  Open flow network is a weighted directed graph with a source and a sink,
depicting flux distributions on networks in the steady state of an open flow
system. Energetic food webs, economic input-output networks, and international
trade networks, are open flow network models of energy flows between species,
money or value flows between industrial sectors, and goods flows between
countries, respectively. Flow distances (first-passage or total) between any
given two nodes $i$ and $j$ are defined as the average number of transition
steps of a random walker along the network from $i$ to $j$ under some
conditions. They apparently deviate from the conventional random walk distance
on a closed directed graph because they consider the openness of the flow
network. Flow distances are explicitly expressed by underlying Markov matrix of
a flow system in this paper. With this novel theoretical conception, we can
visualize open flow networks, calculating centrality of each node, and
clustering nodes into groups. We apply flow distances to two kinds of empirical
open flow networks, including energetic food webs and economic input-output
network. In energetic food webs example, we visualize the trophic level of each
species and compare flow distances with other distance metrics on graph. In
input-output network, we rank sectors according to their average distances away
other sectors, and cluster sectors into different groups. Some other potential
applications and mathematical properties are also discussed. To summarize, flow
distance is a useful and powerful tool to study open flow systems.
