Ultrametric organization of energy landscapes on random Erd\H{o}s--R\'enyi graphs: topological origin of barrier hierarchy

Abstract

In the present work, we investigate the hypothesis of the ultrametric organization of abstract energy landscapes defined on random Erd\H{o}s--R\'enyi graphs. Within the proposed model, each vertex of the graph is assigned a random free energy uniformly distributed on a given interval, and the kinetics of the system is described by a Markov process with a Kramers transition rate matrix. Based on the spectral decomposition of this matrix, a kinetic metric between basins of attraction is constructed. Computational experiments for graphs with $V=5000$ vertices and $E=5000$ edges (average degree $\langle k\rangle=2.0$) demonstrate that this metric reveals a high degree of nontrivial ultrametricity, which monotonically increases from $\approx42\%$ at an energy interval width of $\Delta F=10$ kJ/mol to $\approx96\%$ at $\Delta F=1000$ kJ/mol. For a rigorous mathematical justification of this phenomenon, a limit theorem is formulated and proved, establishing that as the width of this interval tends to infinity, the logarithmic asymptotics of the proposed metric converges pointwise to the classical single-linkage ultrametric. The analysis of computational experiments shows that for finite values of the interval width, suboptimal paths and thermal fluctuations introduce corrections that violate the strong triangle inequality; however, in the regime of large energy fluctuations, their contribution is exponentially suppressed, and consequently the metric asymptotically acquires ultrametric properties. The obtained results indicate that the ultrametric organization of energy landscapes represents a universal asymptotic property of networks of arbitrary topology with a strongly rugged relief, which for sparse, locally tree-like graphs manifests itself already at moderate energy spreads.