Tim Oren critiques Ross Mayfield’s review of the various valuation models for social network topologies: Sarnoff, Metcalfe, and Reed’s laws. Oren notes that these models all predict untrammeled growth, but that is not what we see. He explains this contradiction by noting that these oversimplified models don’t take into account commons failure (think spam), scaling issues (moderation and representation systems), and competition among various networks.
There is a simpler explanation. When economists talk about value (or price), they are generally talking about marginal value. In the case of a network, this marginal value would be the highest amount the next person would pay to join the network or the lowest amount you could pay someone to leave it. Assuming that the demand curve for these networks has a typical reverse S-shape to it, even a network whose value grows exponentially with an increase in membership will run into demand that decreases exponentially as prior parts of the curve are saturated.
Even this demand curve model is actually optimistic about customer adoption. It assumes a smooth distribution of demand. In practice, market tastes are frequently clustered. A network that connects physicists together is of little interest to Britney spears fans and vice-versa. Patrick Ball’s paper on “circuits” of reported killings in Kosovo shows that not all social groups have overlapping membership, meaning that increased participation by one group is of no value to members of other groups.
Lastly I would note that all of these models fail to take into account time. If the cost of joining is going down sufficiently rapidly (think Moore’s Law), I may be willing to trade the present value of the network for the discount in the future. If a large number of potential members choose this route, this decision becomes self-justifying.