# What If Metcalfe’s Law Is Wrong?

Networks — be they telecom, social, transportation or otherwise — are the fabric of modern society. They provide immense value to consumers and businesses alike, enhancing mutual relationships and enabling the distribution of goods, services and information. But does this value grow as the size of the networks grow? And if so, how much?

Metcalfe’s Law” has long been accepted as characterizing the value — and value growth — of fully connected networks. It states that the value of a network is proportional to the square of the number of its nodes, which may take the form of devices — such as computers — or users, in which case a network connection is represented by a “friend” or “follower.” But there are times when the “law,” which has been used to explain network effects and justify mergers and acquisitions, appears to overstate a network’s value. And if that’s the case, what can service providers do about it?

While the number of possible connections in a network is indeed proportional to the square of the number of nodes, value is not necessarily equivalent to number. After all, I may have 10 bills in my wallet, but it matters a lot whether they are \$1 or \$10,000 denominations.

As I’ve previously observed (at Telecosm and via some math (PDF)), there are several reasons that Metcalfe’s Law can overestimate the value of a network. First, typically only a fraction of the possible connections have value. Second, there are natural limits to consumption of that value. And third, the value of the entire network may decline over time.

Convergent Value Distributions

The number of links in a fully connected network is certainly proportional to the square of the number of nodes. If each connection had the same value as any other, then Metcalfe’s Law would be correct. What would that mean in practice? It would mean that you would spend equal time on the phone with each of the nearly 7 billion people in the world, that they would all friend you or follow you, and you would reciprocate. But humans don’t behave that way.

In 1992, anthropologist Robin Dunbar posited that primates have neurobiologically-based limits to the size of their social networks. For humans, “Dunbar’s Number” is 150. This is exemplified by the fact that the most popular social networking site on the planet now has more than 400 million users, yet the average number of “friends” a person has is only 130 and only a small percentage of those “friends” actually communicate with one another. And although there are a variety of ways to slice the data, the largest microblogging site has close to 100 million users but the average number of followers is roughly 126. Even if we were to assume that tweets have the same “value” as intimate face-to-face interactions and that electronic media might expand Dunbar’s number in some way, there is still an upper bound to the number of relationships, or even weak ties, that can be maintained. If the total value of such social media is related to engagement, and engagement is related to the number of friends, such value would in these cases be linearly proportional to the size of the network, rather than the square of its size.

Intrinsic Limits of Consumption

Suppose you did have equal social interest in the nearly 7 billion people on the planet, or the more than 100 million shared video clips or even the more than 100,000 touchscreen phone apps out there. You then would run into intrinsic limits to your ability to benefit from all those relationships or consume all that content. Perhaps in the early days of TV it would have been possible for a single individual to consume all the content produced. Currently, however, nearly a day’s worth of content is uploaded to YouTube every minute. Assuming that all those clips did have equal value, even a multitasking insomniac couldn’t keep up. All networks have intrinsic upper limits of consumption, be they bandwidth or dollars or time or attention span.

Holistic Network Value Declines

Even if all nodes were of equal value, and there were no limits to consumption, well, people get jaded. Emotional rewards from novel stimuli are processed by dopamine receptors in the striatum, but the brain is designed to habituate, that is, not get so excited by repeated stimuli. What this means is that an entire social or content network may “grab” you at first, or even for a couple of years, but this infatuation may eventually wear off, and you’ll depart in search of the next new thing. Technological progress can also cause the value of the entire network to decay — consider what the web and email have done to the value of fax networks.

Strategies

There are ways to manage these three effects, however.

If the network node values follow a convergent distribution, ensure that whatever value is present can be extracted by reducing or eliminating core bottlenecks and enhancing the process of discovery. Specific approaches such as scalable non-blocking network infrastructure, content delivery networks, tagging, recommendation and search engines can help.

To extract maximum value when there are intrinsic limits of consumption, not only is removing access bottlenecks effective, but so are personalization, richness, context sensitivity and multitasking facilitation.

