# k-hop Centrality

#### Definition

This method is a generalization of degree centrality. It is calculated in a localized manner and is complexity-scalable by adjusting the value of k, thus suitable for dynamically changing, large social networks. This centrality is calculated by:

$$C_K(v)={\underset{i=1}{\overset{k}{\sum}}} {n_i(v)\over \alpha^{i-1}}$$

were the $C_K(v)$ denotes the k-hop centrality of node $v$, $n_i(v)$ is the number of nodes whose path lengths from node $v$ is at most $i$, and $\alpha$ is the average degree of a network.

The $k$-hop centrality measure is locally calculated and complexity-scalable, and therefore suitable for large-scale dynamic networks. The $k$-hop index outperforms the betweenness and $k$ shell indices in terms of infection ratios (spreading influence) and time complexity.

$$C_K(v)={\underset{i=1}{\overset{k}{\sum}}} {n_i(v)\over \alpha^{i-1}}$$

were the $C_K(v)$ denotes the k-hop centrality of node $v$, $n_i(v)$ is the number of nodes whose path lengths from node $v$ is at most $i$, and $\alpha$ is the average degree of a network.

The $k$-hop centrality measure is locally calculated and complexity-scalable, and therefore suitable for large-scale dynamic networks. The $k$-hop index outperforms the betweenness and $k$ shell indices in terms of infection ratios (spreading influence) and time complexity.

#### References

- Niu J., Fan J., Wang L., Stojinenovic M., 2014. K-hop centrality metric for identifying influential spreaders in dynamic large-scale social networks. 2014 IEEE Global Communications Conference, GLOBECOM 2014, , pp.2954-2959. DOI: 10.1109/GLOCOM.2014.7037257