# Holevo Quantity - Vertex Centrality

#### Definition

This method measures vertex centrality based on the quantum information theoretical concept of Holevo quantity. More specifically, it measures the importance of a vertex in terms of the variation in graph entropy before and after its removal from the graph

$$\begin{align*}

HC(v)=x(\{({|E_{\bar v}|\over |E|}, {G_{\bar v})},({|E_{\bar v}|\over |E|},G_v)\})\\

=S(p(G))-({|E_{\bar v}|\over |E|} S(p(G_{\bar v}))+{|E_{v}|\over |E|} S(p(G_{v})))

\end{align*}$$

were $G_v = (V,E_v)$ denote the subgraph with vertex set $V$ and edge set $E_v$ and $G_{\bar v}=V,E_{\bar v}$ is the subgraph with vertex set $V$ and edge set $E_v$. And $S(p(G_{v})$ is the entropy of $ρ(G_v)$.

Considering this method centrality of a vertex $v$ can be broken down in two parts, one which is negatively correlated with the degree centrality of $v$, and one which depends on the emergence of non-trivial structures in the graph when $v$ is disconnected from the rest of the graph

$$\begin{align*}

HC(v)=x(\{({|E_{\bar v}|\over |E|}, {G_{\bar v})},({|E_{\bar v}|\over |E|},G_v)\})\\

=S(p(G))-({|E_{\bar v}|\over |E|} S(p(G_{\bar v}))+{|E_{v}|\over |E|} S(p(G_{v})))

\end{align*}$$

were $G_v = (V,E_v)$ denote the subgraph with vertex set $V$ and edge set $E_v$ and $G_{\bar v}=V,E_{\bar v}$ is the subgraph with vertex set $V$ and edge set $E_v$. And $S(p(G_{v})$ is the entropy of $ρ(G_v)$.

Considering this method centrality of a vertex $v$ can be broken down in two parts, one which is negatively correlated with the degree centrality of $v$, and one which depends on the emergence of non-trivial structures in the graph when $v$ is disconnected from the rest of the graph