Graph-theoretic visualization of ion conduction pathways: Concepts and a tutorial

Ionic conductivity underpins the operation of technologies ranging from fuel cells to solid-state batteries. A central challenge is to visualize and interpret the pathways that enable ion transport. Several complementary strategies have been developed: (1) geometric analyses of binding sites and channels, (2) conduction graph analysis, and (3) statistical distributions from trajectory generating methods. These approaches are deeply interconnected: geometric features define the vertices and edges of conduction graphs, while dynamic simulations provide probability distributions that enrich static connectivity maps. This article surveys these visualization strategies with an emphasis on graph-theoretical methods, highlighting how centrality measures from network theory and pathfinding algorithms can reveal critical conduction routes that include both static and time-dependent features. To bridge theory and practice, we provide a tutorial with scripts that implement three representative methods: flowthrough centrality, path analysis, and kinetic Monte Carlo, on a previously studied model system. By combining conceptual overview with practical tools, we aim to equip researchers with accessible techniques for mapping and interpreting ion conduction pathways across diverse materials. Flow-IN and Flow-OUT centralities are the sum of the rate constants flowing in (a) or out (b) of a specific site. Ions return quickly to high return flow sites as shown in (c) either because they are part of trapped regions or because they are on fast ion periodic highways. Sites with high flow-through centrality have shorter travel times on average for the many paths flowing through them. These paths shown in (d) can be trapped region paths or long-range paths. Note that no periodicity is required for flow-through centrality paths.