  Network science enables the effective analysis of real interconnected
systems, characterized by a complex interplay between topology and
interconnections strength. It is well-known that the topology of a network
affects its resilience to failures or attacks, as well as its functions.
Exchanging information is crucial for many real systems: the internet,
transportation networks and the brain are key examples. Despite the
introduction of measures of efficiency to analyze network flows, i.e.
topologies characterized by weighted connectivity, here we show that they fail
to capture combined information of link existence and link weight. In this
letter we propose a physically-grounded estimator of flow efficiency which can
be computed for every weighted network, regardless from the scale and nature of
weights and from any (missing) metadata. Remarkably, results show that our
estimator captures the heterogeneity of flows along with topological
differences and its complement information obtained from percolation analysis
of several empirical systems, including transportation, trade, migrations, and
brain networks. We show that cutting the heaviest connections may increase the
average communication efficiency of the system and hence, counterintuively, a
sparser network is not necessarily less efficient. Remarkably, our estimator
enables the comparison of communication efficiency of networks arising from
different fields, without the possible pitfalls deriving from the scale of
flow.
