Editing Metcalfe's law (section)

== History and derivation ==
Metcalfe's law was conceived in 1983 in a presentation to the [[3Com]] sales force.<ref>{{Cite journal |last=Metcalfe |first=Bob |date=December 2013 |title=Metcalfe's Law after 40 Years of Ethernet |url= https://ieeexplore.ieee.org/document/6636305 |journal=Computer |volume=46 |issue=12 |pages=26–31 |doi=10.1109/MC.2013.374 |s2cid=206448593 |issn=1558-0814}}</ref> It stated {{var|V}} would be proportional to the total number of possible connections, or approximately {{var|n}}-squared.

The original incarnation was careful to delineate between a linear cost ({{var|Cn}}), non-linear growth({{var|n}}<sup>2</sup>) and a non-constant proportionality factor affinity ({{var|A}}). The [[break-even point]] point where costs are recouped is given by:<math display="block">C \times n=A\times n(n-1)/2</math>At some size, the right-hand side of the equation {{var|V}}, value, exceeds the cost, and {{var|A}} describes the relationship between size and net value added. For large {{var|n}}, net network value is then:<math display="block">\Pi=n(A \times  (n-1)/2 - C)</math>Metcalfe properly dimensioned {{var|A}} as "value per user". Affinity is also a function of network size, and Metcalfe correctly asserted that {{var|A}} must decline as {{var|n}} grows large. In a 2006 interview, Metcalfe stated:<ref>{{Cite web |last=Metcalfe |first=Robert |date=18 August 2006 |title=Guest Blogger Bob Metcalfe: Metcalfe's Law Recurses down the Long Tail of Social Networks |url= https://vcmike.wordpress.com/2006/08/18/metcalfe-social-networks/ |website=VC Mike's Blog}}</ref>

{{blockquote|1=There may be diseconomies of network scale that eventually drive values down with increasing size. So, if ''V''&nbsp;=&nbsp;''An''<sup>2</sup>, it could be that ''A'' (for “affinity,” value per connection) is also a function of ''n'' and heads down after some network size, overwhelming&nbsp;''n''<sup>2</sup>.}}

=== Growth of {{var|n}} ===
Network size, and hence value, does not grow unbounded but is constrained by practical limitations such as infrastructure, access to technology, and [[bounded rationality]] such as [[Dunbar's number]]. It is almost always the case that user growth {{var|n}} reaches a saturation point. With technologies, substitutes, competitors and [[technical obsolescence]] constrain growth of {{var|n}}. Growth of n is typically assumed to follow a [[sigmoid function]] such as a [[Logistic function|logistic curve]] or [[Gompertz function|Gompertz curve]].

=== Density ===
''A'' is also governed by the connectivity or ''density'' of the network topology. In an undirected network, every ''edge'' connects two nodes such that there are 2''m'' nodes per edge. The proportion of nodes in actual contact are given by <math> c=2m / n </math>.

The maximum possible number of edges in a simple network (i.e. one with no multi-edges or self-edges) is <math> \binom{n}{2}=n(n-1)/2</math>. 
Therefore the density ''ρ'' of a network is the faction of those edges that are actually present is:

{{block indent|1=<math> \rho=c/(n-1) </math>}}

which for large networks is approximated by <math> \rho=c/n </math>.<ref>{{Cite book |last=Newman |first=Mark E. J. |title="Mathematics of Networks" in Networks |publisher=Oxford University Press |date=2019 |isbn=9780198805090 |pages=126–128}}</ref>