Ultrametric organization of energy landscapes on random Erd\H{o}s--R\'enyi 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 $\Delta F$, 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 $\approx42\%$ for $\Delta F=10$ kJ/mol to $\approx96\%$ for $\Delta F=1000$ kJ/mol.

We prove a limit theorem: as $\Delta F\to\infty$, the logarithmic asymptotics of this metric converge pointwise to the classical single-linkage ultrametric.

For finite $\Delta F$, corrections from suboptimal paths are exponentially suppressed with increasing $\Delta F$, 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.