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.

Open flow network is a weighted directed graph with source and sink to depict flux distributions in the steady state of an open flow system. Energetic food webs, global trade networks, input-output networks and clickstreams, are open flow network models of energy flows, goods flows, money flows, and collective attention flows respectively. Based on the Markov chain techniques, a set of quantities, such as influx, total through-flow, dissipation flow, first-passage flow distances, and first-passage flows can be defined to characterize flows and interactions between nodes. Under this framework, some universal patterns have been found in open flow networks, such as allometric scaling law, generalized Kleiber law, dissipation law, and gravity law, etc. We suppose constructal law cannot only be applied to flow systems with explicit spatial structures like rivers, vascular networks, animal movements, but also can be applied to open flow networks without explicit spatial structures such as energetic food webs, input-output networks, and clickstreams of attention flows. We try to formulate constructal law in the open flow network framework and test it by the real data.

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