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.
The number next to each node is the distance from that node to the square red node as measured by the length of the shortest path. The green edges illustrate one of the two shortest paths between the red square node and the red circle node. The closeness of the red square node is therefore 5/(1+1+1+2+2) = 5/7.
Closeness was defined by Bavelas (1950) as the reciprocal of the farness,[1][2] that is:
where is the distance (length of the shortest path) between vertices and . This unnormalised version of closeness is sometimes known as status.[3][4][5] 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 , where is the number of nodes in the graph resulting in:
The normalization of closeness simplifies the comparison of nodes in graphs of different sizes. For large graphs, the minus one in the normalisation becomes inconsequential and it is often dropped.
As one of the oldest centrality measures, closeness is often given in general discussions of network centrality measures in introductory texts[6][7][8] or in articles comparing different centrality measures.[9][10][11][12] The values produced by many centrality measures can be highly correlated.[9][13][11] In particular, closeness and degree have been shown[12] to be related in many networks through an approximate relationship
where is the degree of vertex while and β are parameters found by fitting closeness and degree to this formula. The z parameter represents the branching factor, the average degree of nodes (excluding the root node and leaves) of the shortest-path trees used to approximate networks when demonstrating this relationship.[12] This is never an exact relationship but it captures a trend seen in many real-world networks.
Closeness is related to other length scales used in network science. For instance, the average shortest path length , the average distance between vertices in a network, is simply the average of the inverse closeness values
.
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 links, but low closeness centrality from incoming links).
^Bavelas, Alex (1950). "Communication Patterns in Task‐Oriented Groups". The Journal of the Acoustical Society of America. 22 (6): 725–730. Bibcode:1950ASAJ...22..725B. doi:10.1121/1.1906679.
^Sabidussi, G (1966). "The centrality index of a graph". Psychometrika. 31 (4): 581–603. doi:10.1007/bf02289527. hdl:10338.dmlcz/101401. PMID 5232444. S2CID 119981743.
^Harary, Frank (1959). "Status and Contrastatus". Sociometry. 22 (1): 23–43. doi:10.2307/2785610. JSTOR 2785610.
^Hage, Per; Harary, Frank (1995). "Eccentricity and centrality in networks". Social Networks. 17 (1): 57–63. doi:10.1016/0378-8733(94)00248-9.
^Wuchty, Stefan; Stadler, Peter F. (2003). "Centers of complex networks". Journal of Theoretical Biology. 223 (1): 45–53. Bibcode:2003JThBi.223...45W. doi:10.1016/S0022-5193(03)00071-7. PMID 12782116.
^Newman, M. E. J. (2010). Networks : an introduction. Oxford: Oxford University Press. ISBN 978-0-19-920665-0. OCLC 456837194.
^Latora, Vito (2017). Complex Networks : principles, methods and applications. Vincenzo Nicosia, Giovanni Russo. Cambridge, United Kingdom. ISBN 978-1-316-21600-2. OCLC 1004620089.{{cite book}}: CS1 maint: location missing publisher (link)
^Cosia, Michele (2021). The Atlas for the Aspiring Network Scientist. arXiv:2101.00863. ISBN 9788797282403.
^ abBolland, John M (1988). "Sorting out centrality: An analysis of the performance of four centrality models in real and simulated networks". Social Networks. 10 (3): 233–253. doi:10.1016/0378-8733(88)90014-7.
^ abSchoch, David; Valente, Thomas W.; Brandes, Ulrik (2017). "Correlations among centrality indices and a class of uniquely ranked graphs". Social Networks. 50: 46–54. doi:10.1016/j.socnet.2017.03.010. S2CID 10932381.
^ abcEvans, Tim S.; Chen, Bingsheng (2022). "Linking the network centrality measures closeness and degree". Communications Physics. 5 (1): 172. arXiv:2108.01149. Bibcode:2022CmPhy...5..172E. doi:10.1038/s42005-022-00949-5. ISSN 2399-3650. S2CID 236881169.
^Valente, Thomas W.; Coronges, Kathryn; Lakon, Cynthia; Costenbader, Elizabeth (2008-01-01). "How Correlated Are Network Centrality Measures?". Connections (Toronto, Ont.). 28 (1): 16–26. ISSN 0226-1766. PMC 2875682. PMID 20505784.
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