And to keep a given network exciting and the dopamine system revved up, new features, content or applications can help. Even if the core “network”— whether social site or app store — remains the same, using a platform for new content or apps can continue to trigger the pleasure receptors associated with novelty, maximizing value and engagement.

Overall, the behavior of real-world networks isn’t always as simple as what’s represented by Metcalfe’s Law. However, understanding their underlying characteristics can help users and service providers maximize their value as well as create new business opportunities.

### 16 Responses to “What If Metcalfe’s Law Is Wrong?”

1. I think it all gets back to signal vs. noise.

These issues hint that ideas, like Neal Stephenson’s notions of future societal and governmental bodies dedicated to this issue, as portrayed in Anathem, as extremely insightful, if not outright predictive.

Also, if you haven’t seen it already, Gary Flake’s TED on Pivot (a MS Live Labs project) is right on point.

2. The value of the entire network may change based on the activity within it. Metcalfe’s Law was defined when the main goal was being able to access other nodes, for general communications, either an email or a transaction.

With social networks that equation is driven to being able to include anybody within the entire network into your sub-group. I’m not familiar enough with Reed’s Law to determine if that is the correct formula for this situation. However, increasing value within small groups add another layer of value to the overall network. Being able to retrieve information from a person within this small network may not require a “relationship” that puts them within the Dunbar constraints.

The math to correlate the entire network value, in addition to the sum of the value of the smaller networks, is beyond me. Of course all of this varies based upon your definition of value.

3. value growth â€” of fully connected networks.a network connection is represented by a â€œfriendâ€ or â€œfollower.â€there are several reasons that Metcalfeâ€™s Law can overestimate the value of a network. First, typically only a fraction of the possible connections have value. Second, there are natural limits to consumption of that value. And third, the value of the entire network may decline over time.

4. Eideard: the fact that some stimuli may not cause tolerance or habituation does not negate my point, which is that habituation, the functioning of the anterior cingulate cortex, the dopamine system, “prediction errors,” technology substitution, need for uniqueness, fashion, Technology Adoption Lifecycles, and/or group dynamics cause shifts in usage and value of everything from Nehru jackets and tie-dyed clothing to large scale networks. We can all think of various bulletin boards, online providers, and social networks that have peaked in popularity.

Ronald, I agree that brain function is more complex than my one sentence described, and that we are just beginning to plumb the basics thanks to capabilities such as functional Magnetic Resonance Imaging. I’m not sure that I agree with the rest of your reply: web crawling and query processing for search against index shards are both done on a parallel basis, as are many other computational tasks.

Dan, it may be a metaphor to you, but many other folks treat it as more. In any event, it’s easy to formalize by defining a value function that maps pairs of integers (node indices in a fully meshed network) to the reals, and then ask which value distributions cause which aggregate network value growth effects. U. Minn professor Andrew Odlyzko and colleagues critiqued the law in http://spectrum.ieee.org/computing/networks/metcalfes-law-is-wrong , and came to the conclusion that aggregate value is of order n * log(n) rather than n * n. However, as I argued in http://www.joeweinman.com/Resources/WeinmanMetcalfe.pdf , that conclusion can only be reached if the value distribution is based on a Zipf or Harmonic series…if the series is finite or convergent, as I point out above occurs in many real world networks, then one can, surprisingly, end up with order(n) aggregate value, rather than order(n) squared.

Greg, you are correct. The key point however is the number of active communicators, where the low average is suggestive that very few people pass the Dunbar (hypothesized) “limit.”

Martinking and RS500guy, I agree that human networks are different than machine networks. However, abstractly, both are networks, and unless we define “value” trivially as the existence of a connection, there will be limits. If the machine world is viewed as a group of cooperating processes which may make remote calls to other processes, the same effects hold. Alfred-Laszlo Barabasi and his students have conducted empirical Internet studies (e.g., http://arxiv.org/PS_cache/cond-mat/pdf/9907/9907038v2.pdf) showing the unequal value distributions that martinking commented on.

Finally, Nathan and Dennis, I resonate very much with your comments. A complete model of network value would include not just positive pairwise value, but negative value, asymmetric value (spammers), and also include how infrastructure costs scale to provide the foundation for those value interactions. And to Ronald’s point, value is not just a single value, but a time-varying one.

