发表机构
University of Oxford; Center for Systems Biology Dresden（牛津大学; 德累斯顿系统生物学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对复边权重二分网络，提出规范不变聚类系数，可分解为拓扑与相位贡献，推导相关闭式表达式并数值验证，发现相位无序会增加相干距离。
AI 中文摘要
聚类系数和平均最短路径长度等结构度量可刻画网络的组织方式，这些度量通常针对实值非负边权重构建。一类量子和光子架构具有复边权重，其相位决定了替代路径是相长干涉还是相消干涉，且这类架构为二分结构，不存在三角形，最小闭合环是四节点正方形。现有度量分别处理复权重和二分结构：二分聚类系数通过四节点正方形量化聚类，但不包含相位信息；干涉度量保留相位，但定义在三角形上。本研究定义了一种适用于复杂加权二分网络的聚类系数，当相位消失时，该系数退化为经典二分系数；当替代路径抵消时，该系数为负；可通过单次稀疏矩阵乘积得到所有节点的该系数。研究表明，正方形周围积累的相位是二分网络中最小的规范不变结构相位信息载体，该聚类系数可分解为拓扑贡献和相位贡献，使二分Watts--Strogatz系综具有解析可处理性。对于在[-Δ,Δ]上均匀分布的相位，平均相位贡献为(sinΔ/Δ)^4，与节点、度和拓扑无关。此外，本研究还推导了聚类系数、开放路径可见性和相位方差的闭式表达式，通过数值验证了这些预测，并进一步表明相位无序会增加相干距离。
英文摘要
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.
Comments22 pages (15 main), 6 figures
