The ability to reroute and control flow is vital to the function of venation networks across a wide range of organisms. By modifying individual edges in these networks, either by adjusting edge conductances or creating and destroying edges, organisms robustly control the propagation of inputs to perform specific tasks. However, a fundamental disconnect exists between the structure and function: networks with different local architectures can perform the same functions. Here, we answer the question of how changes at the level of individual edges collectively create functionality at the scale of an entire network. Using persistent homology, we analyze networks tuned to perform complex tasks. We find that the responses of such networks encode a hidden topological structure composed of sectors of nearly uniform pressure. Although these sectors are not apparent in the underlying network structure, they correlate strongly with the tuned function. The connectivity of these sectors, rather than that of individual nodes, provides a quantitative relationship between structure and function in flow networks.

中文翻译：

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流网络功能的隐藏拓扑结构

改变路线和控制流量的能力对于各种生物的通风网络的功能至关重要。通过修改这些网络中的各个边缘（通过调整边缘电导或创建和破坏边缘），有机体可以强有力地控制输入的传播以执行特定任务。但是，结构和功能之间存在根本的脱节：具有不同本地体系结构的网络可以执行相同的功能。在这里，我们回答以下问题：单个边缘级别的更改如何在整个网络规模上共同创建功能。使用持久性同源性，我们分析调整为执行复杂任务的网络。我们发现，此类网络的响应编码了由几乎均匀压力的扇区组成的隐藏拓扑结构。尽管这些扇区在底层网络结构中并不明显，但它们与调整后的功能密切相关。这些部门的连通性而不是单个节点的连通性在流动网络中提供了结构和功能之间的定量关系。