Transport Localization at the Percolation Transition Revealed by Betweenness Centrality: Finite-Size Scaling in Site-Diluted Archimedean Lattices

We investigate how transport organizes in two-dimensional Archimedean lattices under random site percolation by analyzing the scaling behavior of betweenness centrality. Transport observables are computed on the connected subgraphs formed by occupied sites as the occupation probability (po) is varied across the percolation transition. We show that the maximization of betweenness centrality defines a transport pseudo-critical point whose position converges to the percolation threshold (pc) in the thermodynamic limit. Near criticality, the maximum betweenness centrality exhibits nontrivial power-law scaling with system size, consistent with the fractal geometry of the incipient infinite cluster. In particular, we find (BCmax∼L2df), while its variance—interpreted as a transport susceptibility—scales as (χ(L)∼Lγ), with (γ≈4df). Finite-size scaling collapses further demonstrate that both the magnitude and the fluctuations of the dominant transport bottleneck follow the universal scaling structure of two-dimensional percolation. These results are consistently observed across square, triangular, kagome, and extended kagome lattices, revealing lattice-independent critical transport behavior. Complementary measures, including the percentile (p90) and inequality indicators of the betweenness distribution, show that transport criticality extends beyond extreme nodes and reflects a collective reorganization of load. Overall, our results establish a direct connection between geometric criticality and transport localization, providing a unified scaling framework for understanding critical transport phenomena in spatial networks.