Metcalfe's law characterizes many of the network effects of communication technologies and networks such as the Internet, social networking and the World Wide Web. Former Chairman of the U.S. Federal Communications Commission Reed Hundt 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 , which is asymptotically proportional to .
The law has often been illustrated using the example of fax 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.
In addition to the difficulty of quantifying the "value" of a network, the mathematical justification for Metcalfe's law measures only the potential number of contacts, i.e., the technological side of a network. However the social utility of a network depends upon the number of nodes in contact. If there are language barriers or other reasons why large parts of a network are not in contact with other parts then the effect may be smaller.
Metcalfe’s law assumes that the value of each node is of equal benefit. If this is not the case, for example because the one fax machines serves 50 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.
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 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 dataEdit
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 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. 
- Carl Shapiro and Hal R. Varian (1999). Information Rules. Harvard Business Press. ISBN 978-0-87584-863-1.
- Simeon Simeonov (July 26, 2006). "Metcalfe's Law: more misunderstood than wrong?". HighContrast: Innovation & venture capital in the post-broadband era.
- James Hendler and Jennifer Golbeck (2008). "Metcalfe's Law, Web 2.0, and the Semantic Web" (PDF).
- Bob Briscoe, Andrew Odlyzko and Benjamin Tilly (July 2006). "Metcalfe's Law is wrong". Retrieved 2010-07-25.
- R. Tongia. "The Dark Side of Metcalfe's Law: Multiple and Growing Costs of Network Exclusion" (PDF). Retrieved 2017-12-19.
- Andrew Odlyzko; Bob Briscoe (1 Jul 2006). "Metcalfe's Law is Wrong". IEEE Spectrum: Technology, Engineering, and Science News. Retrieved 25 November 2016.
- "Guest Blogger Bob Metcalfe: Metcalfe's Law Recurses Down the Long Tail of Social Networks". 18 August 2006. Retrieved 2010-06-20.
- B. Briscoe, A. Odlyzko, and B. Tilly, Metcalfe’s law is wrong, IEEE Spectrum 43:7 (2006), pp. 34–39.
- Rahul Tongia and Ernest Wilson (September 2007). "The Flip Side of Metcalfe's Law: Multiple and Growing Costs of Network Exclusion". Retrieved 2013-01-15.
- Madureira, António; den Hartog, Frank; Bouwman, Harry; Baken, Nico (2013), "Empirical validation of Metcalfe's law: How Internet usage patterns have changed over time", Information Economics and Policy, doi:10.1016/j.infoecopol.2013.07.002
- Metcalfe, Bob (2013). "Metcalfe's law after 40 years of Ethernet". IEEE Computer. 46 (12): 26–31. doi:10.1109/MC.2013.374.
- Zhang, Xing-Zhou; Liu, Jing-Jie; Xu, Zhi-Wei (2015). "Tencent and Facebook Data Validate Metcalfe's Law". Journal of Computer Science and Technology. 30 (2): 246–251. doi:10.1007/s11390-015-1518-1.
- Peterson, Timothy (2019). "Bitcoin Spreads Like a Virus". Working Paper. doi:10.2139/ssrn.3356098.
- Peterson, Timothy (2018). "Metcalfe's Law as a Model for Bitcoin's Value". Alternative Investment Analyst Review. 7 (2): 9–18. doi:10.2139/ssrn.3078248.
- 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."