本研究系统考察了具有功能连接的公共交通网络(PTN)中尚未实现的部分,即其补图结构。无权图 $G$ 的补图 $\bar G$ 是一个标准概念:它与 $G$ 共享同一节点集,且当且仅当某条边在 $G$ 中不存在时,该边才存在于 $\bar G$ 中。相比之下,加权补图无法被唯一确定。然而,若将 PTN 的边权重定义为行程时间,则其可能取值范围受到物理约束。我们提出一种方法,基于节点(代表车站)间的地理距离,并结合网络特异的有效速度分布与候车时间分布,构建运营中 PTN 图表示的加权补图,并据此为边赋予权重。我们发现,加权补图中中心性最高的节点,并非原网络中中心性最低的节点,而是位于网络地理中心但拓扑连通性较弱的节点。在涵盖全球31个地铁网络的数据集上,通过与零模型对比验证表明,该现象本质上是一种空间效应。
In this study, we take a systematic look at the unrealised part of public transport networks (PTNs) with functional connections. We consider their complement graphs and study their structure. The complement graph $\bar G$ of an unweighted graph $G$ is a straightforward concept, yielding a graph on the same set of nodes, and an edge exists in $\bar G$ if and only if it is not present in $G$. In contrast, a weighted complement graph cannot be uniquely determined. However, if we consider PTNs with travel times as edge weights, there are physical constraints on the possible weight ranges. We propose a method to construct weighted complement graphs of operational PTN graph representations based on the geographical distances between nodes (representing stops) and assign weights to edges based on distance, combined with network-specific distributions of effective velocities and waiting times. We observe that the most central nodes in the weighted complement graph do not correspond to the least central nodes in the original network but are, remarkably, those in the geographical centre of the network that lack topological connectedness. Testing against null models on a dataset of 31 metro networks worldwide confirms that this is a fundamentally spatial effect.