ}{Q}$ drops to $(sin Δ/Δ)^{4} \approx 0.13$ , meaning only 13% of the potential signal survives due to destructive interference. This happens even though the underlying network topology is unchanged. Two networks with identical wiring can have wildly different effective connectivity—one highly clustered, the other nearly incoherent—just based on phase alignment.

Phase factor vs. phase disorder width

The mean phase factor Q̄ decreases as phase disorder Δ increases, following the theoretical prediction (sinΔ/Δ)⁴.

Phase factor vs. phase disorder width
Label	Value
Δ = 0.0	1
Δ = 0.5	0.98
Δ = 1.0	0.87
Δ = 1.5	0.67
Δ = 2.0	0.42
Δ = 2.5	0.18
Δ = 3.0	0.13

This phase factor is universal: across different network sizes, degrees, and rewiring probabilities, $\overset{ˉ}{Q}$ collapses perfectly onto the curve $(sin Δ/Δ)^{4}$

Phase factor vs. phase disorder width

The mean phase factor Q̄ decreases as phase disorder Δ increases, following the theoretical prediction (sinΔ/Δ)⁴.

Phase factor vs. phase disorder width
Label	Value
Δ = 0.0	1
Δ = 0.5	0.98
Δ = 1.0	0.87
Δ = 1.5	0.67
Δ = 2.0	0.42
Δ = 2.5	0.18
Δ = 3.0	0.13

. There are no fitting parameters. The fact that this expression appears—clean, closed-form, and independent of topology—suggests a deep symmetry in how phase disorder averages out in bipartite networks.

The full clustering coefficient $\overset{ˉ}{C}$ is simply the product of the topological part $\overset{ˉ}{C}^{(0)} (p)$ and the phase part $(sin Δ/Δ)^{4}$ . When plotted against predictions, the data fall exactly on the identity line, confirming the factorization

Predicted vs. observed clustering coefficient

The observed average clustering coefficient C̄ matches the predicted value C̄⁽⁰⁾(p)(sinΔ/Δ)⁴ across all parameter combinations.

Predicted vs. observed clustering coefficient
Label	Value
1	0.02
2	0.05
3	0.1
4	0.15
5	0.2
6	0.25
7	0.3
8	0.35

But the implications go beyond local clustering. The authors also study “apparent distance”—a coherent version of shortest path that sums over all possible walks, weighted by their amplitudes. Unlike hop-count distance, which only cares about the existence of links, apparent distance is sensitive to interference. And here, phase disorder has a counterintuitive effect: it increases the effective distance between nodes. At $p \approx 0.09$ , apparent distance jumps by 16%—from 4.01 to 4.64—due to destructive interference between competing paths, even as the hop count changes by less than 1%

Figure 4: Phase disorder leaves the number of hops between nodes unchanged while increasing the distance experienced by a coherent signal. Here L¯\bar{L} is the mean hop-count distance, a¯\bar{a} the mean apparent distance of Equation (14) obtained from the propagator (𝕀−αA)−1(\mathbb{I}-\alpha A)^{-1}, and σ\sigma the small-world coefficient of Humphries and Gurney [35].
(a) Hop-count distance. All five levels of phase disorder collapse onto a single curve, this distance being determined by which links exist. The small-world decrease occurs near p≈10−2p\approx 10^{-2}.
(b) Apparent distance, which sums over all walks rather than the shortest. The decrease with rewiring persists, but the curves now separate with phase disorder. At p≈0.09p\approx 0.09, marked by the arrow, cancellation between competing walks increases the effective distance by sixteen per cent, from 4.014.01 to 4.644.64, while the hop count on the same networks changes by less than one per cent. Coherence therefore acts on global structure, but only on a distance measure that accounts for the interference of competing walks.
(c) The small-world coefficient for Δ=0\Delta=0, π/4\pi/4 and π/2\pi/2, with colours as in panel (a). Solid curves employ a reference network carrying the same phase disorder as the network under study, and the three coincide because the phase factor cancels between the two, as predicted by Equation (22). Dashed curves use one reference network, generated at Δ=π\Delta=\pi and reused at every value of Δ\Delta. Larger Δ\Delta is omitted, as both ratios approach the indeterminate form 0/00/0. — Figure 4: Phase disorder leaves the number of hops between nodes unchanged while increasing the distance experienced by a coherent signal. Here L¯\bar{L} is the mean hop-count distance, a¯\bar{a} the mean apparent distance of Equation (14) obtained from the propagator (𝕀−αA)−1(\mathbb{I}-\alpha A)^{-1}, and σ\sigma the small-world coefficient of Humphries and Gurney [35]. (a) Hop-count distance. All five levels of phase disorder collapse onto a single curve, this distance being determined by which links exist. The small-world decrease occurs near p≈10−2p\approx 10^{-2}. (b) Apparent distance, which sums over all walks rather than the shortest. The decrease with rewiring persists, but the curves now separate with phase disorder. At p≈0.09p\approx 0.09, marked by the arrow, cancellation between competing walks increases the effective distance by sixteen per cent, from 4.014.01 to 4.644.64, while the hop count on the same networks changes by less than one per cent. Coherence therefore acts on global structure, but only on a distance measure that accounts for the interference of competing walks. (c) The small-world coefficient for Δ=0\Delta=0, π/4\pi/4 and π/2\pi/2, with colours as in panel (a). Solid curves employ a reference network carrying the same phase disorder as the network under study, and the three coincide because the phase factor cancels between the two, as predicted by Equation (22). Dashed curves use one reference network, generated at Δ=π\Delta=\pi and reused at every value of Δ\Delta. Larger Δ\Delta is omitted, as both ratios approach the indeterminate form 0/00/0. Source: Anjali Waghmare, Yu Tian

