From the beginning of the twentieth century until today, a series of laws has emerged concerning the growth of computers and networks. They all negotiate the value of a network according to the technological leverage of the time. A short introduction to these laws, will prove why social media and collaborative communities can potentially grow into much larger networks than any other system.
Sarnoff’s law, which was interested with the growth of radio and television networks (and said that the system grows according to the amount of users constituting it) was followed by Moore’s law handling the evolution and expansion of computer microchips. Some years later, with the introduction of ARPANET and the first connected computers, Bob Metcalfe proposed a new law describing the value and the growth of networks.[as explained in the blog of VC Mike]
The law suggests that the value of a network grows with the square of the number of nodes (namely devices or people) it connects. It proposes that the number of potential connections between nodes grows faster than the actual number of nodes. If we have for example four nodes in a network, they will have a value of sixteen.
[When N the number of nodes, Metcalfe's law suggests that the network expands in a rate of N²; see Simeon Simeonov's post for more information] // [Value, according to David Reed, takes the form of potential connectivity for transactions. Namely, the number of different access points (users), which any particular user can connect to]
David Reed studied Metcalfe’s law and noticed, that although it describes telephone systems or small ethernet networks very well, it cannot be applied to social, group-forming networks such as the internet. The reason is, that Metcalfe’s law does not take into account the ability of the people in the network to form groups. Considering this, Reed discovered that “…networks that support the construction of communicating groups create value that scales exponentially with network size, i.e. much more rapidly than Metcalfe’s square law”. For example, according to Metcalfe’s law the value of ten users is one hundred (ten to the second power) and according to Reed’s Law 1,024 (two to the tenth power).

Illustration found at http://spectrum.ieee.org/print/4109
We see that both Metcalfe’s and Reed’s laws are associating the linkage of a network with its value. The connections in a system multiply faster the more users are part of it. Reed’s law assigns one more variable to this thesis. It declares that in a networked system, participation can exponentially boost its value. Though at first this increase may be small, it can eventually grow rapidly – and indeed faster than any other power law. Under that perspective, it is logical to assume that the possibility of group forming networks to expand into vast interconnected systems, systems that can function as hubs in the social web, is much higher than any other non-social network.
The transition that we are witnessing in traditional hubs (Amazon.com, google, etc.) verifies this assumption. These sites were providing services, which could be understood under Metcalfe’s law. They were connecting people and sites, but they were not giving them the option to form their own communities. But the rise of group forming networks and their expansion potential forced these sites to change. Amazon is now providing user review services and google several blogging and document sharing functions. These traditional hubs saw that social networking has the ability to create rapidly a vast system; a fact that made ‘fit’ (fitness as expressed by Albert-Làszlo Barabàsi) social websites strong competitors. The only way to stay on top was to add group forming services, which would allow the development of their network and keep users interested.
In Conclusion
There has been a great controversy about the role of Metcalfe’s law in the internet and if it actually can be applied to it. But if we disregard this debate and go a little bit further to Reed’s Law, we will see that a network, which encourages user interaction and group formation is able to grow much faster than other networks and - as I previously noticed - faster than any power law. This may be a mathematical explanation, why the blogosphere has grown so rapidly. It includes the parameter of socializing; blogs do not connect only computers but also people.
But its development makes me nonetheless wonder: In what frequency and degree does interaction between bloggers actually take place? Is the blogosphere also evolving in a ‘one-to-many’ medium or does the social aspect constantly win ground?