The concept of connectedness has been widely used in financial applications, in particular for systemic risk detection. Despite its popularity, at the state of the art, a rigorous definition of connectedness is still missing. In this paper we propose a general definition of connectedness introducing the notion of proper measures of connectedness (PMCs). Based on the classical concept of mean introduced by Chisini, we define a family of PMCs and prove some useful properties. Further, we investigate whether the most popular measures of connectedness available in the literature are consistent with the proposed theoretical framework. We also compare different measures in terms of forecasting performances on real financial data. The empirical evidence shows the forecasting superiority of the PMCs compared to the measures that do not satisfy the theoretical properties. Moreover, the empirical results support the evidence that the PMCs can be useful to detect in advance financial bubbles, crises, and, in general, for systemic risk detection.

连通性的概念已广泛用于金融应用程序中，尤其是用于系统性风险检测。尽管它很受欢迎，但在现有技术中，仍然缺少对连接性的严格定义。在本文中，我们提出了对连接性的一般定义，引入了对连接性的适当度量（PMC）的概念。基于Chisini提出的经典的均值概念，我们定义了PMC族并证明了一些有用的特性。此外，我们调查了文献中可用的最流行的连通性度量是否与提出的理论框架一致。我们还根据实际财务数据的预测效果比较了不同的度量。经验证据表明，与不满足理论特性的措施相比，PMC的预测优越性。此外，经验结果支持以下证据：PMC可用于提前发现金融泡沫，危机，并且通常可用于系统风险检测。