Price of anarchy is maximized at the percolation threshold

Antagonistic Phenomena in Network Dynamics

Recent research on the network modeling of complex systems has led to a convenient representation of numerous natural, social, and engineered systems that are now recognized as networks of interacting parts. Such systems can exhibit a wealth of phenomena that not only cannot be anticipated from merely examining their parts, as per the textbook definition of complexity, but also challenge intuition even when considered in the context of what is now known in network science. Here we review the recent literature on two major classes of such phenomena that have far-reaching implications: (i) antagonistic responses to changes of states or parameters and (ii) coexistence of seemingly incongruous behaviors or properties-both deriving from the collective and inherently decentralized nature of the dynamics. They include effects as diverse as negative compressibility in engineered materials, rescue interactions in biological networks, negative resistance in fluid networks, and the Braess paradox occurring across transport and supply networks. They also include remote synchronization, chimera states and the converse of symmetry breaking in brain, power-grid and oscillator networks as well as remote control in biological and bio-inspired systems. By offering a unified view of these various scenarios, we suggest that they are representative of a yet broader class of unprecedented network phenomena that ought to be revealed and explained by future research.

Price of anarchy on heterogeneous traffic-flow networks

The efficiency of routing traffic through a network, comprising nodes connected by links whose cost of traversal is either fixed or varies in proportion to volume of usage, can be measured by the "price of anarchy." This is the ratio of the cost incurred by agents who act to minimize their individual expenditure to the optimal cost borne by the entire system. As the total traffic load and the network variability-parameterized by the proportion of variable-cost links in the network-changes, the behaviors that the system presents can be understood with the introduction of a network of simpler structure. This is constructed from classes of nonoverlapping paths connecting source to destination nodes that are characterized by the number of variable-cost edges they contain. It is shown that localized peaks in the price of anarchy occur at critical traffic volumes at which it becomes beneficial to exploit ostensibly more expensive paths as the network becomes more congested. Simulation results verifying these findings are presented for the variation of the price of anarchy with the network's size, aspect ratio, variability, and traffic load.