# H-group Closeness Centrality

#### Definition

This method measures how close a node group is to other nodes in a graph, and can be used in numerous applications such as measuring the importance and influence of a group of users in a social network.

The $H$-group closeness of node group $S \subseteq V$ is:

$$C_H(S)\triangleq {\underset{v\in V}{\sum}} g_H (d_{S,v})- f(S),$$

were $f(S)$ is an additive cost function, $d_{S,v}$ measures the distance from node group $S$ to a node $v$ in the graph, and is defined as

$$

d_{S,v} \triangleq

\begin{cases}

\infty, & S= \emptyset,\\

min_{u\in S} dist_{uv} & S\ne \emptyset,

\end{cases}

$$

where $dist_{uv}$ is the shortest path distance from $u$ to $v$, $dist_{uv} = 1$ if $u$ cannot reach $v$ and $dist_{uv}=0$. $H$-group closeness captures features of both group degree and group closeness by choosing proper $g_H$ and $f$.

The $H$-group closeness of node group $S \subseteq V$ is:

$$C_H(S)\triangleq {\underset{v\in V}{\sum}} g_H (d_{S,v})- f(S),$$

were $f(S)$ is an additive cost function, $d_{S,v}$ measures the distance from node group $S$ to a node $v$ in the graph, and is defined as

$$

d_{S,v} \triangleq

\begin{cases}

\infty, & S= \emptyset,\\

min_{u\in S} dist_{uv} & S\ne \emptyset,

\end{cases}

$$

where $dist_{uv}$ is the shortest path distance from $u$ to $v$, $dist_{uv} = 1$ if $u$ cannot reach $v$ and $dist_{uv}=0$. $H$-group closeness captures features of both group degree and group closeness by choosing proper $g_H$ and $f$.

#### References

- Zhao J., Wang P., Lui J.C.S., Towsley D., Guan X., 2017. I/O-efficient calculation of H-group closeness centrality over disk-resident graphs. Information Sciences, 400-401, pp.105-128. DOI: 10.1016/j.ins.2017.03.017