A Refutation of Metcalfe's Law (PDF) and a better estimate for the value of networks and network interconnections by Andrew Odlyzko and Benjamin Tilly.
There have been and continue to be controversies about interconnection policies of ISPs. A particularly sensitive issue is the frequent refusal of large ISPs to peer (roughly speaking, exchange traffic freely without payment) with smaller carriers. (The refusal of AOL to interconnect instant messenger systems is very similar.) This behavior has often been attributed to abusive exploitation of market power. But there may be a more innocent explanation, based on the economic value that interconnection generates. As we show in Section 2, if Metcalfe's Law held, then interconnection would produce equal value for any two network, irrespective of their relative sizes. Hence refusal to interconnect without payment would have to be due to either obtuseness on the part of management or strategic gaming. However, if network value scales like n log(n), as we argue (or by most other rules of this type, the quadratic growth of Metcalfe's Law is very unusual in this regard) then relative gains from interconnection depend on the sizes of the networks. In this case the smaller network gains considerably more than the larger one. This produces an incentive for larger networks to refuse to interconnect without payment, a very common phenomenon in the real economy.
- Metcalfe's Law and Reed's Law both significantly overstate the value of a communication network. In their place we propose another rough rule, that the value of a network of size n grows like n log(n). This rule, while not meant to be exact, does appear to be consistent with historical behavior of networks with regard to interconnection, and it captures the advantage that general connectivity offers over broadcast networks that deliver content. It also helps explain the failure of the dot-com and telecom ventures, since it implies network effects are not as strong as had been hoped for.