# Holevo Quantity - Edge Centrality

#### Definition

This method is a novel edge centrality measure based on the quantum information theoretical concept of Holevo quantity. This is a measure of the difference in Von Neumann entropy between the original graph and the graph where $e$ has been removed. In other words, it can be seen as a measure of the contribution of $e$ to the Von Neumann entropy of $G$

For a graph $G = (V,E)$, the Holevo edge centrality of $e \in E$ is:

$$HC(c)=x (\{ ( {m-1\over m} ,H_{\bar e}),({1\over m} , H_e)\})$$

were $m=|E|$, $H_{\bar e}$ and $H_e$ are the subgraphs over edge sets $\{e\}$ and $E \{e\}$, respectively. $(m− 1)/m$ is constant for all the edges and thus can be safely ignored.

For a graph $G = (V,E)$, the Holevo edge centrality of $e \in E$ is:

$$HC(c)=x (\{ ( {m-1\over m} ,H_{\bar e}),({1\over m} , H_e)\})$$

were $m=|E|$, $H_{\bar e}$ and $H_e$ are the subgraphs over edge sets $\{e\}$ and $E \{e\}$, respectively. $(m− 1)/m$ is constant for all the edges and thus can be safely ignored.