Editing Centrality (section)

===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>