2026/07/17 by A. P. Zubarev
#cond-mat.stat-mech #cond-mat.dis-nn #cs.NA #math.NA
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 ΔF, 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 ≈42% for ΔF=10 kJ/mol to ≈96% for ΔF=1000 kJ/mol. We prove a limit theorem: as ΔF→∞, the logarithmic asymptotics of this metric converge pointwise to the classical single-linkage ultrametric. For finite ΔF, corrections from suboptimal paths are exponentially suppressed with increasing ΔF, 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.