Revised Centrality Measures Tell a Robust Story of Ion Conduction in Solids

The three most commonly used centrality measures in network theory have been adapted to consider ion conduction time rather than the number of steps. Flow-IN centrality highlights sites with the largest flow of ions from the nearest neighbor sites. Return-flow centrality highlights sites with a fast rate of first returns for the conducting ion. Flow-through centrality highlights which sites support significant flow of conducting ions and appears more robust to removal of the most central vertices. Exploring these centrality measures with the sample system of proton conduction in yttrium doped barium zirconate shows flow-through centrality to provide a robust picture with high contrast between sites involved in the most probable long-range periodic conduction paths and kinetic Monte Carlo trajectories versus sites rarely visited. The flow-through centrality, including all paths further highlights that when the most central proton site is filled, the remaining highest flow-through centrality sites are nearby, corroborating earlier studies suggesting proton pair motion. Finally, while both return-flow and flow-through centrality measure images deteriorate with noise, image restoration is possible when a detailed balance is used to calculate the smaller rate constant in a forward/backward pair.