AI 中文总结
研究稀疏随机厄多斯-雷尼图上能量景观的超度量组织，通过谱分解构建动力学马氏度量，计算实验表明非平凡超度量性随ΔF增加，证明极限定理，结果显示超度量性是此类网络在大能量扩散极限下的普遍属性。
AI 中文摘要
我们研究了定义在稀疏随机厄多斯-雷尼图上的能量景观的超度量组织。为每个图顶点从宽度为ΔF的区间上的均匀分布中分配一个随机自由能，并通过具有克莱默斯跃迁速率的马尔可夫过程对动力学进行建模。利用速率矩阵的谱分解，我们在吸引盆地之间构建了一个动力学马氏度量。对具有5000个顶点和5000条边的图的计算实验表明，非平凡超度量性的程度从ΔF = 10 kJ/mol时的约42%单调增加到ΔF = 1000 kJ/mol时的约96%。我们证明了一个极限定理：当ΔF→∞时，该度量的对数渐近性逐点收敛到经典的单链超度量。对于有限的ΔF，次优路径的修正随着ΔF的增加而指数抑制，因此该度量渐近地成为超度量。我们的结果表明，在大能量扩散的极限下，超度量性是具有崎岖能量景观的稀疏、局部树状网络的普遍属性。
英文摘要
In the present work, we investigate the hypothesis of the ultrametric organization of abstract energy landscapes defined on random Erdős--Rényi 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 $ΔF=10$ kJ/mol to $\approx96\%$ at $ΔF=1000$ 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.
Comments34 pages, 2 tables
