In the financial framework, the concepts of connectedness and diversification have been introduced and developed respectively in the context of systemic risk and portfolio theory. In this paper we propose a theoretical approach to bring to light the relation between connectedness and diversification. Starting from the respective axiomatic definitions, we prove that a class of proper measures of connectedness verifies, after a suitable functional transformation, the axiomatic requirements for a measure of diversification. The core idea of the paper is that connectedness and diversification are so deeply related that it is possible to pass from one concept to the other. In order to exploit such correspondence, we introduce a function, depending on the classical notion of rank of a matrix, that transforms a suitable proper measure of connectedness in a measure of diversification. We point out general properties of the proposed transformation function and apply it to a selection of measures of connectedness, such as the well-known Variance Inflation Factor.

在金融框架中，已经在系统风险和投资组合理论的背景下分别引入和发展了连通性和多元化的概念。在本文中，我们提出了一种理论方法来揭示连通性和多元化之间的关系。从相应的公理定义出发，我们证明了一类适当的连接性度量，在经过适当的功能转换后，可以证明多样化的公理要求。本文的核心思想是联系性和多样化之间的联系如此紧密，以至于有可能从一个概念转移到另一个概念。为了利用这种对应关系，我们根据矩阵秩的经典概念引入了一个函数，从而将多元化的衡量标准转化为合适的连通性衡量标准。我们指出了拟议的变换函数的一般性质，并将其应用于选择的连通性度量，例如众所周知的方差膨胀因子。