Identifying the Limits to Network Growth

This article draws heavily on ideas from Jeffrey M. Stibel's (@Stibel) book Wired for Thought: How the Brain Is Shaping the Future of the Internet. I will write a book review in the next few days. However, the ideas presented here are too broad to be discussed in the review only.

Let's have a look at the growth of networks, be it the internet, the neural network of the brain or the social network of your friends and family members (online or offline). Metcalfe's law - formulated by Bob Metcalfe (@BobMetcalfe) - states that the value of a telecommunications network is proportional to the square of the number of connected users of this network (n^2). While this law has been critized by several authors (e.g. here), its general idea has proven to be quite robust in the course of time despite postulating unlimited growth.

However, Metcalfe (@BobMetcalfe) asked in a 2006 blogpost whether this is actually true. It might be the case that the value of a network actually starts going down after some size. This is exactly what Jeff Stibel (@Stibel) proposes when he writes:

My theory is that Metcalfe's law works until a network reaches the point of critical mass. But at that point - where cost exceeds value - the value curve stops increasing. The value curve almost inevitably follows the cost curve from that point forward. (p. 113)

He reasons that this critical mass is determined by a wide range of external factors that determine something that might be termed "carrying capacity". For an ant colony (which is basically a network) such a factor might for example be food availability. If Stibel (@Stibel) is correct and such a carrying capacity exists for the internet, we can use systems theory to give an idea of how this slowing down of growth might happen:

Systems facing a carrying capacity constraint usually exhibit S-shaped growth. In their beginnings they are dominated by economies of scale and grow at an ever increasing rate (similar to Metcalfe's law). However, when their size approaches the carrying capacity and resources become scarce, growth slows down and the system eventually settles in an equilibrium (for a generic explanation of the S-shaped growth behavior visit the website of the System Dynamics Society).

It is quite difficult to forecast such S-shaped growth patterns because a number of mathematical models can be used (e.g. the logistic model or the Gompertz model), that might fit historical data equally well. However, it is particularly important to pay attention to the inflection point, at which the growth rate starts to decrease. Furthermore it is important to identify the factors determining the carrying capacity of a particular network.

The identification of these factors can not only improve the valuation of internet companies which might prevent another dot-com bubble; it can also help to extend the carrying capacity of the internet and to ensure its continuous growth. Any ideas on what these factors might be?

I will soon publish a more comprehensive review of Wired for Thought: How the Brain Is Shaping the Future of the Internet. Follow me on Twitter and I will let you know!