Flow distances on open flow networks

An open flow network is a weighted directed graph with a source and a sink, depicting flux distributions on networks in the steady state mode 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. An open flow network is different from a closed flow network because it considers the flows from or to the environment (the source and the sink). For instance, in energetic food webs, species obtain energy not only from other species but also from the environment (sunlight), and species also dissipate energy to the environment. Flow distances between any 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. The conventional method for the calculation of the random walk distance on closed flow networks cannot be applied to open flow networks. Therefore, we derive novel explicit expressions for flow distances of open flow networks according to their underlying Markov matrix of the network in this paper. We apply flow distances to two types of empirical open flow networks, including energetic food webs and economic input–output networks. In energetic food webs, we visualize the trophic level of each species and compare flow distances with other distance metrics on the graph. In economic input–output networks, we rank sectors according to their average flow distances and cluster sectors into different industrial groups with strong connections. Other potential applications and mathematical properties are also discussed. To summarize, flow distance is a useful and powerful tool to study open flow systems.