# EDCC - Effective Distance Closeness Centrality

#### Definition

EDCC consider not only the global structure of the network but also the local information of nodes. Closeness centrality EDCC of node $i$, denoted as $C_i^{eff}$

$$C_i^{eff}= {\left[{\underset{j}{\overset{N}{\sum}}}\right]}^{-1} D_{ji}$$

where $D_{ji}$ denotes the effective distance from node $i$ to node $j$. $D_{ji}$ is defined from an arbitrary reference node $i$ to another node $j$ in the network by the length of the shortest path from $i$ to $j$.

$$D_{ji}= {\underset{\Gamma}{\min}} \lambda(\Gamma)$$

$\lambda(\gamma)$ is the directed length of an ordered path $\gamma=\{i_1,...,i_L\}$ is defined, as the sum of effective lengths along the legs of the path

Compared with the classical $CC$, the main difference of $EDCC$ is that using effective distance to replace the conventional geographic distance or binary distance. Effective distance is obtained by the node’s input flow (in-degree in unweighted networks) and its neighbor’s output flow (outdegree in unweighted networks), and the EDCC carries more information about its neighbors than other centrality measures.

$$C_i^{eff}= {\left[{\underset{j}{\overset{N}{\sum}}}\right]}^{-1} D_{ji}$$

where $D_{ji}$ denotes the effective distance from node $i$ to node $j$. $D_{ji}$ is defined from an arbitrary reference node $i$ to another node $j$ in the network by the length of the shortest path from $i$ to $j$.

$$D_{ji}= {\underset{\Gamma}{\min}} \lambda(\Gamma)$$

$\lambda(\gamma)$ is the directed length of an ordered path $\gamma=\{i_1,...,i_L\}$ is defined, as the sum of effective lengths along the legs of the path

Compared with the classical $CC$, the main difference of $EDCC$ is that using effective distance to replace the conventional geographic distance or binary distance. Effective distance is obtained by the node’s input flow (in-degree in unweighted networks) and its neighbor’s output flow (outdegree in unweighted networks), and the EDCC carries more information about its neighbors than other centrality measures.

#### References

- Du, Y., Gao, C., Chen, X., Hu, Y., Sadiq, R. and Deng, Y., 2015. A new closeness centrality measure via effective distance in complex networks. Chaos: An Interdisciplinary Journal of Nonlinear Science, 25(3), p.033112. DOI: 10.1063/1.4916215