We introduce the \emph{Topological Stability Index} (TSI), a variance-based scalar measure for persistence barcodes that quantifies the dispersion of persistence lifetimes. Unlike persistent entropy, which depends only on normalized weights, the TSI captures absolute variability and is sensitive to heterogeneous feature scales. We establish fundamental properties of the TSI, including its scaling behavior, invariance under lifetime translation and explicit update formulas under insertion and deletion of bars. We also consider a complementary first-moment-type quantity, the Topological Signal Index (TSigI), which captures the typical scale of persistence lifetimes and provides additional interpretability alongside the TSI. We further introduce a normalized version, $cv\text{TSI}$, which is scale invariant and admits an explicit algebraic relation to the Rényi entropy of order two. In particular, $cv\text{TSI}$ is an affine function of the collision probability $\sum_i p_i^2$, and therefore a monotone reparametrization of the Rényi entropy, providing a direct link between variance-based and entropy-based summaries in topological data analysis. Numerical experiments on synthetic data and stochastic time series demonstrate that the TSI captures structural variability complementary to entropy: it is relatively insensitive to deterministic trends, while responding strongly to stochastic fluctuations and variations in persistence magnitude.

翻译：我们提出拓扑稳定性指数（TSI），这是一种基于方差的标量度量，用于量化持续条码中持久寿命的离散程度。与仅依赖于归一化权重的持续熵不同，TSI捕获绝对变异性，并对异质特征尺度敏感。我们建立了TSI的基本性质，包括其缩放行为、在寿命平移下的不变性以及在条插入与删除下的显式更新公式。我们还考虑了一种互补的一阶矩量——拓扑信号指数（TSigI），它捕获持久寿命的典型尺度，并与TSI一同提供额外的可解释性。进一步，我们引入归一化版本$cv\text{TSI}$，该量具有尺度不变性，并允许与二阶Rényi熵建立显式代数关系。特别地，$cv\text{TSI}$是碰撞概率$\sum_i p_i^2$的仿射函数，因此是Rényi熵的单调重参数化，为拓扑数据分析中基于方差和基于熵的摘要之间提供了直接联系。对合成数据和随机时间序列的数值实验表明，TSI捕获了与熵互补的结构变异性：它对确定性趋势相对不敏感，而对随机波动和持久幅度变化反应强烈。