Betweenness centrality

In graph theory, betweenness centrality is a measure of centrality in a graph based on shortest paths. For every pair of vertices in a connected graph, there exists at least one shortest path between the vertices such that either the number of edges that the path passes through (for unweighted graphs) or the sum of the weights of the edges (for weighted graphs) is minimized. The betweenness centrality for each vertex is the number of these shortest paths that pass through the vertex.

Betweenness centrality finds wide application in network theory: it represents the degree of which nodes stand between each other. For example, in a telecommunications network, a node with higher betweenness centrality would have more control over the network, because more information will pass through that node. Betweenness centrality was devised as a general measure of centrality: it applies to a wide range of problems in network theory, including problems related to social networks, biology, transport and scientific cooperation.

Although earlier authors have intuitively described centrality as based on betweenness, Freeman (1977) gave the first formal definition of betweenness centrality. The idea was earlier proposed by mathematician J. Anthonisse, but his work was never published.

The betweenness centrality of a node ${\displaystyle v}$ is given by the expression:

where ${\displaystyle \sigma _{st}}$ is the total number of shortest paths from node ${\displaystyle s}$ to node ${\displaystyle t}$ and ${\displaystyle \sigma _{st}(v)}$ is the number of those paths that pass through ${\displaystyle v}$.

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