Closeness Centrality

From SONIVIS:Wiki

This metric is implemented in every version.

The Closeness Centrality is the normalized number of steps required to access every other node from a given node in a network.

Objective

The Closeness Centrality uses not only neighbors of a node (see Degree Centrality) to determine its centrality, albeit the neighbors of the neighbors are considered, and therefore also nodes that are not directly connected to this given node. Nodes that are not directly connected with this given node receive a lower weight because the intensity of their relation or their influence is lower.

In a social network the idea of the closeness is if one author is close to other authors in the network then this author can interact more quickly with all other nodes in the network. Therefore, the main question is how close is one node to each other node in the network.^[1] The simplest way to realize this idea is to consider geodesics of one node to each other node. ^[1]

Generally speaking, the Closeness Centrality describes the connectedness of a node in undirected networks ^[1]. A node that is connected by a lot of short paths to other nodes, can be interpreted as relatively autonomous in opposite to all nodes that are less connected by short paths. For calculating the Closeness Centrality the geodesic distance of every node to every other node is computed.

In directed networks the InCloseness Centrality and the OutCloseness Centrality are available.

Calculation

The closeness of a node $n_i\,\!$ describes the total distance of $n_i\,\!$ to all other nodes in the network.^[1] For calculation sum the shortest paths $d(n_{i},n_{j})\,\!$ (where $i \neq j\,\!$ ) of a node $n_{i}\,\!$ to all other nodes in the network.

$\sum_{i=1}^{g}d(n_i, n_j)$ , where $i \neq j\,\!$ , $d(n_i, n_j)\,\!$ shortest path between nodes $n_i\,\!$ and $n_j\,\!$ , and $g\,\!$ the total number of nodes in the network.

If a node $n_{i} \,\!$ is connected to all other nodes $n_{j}:j=1,...,m \,\!$ by its shortest paths $d(n_{i},n_{j})\,\!$ (where $j\neq i$ ) then this node $n_{i} \,\!$ is central regarding the Closeness Centrality. The calculation is standardized on the size of the network: