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