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Understanding how different networks relate to each other is key forobtaining a greater insight into complex systems. Here, we introduce anintuitive yet powerful framework to characterise the relationship between twonetworks comprising the same nodes. We showcase our framework by decomposingthe shortest paths between nodes as being contributed uniquely by one or theother source network, or redundantly by either, or synergistically by the twotogether. Our approach takes into account the networks' full topology, and italso provides insights at multiple levels of resolution: from globalstatistics, to individual paths of different length. We show that this approachis widely applicable, from brains to the London public transport system. Inhumans and across 123 other mammalian species, we demonstrate that reliance onunique contributions by long-range white matter fibers is a conserved featureof mammalian structural brain networks. Across species, we also find thatefficient communication relies on significantly greater synergy betweenlong-range and short-range fibers than expected by chance, and significantlyless redundancy. Our framework may find applications to help decide how totrade-off different desiderata when designing network systems, or to evaluatetheir relative presence in existing systems, whether biological or artificial.

Quantifying synergy and redundancy in multiplex networks