Supporting Information of “A Geometric Representation of Collective Attention Flows”

Distance on graph is a very useful concept [1]. Both the shortest path distance [2], resistance distance [3] and the mean first-passage distance of a random walker [4-7] can reflect the intrinsic properties of the graph. However, conventional first-passage distance on a graph is based on the basic assumption that the whole network is closed, which means the random walker cannot escape from the network, thus the total number of walkers on the graph is conservative. Nevertheless, the open flow network is an open system. Random walkers can flow into the system from the source and flow out to the sink despite the total number of walkers staying in the network can be also conservative if the flow system is in a steady state. Therefore, the traditional method for closed system cannot be simply extended to open flow networks. It is necessary to extend the distance notions for open flow networks.