Summary
Networks often have connections with strengths represented by simple numbers, but some advanced systems, like quantum or photonic setups, have connections described by complex numbers that include phase information. The authors created a new way to measure clustering—the tendency of nodes to form close loops—in networks that are both bipartite (two types of nodes with no triangles) and have these complex-weighted connections. Their method can detect when routes interfere negatively or positively due to phases and works efficiently with a single matrix operation. It matches standard measures when phases aren’t involved and offers new insights about how phase disorder affects network behavior.
What this means in practice
- •For quantum hardware developers: Quantify interference effects of complex-weighted connections in bipartite quantum or photonic network designs to assess circuit coherence and routing reliability.
- •For communication network designers: Analyze signal interference patterns in bipartite networks with phase-encoded weights to optimize network layouts that exploit or mitigate phase cancellations.
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.