Closeness centrality

In a connected graph, closeness centrality (or closeness) of a node is a measure of centrality in a network, calculated as the reciprocal of the sum of the length of the shortest paths between the node and all other nodes in the graph. Thus, the more central a node is, the closer it is to all other nodes.

Closeness was defined by Bavelas (1950) as the reciprocal of the farness, that is:

$C(x)={\frac {1}{\sum _{y}d(y,x)}}.$ where $d(y,x)$ is the distance between vertices $x$ and $y$ . When speaking of closeness centrality, people usually refer to its normalized form which represents the average length of the shortest paths instead of their sum. It is generally given by the previous formula multiplied by $N-1$ , where $N$ is the number of nodes in the graph. For large graphs this difference becomes inconsequential so the $-1$ is dropped resulting in:

$C(x)={\frac {N}{\sum _{y}d(y,x)}}.$ Normalization allows comparisons between nodes of graphs of different sizes.

Taking distances from or to all other nodes is irrelevant in undirected graphs, whereas it can produce totally different results in directed graphs (e.g. a website can have a high closeness centrality from outgoing link, but low closeness centrality from incoming links).

In disconnected graphs

When a graph is not strongly connected, Beauchamp introduced in 1965 the idea of using the sum of reciprocal of distances, instead of the reciprocal of the sum of distances, with the convention $1/\infty =0$ :

$H(x)=\sum _{y\neq x}{\frac {n-1}{d(y,x)}}.$ Beauchamp's modification follows the (much later in time) general principle proposed by Marchiori and Latora (2000) that in graphs with infinite distances the harmonic mean behaves better than the arithmetic mean. Indeed, Bavelas's closeness can be described as the denormalized reciprocal of the arithmetic mean of distances, whereas Beauchamp's centrality is the reciprocal of the harmonic mean of distances.

This idea has resurfaced several time in the literature, often without the normalization factor $n-1$ : for undirected graphs under the name valued centrality by Dekker (2005) and under the name harmonic centrality by Rochat (2009); if was axiomatized by Garg (2009) and proposed once again later by Opsahl (2010). It was studied on general directed graphs by Boldi and Vigna (2014). This idea is also quite similar to market potential proposed in Harris (1954) which now often goes by the term market access.

Variants

Dangalchev (2006), in a work on network vulnerability proposes for undirected graphs a different definition:

$D(x)=\sum _{y\neq x}{\frac {1}{2^{d(y,x)}}}.$ This definition is used effectively for disconnected graphs and allows to create convenient formulae for graph operations. For example:

If a graph $G_{1}+G_{2}$ is created by linking node $p$ of graph $G_{1}$ to node $q$ of graph $G_{2}$ then the combined closeness is:

$D(G_{1}+G_{2})=D(G_{1})+D(G_{2})+(1+D(p))(1+D(q));$ if a graph $G_{1}+G_{2}$ is created by collapsing node $p$ of graph $G_{1}$ and node $q$ of graph $G_{2}$ into one node then the closeness is:

$D(G_{1}+G_{2})=D(G_{1})+D(G_{2})+2D(p)D(q).$ If graph $T(G)$ is the thorn graph of graph $G$ , which has $n$ nodes, then $T(G)$ closeness is:

$D(T(G))={\frac {9}{4}}D(G)+n.$ The natural generalization of this definition is:

$D(x)=\sum _{y\neq x}\ {\alpha ^{d(y,x)}},$ where $\alpha$ belongs to (0,1). As $\alpha$ increases from 0 to 1, the generalized closeness changes from local characteristic (degree) to global (number of connected nodes).

The information centrality of Stephenson and Zelen (1989) is another closeness measure, which computes the harmonic mean of the resistance distances towards a vertex x, which is smaller if x has many paths of small resistance connecting it to other vertices.

In the classic definition of the closeness centrality, the spread of information is modeled by the use of shortest paths. This model might not be the most realistic for all types of communication scenarios. Thus, related definitions have been discussed to measure closeness, like the random walk closeness centrality introduced by Noh and Rieger (2004). It measures the speed with which randomly walking messages reach a vertex from elsewhere in the graph—a sort of random-walk version of closeness centrality. Hierarchical closeness of Tran and Kwon (2014) is an extended closeness centrality to deal still in another way with the limitation of closeness in graphs that are not strongly connected. The hierarchical closeness explicitly includes information about the range of other nodes that can be affected by the given node.