Over the years, various network pioneers have attempted to model how the growth of a network increases its value. In other words, they tried to describe the power of network effects. As time went on, each new law discovered that the value of networks and network growth had been significantly underestimated in the past.
These laws are not true laws in the same way that the law of gravity is a scientifically proven law. They're simply math concepts that describe the relationships between different types of networks and the value of those networks. They've been called laws because it sounds cool. Sometimes you can have aspects of all these "laws" applying to the same network simultaneously.
David Sarnoff was a titan of broadcast era radio and TV, who led the Radio Corporation of America (which created NBC) from 1919 until 1970. It was one of the largest networks in the world during those years. Sarnoff observed that the value of his network seemed to increase in direct proportion to the size of the network - proportional to N, where N is the total number of users on the network.
As it turned out, Sarnoff's description of network value ended up being an underestimate for some types of networks, although it was an accurate description of broadcast networks with a few central nodes broadcasting to many marginal nodes (a radio or television audience).
Metcalfe's Law states the value of a communications network grows in proportion to the square of the number of users on the network (N2 where N is the total number of users on the network).
The formulation of this concept, which dates to about 1980, is attributed to Robert Metcalfe, who was one of the inventors of the Ethernet standard.
Metcalfe's Law seems to hold because the number of links between nodes on a network increase mathematically at a rate of N2, where N is the number of nodes. Although originally formulated to describe communication networks like Ethernet, fax, or phone networks, with the arrival of the internet it has evolved to describe social networks and marketplaces as well.
Reed's Law was published by David P. Reed of MIT in 1999. While Reed acknowledged that "many kinds of value grow proportionally to network size" and that some grow as a proportion to the square of network size, he suggested that "group-forming networks" that allow for the formation of clusters (as described above) scale value even faster than other networks.
Group-forming networks, according to Reed, increase in value a rate of 2N, where N is the total number of nodes on the network.
The reason why Reed suggested a formula of 2N instead of N2 is because the number of possible groups within a network that "supports easy group communication" is much higher than 1, so that the total number of connections in the network (the network density) is not just a function of the total number of nodes (N2). In reality it's a function of the total number of nodes plus the total number of possible sub-groupings or clusters, which scales at a much faster rate with the addition of more users to the network.
Since most online networks allow for the formation of clusters, they will likely behave at least somewhat as Reed's Law suggests and grow in value at a much faster rate than either Metcalfe's Law or Sarnoff's Law suggest.