a connected graph, the normalized closeness centrality (or closeness) of a node is the average length of the shortest path 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 Alex Bavelas (1950) as the reciprocal of the farness, that is where is the distance between vertices u and v. However, when speaking of closeness centrality, people usually refer to its normalized form, given by the previous formula multiplied by , where is the number of nodes in the graph This normalisation allows comparisons between nodes of graphs of different sizes. For many graphs, there is a strong correlation between the inverse of closeness and the logarithm of degree, where is the degree of vertex v while α and β are constants for each network. 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). Harmonic centrality In a (not necessarily connected) graph, the harmonic centrality reverses the sum and reciprocal operations in the definition of closeness centrality: where if there is no path from u to v. Harmonic centrality can be normalized by dividing by , where is the number of nodes in the graph. Harmonic centrality was proposed by Marchiori and Latora (2000) and then independently by Dekker (2005), using the name "valued centrality," and by Rochat (2009). Betweenness centrality Betweenness is a centrality measure of a vertex within a graph (there is also edge betweenness, which is not discussed here). Betweenness centrality quantifies the number of times a node acts as a bridge along the shortest path between two other nodes. It was introduced as a measure for quantifying the control of a human on the communication between other humans in a social network by Linton Freeman. In his conception, vertices that have a high probability to occur on a randomly chosen shortest path between two randomly chosen vertices have a high betweenness. The betweenness of a vertex in a graph with vertices is computed as follows: For each pair of vertices (s,t), compute the shortest paths between them. For each pair of vertices (s,t), determine the fraction of shortest paths that pass through the vertex