Ultrametric Organization of Energy Landscapes on Random Erdös-Rényi Graphs: Topological Origin of Barrier Hierarchy

This paper investigates the hypothesis of ultrametric organization of abstract energy landscapes defined on random Erdös-Rényi graphs. Within the proposed model, each vertex of the graph is assigned a random free energy uniformly distributed over a given interval, and the kinetics of the system is described by a Markov process with a Kramers transition rate matrix. Based on an exact macroscopic coarse-graining procedure and the spectral decomposition of the effective generator, a kinetic metric between basins of attraction is constructed. Computational experiments for graphs with $$V=5000$$ vertices and $$E=15000$$ edges (average degree $$\langle k\rangle=6.0$$ ) demonstrate that this metric reveals a high degree of nontrivial ultrametricity, which monotonically increases from $$\approx21\%$$ at an energy interval width of $$\Delta F=10$$ kJ/mol to $$\approx92\%$$ 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. Analysis of computational experiments shows that at 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, due to which the metric asymptotically acquires ultrametric properties. The results obtained indicate that the ultrametric organization of energy landscapes is a universal asymptotic property of networks of arbitrary topology with a highly rugged landscape, which for graphs with moderate edge density manifests itself already at sufficiently large energy spreads.