  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.
