In models of excitable dynamics on graphs, excitations can travel in both directions of an undirected link. However, as a striking interplay of dynamics and network topology, excitations often establish a directional preference. Some of these cases of “link-usage asymmetry” are local in nature and can be mechanistically understood, for instance, from the degree gradient of a link (i.e., the difference in node degrees at both ends of the link). Other contributions to the link-usage asymmetry are instead, as we show, self-organized in nature, and strictly nonlocal. This is the case for excitation waves, where the preferential propagation of excitations along a link depends on its orientation with respect to a hub acting as a source, even if the link in question is several steps away from the hub itself. Here, we identify and quantify the contribution of such self-organized patterns to link-usage asymmetry and show that they extend to ranges significantly longer than those ascribed to local patterns. We introduce a topological characterization, the hub-set-orientation prevalence of a link, which indicates its average orientation with respect to the hubs of a graph. Our numerical results show that the hub-set-orientation prevalence of a link strongly correlates with the preferential usage of the link in the direction of propagation away from the hub core of the graph. Our methodology is embedding-agnostic and allows for the measurement of wave signals and the sizes of the cores from which they originate.

在图上的可激励动力学模型中，激励可以在无向链路的两个方向上传播。然而，作为动力学和网络拓扑的显着相互作用，激励通常会建立方向偏好。其中一些“链路使用不对称”的情况本质上是局部的，并且可以从机械上理解，例如，从链路的度梯度（即链路两端的节点度的差异）。正如我们所展示的，对链接使用不对称性的其他贡献本质上是自组织的，并且是严格非局部的。激励波就是这种情况，其中激励沿链路的优先传播取决于其相对于充当源的集线器的方向，即使所讨论的链路距离集线器本身有几步之遥。在这里，我们识别并量化了这种自组织模式对链接使用不对称性的贡献，并表明它们延伸的范围比归因于本地模式的范围要长得多。我们引入了拓扑特征，即链接的中心集方向普遍性，它指示了其相对于图的中心的平均方向。我们的数值结果表明，链路的集线器集方向普遍性与远离图的集线器核心的传播方向上的链路的优先使用密切相关。我们的方法与嵌入无关，可以测量波信号及其来源的核心尺寸。