Ultrametric organization of energy landscapes on random Erdős--Rényi graphs: topological origin of barrier hierarchy

We investigate the ultrametric organization of energy landscapes defined on sparse random Erdős–Rényi graphs. Each graph vertex is assigned a random free energy from a uniform distribution over an interval of width $\mathrm{\Delta }F$, and the kinetics are modeled by a Markov process with Kramers transition rates. Using spectral decomposition of the rate matrix, we construct a kinetic Mahalanobis metric between basins of attraction. Computational experiments for graphs with $V=5000$ vertices and $E=5000$ edges show that the degree of nontrivial ultrametricity increases monotonically from $\approx 42\mathrm{\%}$ for $\mathrm{\Delta }F=10$ kJ/mol to $\approx 96\mathrm{\%}$ for $\mathrm{\Delta }F=1000$ kJ/mol. We prove a limit theorem: as $\mathrm{\Delta }F\to \mathrm{\infty }$, the logarithmic asymptotics of this metric converge pointwise to the classical single-linkage ultrametric. For finite $\mathrm{\Delta }F$, corrections from suboptimal paths are exponentially suppressed with increasing $\mathrm{\Delta }F$, so that the metric becomes asymptotically ultrametric. Our results suggest that ultrametricity is a universal property of sparse, locally tree-like networks with rugged energy landscapes in the limit of large energy spreads.