Understanding how different networks relate to each other is key for obtaining a greater insight into complex systems. Here, we introduce an intuitive yet powerful framework to characterise the relationship between two networks, comprising the same nodes. We showcase our framework by decomposing the shortest paths between nodes as being contributed uniquely by one or the other source network, or redundantly by either, or synergistically by the two together. Our approach takes into account the networks' full topology, but it also provides insights at multiple levels of resolution: from global statistics, to individual paths of different length. We show that this approach is widely applicable, from brains to the London transport system. In humans and across $123$ other species, we demonstrate that reliance on unique contributions by long-range white matter fibers is a conserved feature of mammalian structural connectomes. Across species, we also find that efficient communication relies on significantly greater synergy between long-range and short-range fibers than expected by chance, and significantly less redundancy. Our framework may find applications to help decide how to trade-off different desiderata when designing network systems, or to evaluate their relative presence in existing systems, whether biological or artificial.

翻译：理解不同网络之间的相互关系是深入洞察复杂系统的关键。本文提出一个直观而强大的框架，用于刻画包含相同节点的两个网络之间的关系。通过分解节点间的最短路径，我们展示该框架如何将路径贡献归因于单一源网络的独特贡献、两者任意之一的冗余贡献，或两者协同作用下的协同贡献。该方法不仅考虑了网络的完整拓扑结构，还能从全局统计到不同长度的单个路径等多个分辨率层面提供洞见。本文表明该框架具有广泛适用性，从脑网络到伦敦交通系统均可应用。在人类及123个其他物种中，我们证实依赖远程白质纤维的独特贡献是哺乳动物结构连接组的保守特征。跨物种分析还发现，高效通信依赖于远程和短程纤维之间显著高于随机预期的协同作用，而冗余性则显著低于随机预期。该框架可应用于网络系统设计时不同目标之间的权衡决策，或评估现有生物或人工系统中这些特性的相对程度。