different fields or to measure the impact of an author on a field and their social capital. When used to select potential leads in customer data, closeness has been seen to lead to a significant gain in the success rate. The closeness of a city in an air transport network has been shown to be highly correlated with socio-economic indicators such as gross regional domestic product. Closeness has also been applied to biological networks where, for instance, this was used to identify more than 50% of the global regulators within the top 2% of the ranked genes or essential genes were found to have higher closeness than nonessential genes in protein-interaction networks. In a metabolic network the closeness of nodes can identify the most important metabolites. 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 : 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 : 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: This definition is used effectively for disconnected graphs and allows to create convenient formulae for graph operations. For example: If a graph is created by linking node of graph to node of graph then the combined closeness is: if a graph is created by collapsing node of graph and node of graph into one node then the closeness is: If graph is the thorn graph of graph , which has nodes, then closeness is: The natural generalization of this