基于增量嵌套熵的尺度划分：一种面向测度的多尺度结构理论
Scale Partitioning by Incremental Nested Entropy: A Measure-Oriented Theory of Multiscale Structure
AI总结：
该研究提出无需预设输入的SPINE框架，通过增量嵌套熵识别多尺度结构，验证其临界关系并在双涡旋流中识别特定结构，适用于多类测度。
AI中文摘要：
在复杂系统科学领域，网络、空间结构、种群、频谱和动力学流等通常由连接性、频率、几何孤立性、谱强度或变形等对象级测度表示。要确定这些值是否包含不同尺度，通常需要选定的截断值、规定的分组数量或假设的普遍性。我们提出了基于增量嵌套熵的尺度划分（SPINE），这是一种无需上述输入即可识别尺度结构的确定性框架。核心理论将有序测度值的任意正前缀简化为两个熵有效描述符：贡献分量的有效数量和特征测度尺度。这种简化为下一个值何时产生熵支持的尺度转变提供了精确的有限尺寸准则。随着有效规模的增长，临界比率收敛到\textrm{e}，其源于多样性增加与主导性增加之间的平衡，而非阈值校准。相同的临界关系还确定了检测到的边界的精确对齐扰动裕度。连续的局部转变产生数据支持的尺度层数量，而无明确分离的测度则返回单个层。数值实验在异质前缀上验证了临界和稳定性关系的数值精度，且在分离足够时无需提供尺度数量即可恢复生成的2、3、4个尺度。在双涡旋流中，SPINE识别出占域12.85%的高膨胀结构，且无需规定保留的百分位。该框架适用于几何、分类、网络、频谱和动力学测度。
英文摘要：
Across complex systems science, networks, spatial structures, populations, spectra, and dynamical flows are often represented by object-level measures such as connectivity, frequency, geometric isolation, spectral strength, or deformation. Determining whether these values contain distinct scales commonly requires a chosen cutoff, a prescribed number of groups, or an assumed prevalence. We introduce Scale Partitioning by Incremental Nested Entropy (SPINE), a deterministic framework that identifies scale structure without these inputs. The central theory reduces an arbitrary positive prefix of ordered measure values to two entropy-effective descriptors: an effective number of contributing components and a characteristic measure scale. This reduction yields an exact finite-size criterion for when the next value produces an entropy-supported scale transition. As the effective size grows, the critical ratio converges to \(e\), which emerges from the balance between increasing diversity and increasing dominance rather than from threshold calibration. The same critical relation also determines the exact aligned perturbation margin of a detected boundary. Successive local transitions produce a data-supported number of scale strata, while a measure with no resolvable separation returns a single stratum. Numerical experiments verify the critical and stability relations to numerical precision across heterogeneous prefixes and recover two, three, and four generated scales without being supplied their number once separation is sufficient. In a double-gyre flow, SPINE identifies a high-expansion structure occupying \(12.85\%\) of the domain without prescribing a retained percentile. The framework applies to geometric, categorical, network, spectral, and dynamical measures.
