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, and 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 public transport system. In humans and across 123 other mammalian species, we demonstrate that reliance on unique contributions by long-range white matter fibers is a conserved feature of mammalian structural brain networks. 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对于深入洞察复杂系统至关重要。本文提出一个直观而强大的框架，用以描述由相同节点组成的两个网络之间的关系。我们通过将节点间最短路径分解为仅由其中一个源网络单独贡献、由任一源网络冗余贡献、或由两者协同贡献的方式，展示了该框架的应用。该方法不仅考虑了网络的完整拓扑结构，还能在多个分辨率层次上提供见解：从全局统计特征到不同长度的个体路径。我们证明，该框架具有广泛适用性，可应用于从大脑到伦敦公共交通系统的多种场景。在人类及123种其他哺乳动物物种中，我们发现对长程白质纤维独特贡献的依赖是哺乳动物结构脑网络的一个保守特征。跨物种研究还表明，高效通信依赖于长程与短程纤维之间显著高于随机水平的协同作用，且冗余程度显著低于随机水平。我们的框架可应用于辅助决策：在设计网络系统时如何权衡不同目标，或在评估现有系统（无论是生物系统还是人工系统）中这些目标的相对实现程度。