Metcalfe's law states that the value of a
telecommunications network
A telecommunications network is a group of nodes interconnected by telecommunications links that are used to exchange messages between the nodes. The links may use a variety of technologies based on the methodologies of circuit switching, mes ...
is
proportional to the square of the number of connected users
of the system (''n''
2). First formulated in this form by
George Gilder
George Franklin Gilder (; born November 29, 1939) is an American investor, author, economist, and co-founder of the Discovery Institute. His 1981 book, '' Wealth and Poverty'', advanced a case for supply-side economics and capitalism during the e ...
in 1993, and attributed to
Robert Metcalfe
Robert Melancton Metcalfe (born April 7, 1946) is an engineer and entrepreneur from the United States who helped pioneer the Internet starting in 1970. He co-invented Ethernet, co-founded 3Com and formulated Metcalfe's law, which describes the e ...
in regard to
Ethernet
Ethernet () is a family of wired computer networking technologies commonly used in local area networks (LAN), metropolitan area networks (MAN) and wide area networks (WAN). It was commercially introduced in 1980 and first standardized in 1 ...
, Metcalfe's law was originally presented, c. 1980, not in terms of users, but rather of "compatible communicating devices" (e.g., fax machines, telephones). Only later with the
globalization of the Internet did this law carry over to users and networks as its original intent was to describe Ethernet connections.
Network effects
Metcalfe's law characterizes many of the
network effect
In economics, a network effect (also called network externality or demand-side economies of scale) is the phenomenon by which the value or utility a user derives from a good or service depends on the number of users of compatible products. Net ...
s of communication technologies and networks such as the
Internet
The Internet (or internet) is the global system of interconnected computer networks that uses the Internet protocol suite (TCP/IP) to communicate between networks and devices. It is a '' network of networks'' that consists of private, pub ...
,
social networking and the
World Wide Web
The World Wide Web (WWW), commonly known as the Web, is an information system enabling documents and other web resources to be accessed over the Internet.
Documents and downloadable media are made available to the network through web ...
. Former Chairman of the U.S.
Federal Communications Commission
The Federal Communications Commission (FCC) is an independent agency of the United States federal government that regulates communications by radio, television, wire, satellite, and cable across the United States. The FCC maintains jurisdicti ...
Reed Hundt
Reed Eric Hundt (born March 3, 1948) is an American attorney who served as chairman of the United States Federal Communications Commission from November 29, 1993 to November 3, 1997. Appointed by President Bill Clinton, he served for most of C ...
said that this law gives the most understanding to the workings of the Internet. Metcalfe's Law is related to the fact that the number of unique possible connections in a network of
nodes can be expressed mathematically as the
triangular number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots i ...
, which is
asymptotically proportional to
.
The law has often been illustrated using the example of
fax
Fax (short for facsimile), sometimes called telecopying or telefax (the latter short for telefacsimile), is the telephonic transmission of scanned printed material (both text and images), normally to a telephone number connected to a printer o ...
machines: a single fax machine is useless, but the value of every fax machine increases with the total number of fax machines in the network, because the total number of people with whom each user may send and receive documents increases. Likewise, in social networks, the greater the number of users with the service, the more valuable the service becomes to the community.
History and Derivation
Metcalfe’s law was conceived in 1983 in a presentation to the 3Com sales force. It stated ''V'' would be proportional to the total number of possible connections, or approximately ''n''-squared.
The original incarnation was careful to delineate between a linear cost (''Cn''), non-linear growth, ''n''
2, and a non-constant proportionality factor ''A'' “Affinity.” The breakeven point where costs are recouped is given by
At some size, the right-hand side of the equation ''V'' “Value” exceeds the cost, and ''A'' describes the relationship between size and net value added. For large ''n'', net network value is then
Metcalfe properly dimensioned ''A'' as “value per user”. Affinity is also a function of network size, and Metcalfe correctly asserted that ''A'' must decline as ''n'' grows large. In a 2006 interview, Metcalfe stated
''“There may be diseconomies of network scale that eventually drive values down with increasing size. So, if V=A*n
2, it could be that A (for “affinity,” value per connection) is also a function of n and heads down after some network size, overwhelming n
2.”''
Growth of ''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
Dunbar's number is a suggested cognitive limit to the number of people with whom one can maintain stable social relationships—relationships in which an individual knows who each person is and how each person relates to every other person. This ...
. It is almost always the case that user growth ''n'' reaches a saturation point. With technologies, substitutes, competitors and
technical obsolescence constrain growth of ''n''. Growth of n is typically assumed to follow a
sigmoid function
A sigmoid function is a mathematical function having a characteristic "S"-shaped curve or sigmoid curve.
