# Metcalfe’s Law, Accessibility, and Zipf

Bob Metcalfe, Inventor of the Ethernet, famously proposed that the value of a communications network is given by n^2, where is n is the number of members on the network. This has been dubbed Metcalfe’s Law.
In an article published in IEEE Spectrum titled Metcalfe’s Law is Wrong, my colleague Andrew Odlyzko with Bob Briscoe and Benjamin Tilly reason from Zipf’s Law (using Zipf’s Law applied to word frequency, but as transportationists, we could just as easily use Zipf’s Law as applied to city size distribution) why this is not the case, and that n log(n) is a better estimate. In short, not every connection is equally valuable. This is something well understood in transportation, where accessibility measures discount connections by a function of their travel impedance. However this article suggests there is something else going on, that there are, in a sense, diminishing returns to connections. The first connection is more valuable than the second.
One could organize this over time instead of just network size, and suggest that network value grows at a decreasing rate as all the best connections are made first, then the next best connections, and so on.
If this is the case, this generates the hypothesis (which I have not yet tested) that in a hedonic model of price (value) of real estate, accessibility measured as a product of the log of activities will give a better fit than one which just uses activities straight. (Results of hedonic models suggest accessibility is a significant factor in explaining house price, see Access to Destinations: Development of Accessibility Measures (esp. Chapter 5) for an example ).
Traditionally we represent Accessibility (Hansen’s Accessibility Measure) at point i (Ai) as proportional to Destinations at j (say employment Ej) multiplied by f(Cij) where Cij is a travel cost, and f(Cij) is a travel impedance function (e.g. I/Cij^2) in the classic gravity model or e^(B*Cij) using a negative exponential form B<0).
Ai = ∑ Ej * f(Cij)
but the n log(n) argument suggests
Ai = log(∑Ej * f(Cij) )
might give a better fit in a behavioral or hedonic model dependent on accessibility.
(in short we discount the job for its difficulty to reach before we discount it because of diminishing returns. )