Evolution equations on graphs: analysis and applications

Abstract

Evolutionary models on graphs arise throughout the sciences, from physics to economics, biology and machine learning. Such models describe a range of phenomena, expressing how attributes of nodes or even the graph itself change over time. On one hand, many evolutionary models on graphs are inspired by analogues in the spatially continuous setting and can be shown to converge to these continuum limits as the graph is refined. On the other hand, the intrinsically discrete nature of graphs often requires new notions of differential equations, which can be flexible enough to apply to a wide range of graph structures, while still recovering the correct behaviour in the continuum limit. This special issue highlights recent research in the analysis and applications of evolutionary equations on graphs, as well as extensions to more general settings. The collected works cover diverse topics, from homogenisation of gradient flows to differential equations describing the flow of a gas through a network or heterogeneous interactions in multi-agent systems. Furthermore, the issue also considers a broad range of applications, including energy infrastructure, community detection in networks and data clustering methods. Whether describing the flow of gas through pipeline networks, the spread of information or disease across social networks, the evolution of opinions in adaptive communities or the geometry of high-dimensional data, graphs provide a flexible language for formulating evolutionary equations that encode both discrete interactions and spatial organisation.