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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.

Universal patterns and constructal law in open flow networks