Numerical Deviations from the Commute Time–Resistance Identity: An Observation Note on a Graph Asymmetry Index

For a connected simple graph G = (V, E) we record the dimensionless asymmetry index d_mu(u, v) = |H(u->v) - H(v->u)| / [H(u->v) + H(v->u)] in [0, 1], where H(u->v) is the expected hitting time of v from u under the simple random walk. By the classical commute-time identity, the denominator equals 2|E| * R_eff(u, v), so d_mu measures the relative asymmetry of hitting times normalized by the symmetric resistance distance, and admits a closed form via the pseudoinverse Laplacian. We observe the behavior of d_mu across nine canonical graph families (paths, cycles, complete graphs, stars, random trees, 2D grids, Erdős–Rényi, Barabási–Albert, and stochastic block model) and report empirical correlations with the algebraic connectivity, the spectral mixing-time proxy, and the Cheeger lower bound. The findings point to two regimes: tree-like or hub-driven graphs concentrate around d_mu_max -> 1, while community-balanced graphs exhibit a decreasing d_mu_max as the system size grows. This is shared as an observation note (research record); source code, data, and figures are included for reproducibility.