Multiscale Field Theory for Network Flows

Network flows are pervasive, including the movement of people, transportation of goods, transmission of energy, and dissemination of information; they occur on a range of empirical interconnected systems, from designed infrastructure to naturally evolved networks. Despite the broad spectrum of applications, because of their domain-specific nature and the inherent analytic complexity, a comprehensive theory of network flows is lacking. We introduce a unifying treatment for network flows that considers the fundamental properties of packet symmetries, conservation laws, and routing strategies. For example, electrons in power grids possess interchangeability symmetry, unlike packages sent by postal mail, which are distinguishable. Likewise, packets can be conserved, such as cars in road networks, or dissipated, such as Internet packets that time out. We introduce a hierarchy of analytical field-theoretic approaches to capture the different scales of complexity required. Mean-field analysis uncovers the nature of the transition through which flow becomes unsustainable upon unchecked growth of demand. Mesoscopic field theory accurately accounts for complicated network structures, packet symmetries, and conservation laws and yet is capable of admitting closed-form solutions. Finally, the full-scale field theory allows us to study routing strategies ranging from random diffusion to shortest path. Our theoretical results indicate that flow bottlenecks tend to be near sources for interchangeable packets and near sinks for distinguishable ones, and that dissipation hinders the maximum sustainable throughput for interchangeable packets but can enhance throughput for distinguishable packets. Finally, we showcase the flexibility of our multiscale theory by applying it in two distinct domains of road networks and the neuronal network. Our work paves the way for a more unifying and comprehensive theory of network flows. Published by the American Physical Society 2025