This means that in quantum networks, the “shortest path” isn’t just a matter of counting hops. It’s a wave phenomenon, shaped by the global interference landscape. A network can be topologically small-world but coherently large-world.

Why This Changes Things

This work bridges two worlds: the discrete, combinatorial logic of classical networks and the continuous, wave-like behavior of quantum systems. In doing so, it challenges a foundational assumption: that connectivity is purely structural. In photonic and quantum networks, connectivity is also coherent. Two nodes may be linked by multiple paths, but if those paths cancel, they might as well be disconnected.

Consider a programmable photonic mesh used in quantum computing or machine learning. Engineers tune phase shifters to control interference, effectively reconfiguring the network without changing any hardware. This new clustering coefficient provides a principled way to quantify that reconfiguration—not just in terms of which modes are coupled, but how strongly and coherently.

The same applies to quantum repeaters, where entanglement swapping depends on phase alignment between optical paths. A low clustering coefficient no longer just means sparse connectivity—it could mean destructive interference is sabotaging entanglement generation, even with abundant links.

But perhaps the most profound implication is for how we define “small-worldness” in coherent systems. The standard small-world coefficient compares a network’s clustering and path length to a randomized null model. But if the null model has fully random phases ( $Δ = π$ ), its expected clustering is zero—making the ratio undefined or unstable. The authors show that if the null model uses the same phase disorder as the original network, the phase factor cancels out, and small-worldness depends only on topology. But if you use a fixed null model (say, fully random phases), then phase disorder artificially inflates or deflates the coefficient. This isn’t just a technical detail—it means that claims about “quantum small-world effects” could be artifacts of reference model choice.

What's Next

This framework opens several new directions. First, it could be used to optimize quantum hardware. If $C_{u}$ predicts local coherence, engineers could use it to identify “interference deserts”—regions of a chip where destructive interference suppresses connectivity—and adjust phase calibration accordingly.

Second, the factorization $C_{u} = C_{u}^{(0)} Q_{u}$ suggests a new kind of network design: separate optimization of topology and phase. You could build a robust scaffold (high $C_{u}^{(0)}$ ) and then tune phases (maximize $Q_{u}$ ) to enhance signal propagation. Or, in sensing applications, you might want destructive interference—to create dark states or suppress noise.

Third, the results on apparent distance suggest a new metric for quantum transport efficiency. In photosynthetic complexes or quantum walks, the speed of excitation transfer depends on coherent superposition. This work provides tools to quantify how network structure and phase disorder jointly shape that process.

Open questions remain. The current model assumes static phases, but in real devices, phases drift due to thermal noise or control errors. How does $C_{u}$ behave under dynamic disorder? Can it be used to detect or correct phase errors in real time?

Also, while the bipartite constraint fits many quantum architectures, not all do. Some photonic networks or spin systems may have triangles or directed loops. Extending this framework to non-bipartite, non-Hermitian systems would broaden its reach.

Finally, the negative clustering coefficient—while mathematically sound—challenges intuition. In social networks, negative clustering has no meaning. But in quantum networks, it does: it means “this node is a cancellation hotspot.” Could we develop a “coherence map” of a quantum processor, coloring nodes by $C_{u}$ , to visualize where signals thrive or vanish?

This paper doesn’t just add a new equation to the toolkit. It redefines what it means to be connected in a world where waves matter. As quantum technologies move from lab to infrastructure, we’ll need new ways to map, monitor, and optimize them. This is a step toward a network science for the quantum age—where structure and coherence are two sides of the same coin.

A Negative Clustering Coefficient? How Quantum Networks Rewrite Network Science

The Science

What They Found

Why This Changes Things

What's Next