The Price of Anarchy in flow networks as a function of node properties

Many real-world systems such as traffic and electrical flow are described as flows following paths of least resistance through networks, with researchers often focusing on promoting efficiency by optimising network topology. Here, we instead focus on the impact of network node properties on flow efficiency. We use the Price of Anarchy $\mathcal{P}$ to characterise the efficiency of least-resistance flows on a range of networks whose nodes have the property of being sources, sinks or passive conduits of the flow. The maximum value of $\mathcal{P}$ and the critical flow volume at which this occurs are determined as a function of the network's node property composition, and found to have a particular morphology that is invariant with network size and topology. Scaling relationships with network size are also obtained, and $\mathcal{P}$ is demonstrated to be a proxy for network redundancy. The results are interpreted for the operation of electrical micro-grids, which possess variable numbers of distributed generators and consumers. The highest inefficiencies in all networks are found to occur when the numbers of source and sink nodes are equal, a situation which may occur in micro-grids, while highest efficiencies are associated with networks containing a few large source nodes and many small sinks, corresponding to more traditional power grids