A common example of a sigmoid function is the logistic function shown in the first figure and defined by the formula:
:S(x) = \frac = \ ...
such as a
logistic curve
A logistic function or logistic curve is a common S-shaped curve (sigmoid function, sigmoid curve) with equation
f(x) = \frac,
where
For values of x in the domain of real numbers from -\infty to +\infty, the S-curve shown on the right is ...
or
Gompertz curve
The Gompertz curve or Gompertz function is a type of mathematical model for a time series, named after Benjamin Gompertz (1779–1865). It is a sigmoid function which describes growth as being slowest at the start and end of a given time period. Th ...
.
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
.
The maximum possible number of edges in a simple network (i.e. one with no multi-edges or self-edges) is
.
Therefore the density ''ρ'' of a network is the faction of those edges that are actually present is
which for large networks is approximated by
.
Limitations
Metcalfe’s law assumes that the value of each node
is of equal benefit. If this is not the case, for example because one fax machine serves 60 workers in a company, the second fax machine serves half of that, the third one third, and so on, then the relative value of an additional connection decreases. Likewise, in social networks, if users that join later use the network less than early adopters, then the benefit of each additional user may lessen, making the overall network less efficient if costs per users are fixed.
Modified models
Within the context of social networks, many, including Metcalfe himself, have proposed modified models in which the value of the network grows as
rather than
. Reed and
Andrew Odlyzko
Andrew Michael Odlyzko (Andrzej Odłyżko) (born 23 July 1949) is a Polish-American mathematician and a former head of the University of Minnesota's Digital Technology Center and of the Minnesota Supercomputing Institute. He began his career in ...
have sought out possible relationships to Metcalfe's Law in terms of describing the relationship of a network and one can read about how those are related. Tongia and Wilson also examine the related question of the costs to those excluded.
Validation in data
Despite many arguments about Metcalfe' law, no real data based evidence for or against was available for more than 30 years. Only in July 2013, Dutch researchers managed to analyze European Internet usage patterns over a long enough time and found
proportionality for small values of
and
proportionality for large values of
. A few months later, Metcalfe himself provided further proof, as he used Facebook's data over the past 10 years to show a good fit for Metcalfe's law (the model is
).
In 2015, Zhang, Liu and Xu parameterized the Metcalfe function in data from
Tencent
Tencent Holdings Ltd. () is a Chinese multinational technology and entertainment conglomerate and holding company headquartered in Shenzhen. It is one of the highest grossing multimedia companies in the world based on revenue. It is also the wo ...
and Facebook. Their work showed that Metcalfe's law held for both, despite differences in audience between the two sites (Facebook serving a worldwide audience and Tencent serving only Chinese users). The functions for the two sites were
and
respectively.
In a working paper, Peterson linked time-value-of-money concepts to Metcalfe value using Bitcoin and Facebook as numerical examples of the proof and in 2018 applied Metcalfe's law to
Bitcoin, showing that over 70% of variance in Bitcoin value was explained by applying Metcalfe's law to increases in Bitcoin network size.
See also
*
Anti-rival good
*
Beckstrom's law
In economics, Beckstrom's law is a model or theorem formulated by Rod Beckstrom. It purports to answer "the decades-old question of 'how valuable is a network'", and states in summary that "The value of a network equals the net value added to each ...
*
List of eponymous laws
This list of eponymous laws provides links to articles on laws, principles, adages, and other succinct observations or predictions named after a person. In some cases the person named has coined the law – such as Parkinson's law. In other ...
*
Matching (graph theory)
In the mathematical discipline of graph theory, a matching or independent edge set in an undirected graph is a set of edges without common vertices. Finding a matching in a bipartite graph can be treated as a network flow problem.
Definiti ...
*
Matthew effect
*
Pareto principle
*
Reed's law
Reed's law is the assertion of David P. Reed that the utility of large networks, particularly social networks, can scale exponentially with the size of the network.
The reason for this is that the number of possible sub-groups of network partici ...
*
Sarnoff's law
David Sarnoff (February 27, 1891 – December 12, 1971) was an American businessman and pioneer of American radio and television. Throughout most of his career, he led the Radio Corporation of America (RCA) in various capacities from shortly afte ...
References
Further reading
*
*.
External links
A Group Is Its Own Worst Enemy Clay Shirky's keynote speech on Social Software at the O'Reilly Emerging Technology conference, Santa Clara, April 24, 2003. The fourth of his "Four Things to Design For" is: "And, finally, you have to find a way to spare the group from scale. Scale alone kills conversations, because conversations require dense two-way conversations. In conversational contexts, Metcalfe's law is a drag."
{{DEFAULTSORT:Metcalfe's Law
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Information theory