Editing Centrality (section)

==Closeness centrality==
{{Main|Closeness centrality}}In a [[Connected component (graph theory)|connected]] [[Graph (discrete mathematics)|graph]], the [[Normalization (statistics)|normalized]] '''closeness centrality''' (or '''closeness''') of a node is the average length of the [[Shortest path problem|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 [[Multiplicative inverse|reciprocal]] of the '''farness''',<ref>Alex Bavelas. Communication patterns in task-oriented groups. ''J. Acoust. Soc. Am'', '''22'''(6):725–730, 1950.</ref><ref>{{cite journal|year=1966|title=The centrality index of a graph|journal=Psychometrika|volume=31|issue=4|pages=581–603|doi=10.1007/bf02289527|pmid=5232444|hdl=10338.dmlcz/101401|last1=Sabidussi|first1=G|s2cid=119981743|hdl-access=free}}</ref> that is <math display="inline">C_B(v)= (\sum_u d(u,v))^{-1}</math> where <math>d(u,v)</math> is the [[Distance (graph theory)|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 <math>N-1</math>, where <math>N</math> is the number of nodes in the graph
: <math>C(v)= \frac{N-1}{\sum_u d(u,v)} .</math>
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,<ref>{{Cite journal |last1=Evans |first1=Tim S. |last2=Chen |first2=Bingsheng |date=2022 |title=Linking the network centrality measures closeness and degree |journal=Communications Physics |language=en |volume=5 |issue=1 |pages=172 |doi=10.1038/s42005-022-00949-5 |arxiv=2108.01149 |bibcode=2022CmPhy...5..172E |s2cid=236881169 |issn=2399-3650|doi-access=free }}</ref> <math>(C(v))^{-1} \approx -\alpha \ln(k_v) + \beta</math> where <math>k_v</math> 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 graph]]s (e.g. a website can have a high closeness centrality from an 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:

: <math>H(v)= \sum_{u | u \neq v} \frac{1}{d(u,v)}</math>

where <math>1 / d(u,v) = 0</math> if there is no path from ''u'' to ''v''. Harmonic centrality can be normalized by dividing by <math>N-1</math>, where <math>N</math> is the number of nodes in the graph.

Harmonic centrality was proposed by [[Massimo Marchiori|Marchiori]] and [[Vito Latora|Latora]] (2000)<ref name="marchiorilatora2000">{{citation| journal = Physica A: Statistical Mechanics and Its Applications  | last1 = Marchiori | first1 = Massimo | last2 = Latora | first2 = Vito | year = 2000 | volume = 285 | issue = 3–4 | pages = 539–546 | title = Harmony in the small-world | doi=10.1016/s0378-4371(00)00311-3| arxiv = cond-mat/0008357 | bibcode = 2000PhyA..285..539M | s2cid = 10523345 }}</ref> and then independently by Dekker (2005), using the name "valued centrality,"<ref>{{cite journal|first1=Anthony|last1=Dekker|title=Conceptual Distance in Social Network Analysis|journal=Journal of Social Structure|volume=6|issue=3|year=2005|url=http://www.cmu.edu/joss/content/articles/volume6/dekker/index.html|access-date=2017-02-18|archive-date=2020-12-04|archive-url=https://web.archive.org/web/20201204145511/https://www.cmu.edu/joss/content/articles/volume6/dekker/index.html|url-status=live}}</ref> and by Rochat (2009).<ref>{{cite conference | author = Yannick Rochat | title = Closeness centrality extended to unconnected graphs: The harmonic centrality index | conference = Applications of Social Network Analysis, ASNA 2009 | url = http://infoscience.epfl.ch/record/200525/files/%5bEN%5dASNA09.pdf | access-date = 2017-02-19 | archive-date = 2017-08-16 | archive-url = https://web.archive.org/web/20170816173227/https://infoscience.epfl.ch/record/200525/files/%5BEN%5DASNA09.pdf | url-status = live }}</ref>