In any event, Metcalfe’s Law may be correct for some networks some of the time, Odlyzko et al may be as well, and even Reed’s Law (subgroup formation) may have a nugget of insight. The point of the article was that there may be reasons that these formulations don’t apply the way that at least some people seem to assume, and understanding the reasons why they may not apply can be used to restore some lost value and create new business opportunities. In a world of nearly unbounded information and digitally-mediated relationships, additional value may not be created as much by yet more information or more relationships as by the tools filter and prioritize this bounty to better match social and cognitive limits.

• Ever tried to teach a machine “all” like a toddler learns it?
Or to explain in a model one-two-many cultures?
A toddler can learn all long before they can count, “many” in one-two-many cultures is also possible without counting past two. Those are real parallel task, there is no foreach,while or whatever loop. Otherwise the building of generalizations would be impossible, and people could count past 2.
We are not really paralleling problems we are partializing serial ones and then run partials on different serial CPUs and coordinate/synchronize everything with locks. If you compare that to any brain, that’s really really primitive.

5. Doesn’t the same network effect also help you filter the information you do have the time / attention for?

6. I did some economic analysis of the value and costs of participating in a network a while back – see http://dbmoore.blogspot.com/2009/07/moores-corrolary-to-metcalfes-law.html for the details. In essence, while the total value of a network increases with additional participation, the total cost increases as well. For any individual in the network, the cost/benefit ratio determines whether it is worthwhile for that individual to remain in the network – when the cost is greater than the value, the individual should (and generally will) leave the network. It is possible to do things to restrict the costs (laws, rules, moderators, subscription lists, alerts, blocking, etc.) to keep the cost/benefit ratio as low as possible. Have a look at http://dbmoore.blogspot.com/2009/07/moores-corrolary-to-metcalfes-law.html for more on this – would appreciate comments on it … thanks!

7. I feel that you are confusing system network with personal network.

The usefulness of the system increases with the number if nodes/connections.

Personal (human) networks are constrained (Dunbar(.

The system network provides the platform and opportunity for personal networks.

I agree with you regarding value – connecting to a single “supernode” like http:// twitter.com/gigaOmcould could have the more value than connecting to thousands of nodes.

8. Metcalf wanted to sell 3Com network equipment, lots of it. Which is fine, and using his law to substantiate what he felt was the value of growing network nodes was his pitch. Great. In a computer network the nodes are individually equal in their value to the whole network. So he could expect to sell many, many 3Com products. This scheme does not translate to social networks.

• Excellent point. I think you nailed it.

To capture social network activity with some semblance of accuracy, these models need to adjust for the declining marginal utility associated with 150+n connected nodes.

Further, there are probably massive variances in the utility (I make no attempt to define utility here!) derived from the connections to the “top” 150 nodes that likely conform to the Pareto Principle, making averages close to useless.

To RS500’s point, in all likelihood these computer hardware-derived network “laws” are ex post facto observations designed to justify valuations by VCs or Wall Street sell-side research or otherwise serve the interests of the manufacturer in question.

9. I agree with dan, but am rather worried that the implication of the article is that there are people make investment decisions based on the simplistic maths – that would be laughable (though I’m not saying it isn’t true – sadly I have seen dumber ways of doing things).

But, did want to pick up that the average number of friends on a social network feels like a poor proxy for Dunbar’s which is much more like a limit than an average. A much better proxy would be the break point/knee in a graph of number of contacts. So, whilst I recognise that the point is that it is the same order of magnitude, not exactly the same, that point is very obvious from the raw mechnics of what such a square law would mean … a point you made perfectly well elsewhere.

10. Fast growth is bound to run into some real world limitation sooner or later (if each atom in the universe were a node, etc…).

Another possible issue as that the growth of the network may itself facilitate the growth of other, possibly unexpected, emergent processes that negate the growing primary value of the network. For example, with more people on a communication network comes more spam, more low-value content masking the good stuff, more of the various manifestations of Information Overload… and they too may grow as N squared, or possibly worse.

11. With all due respect to everyone concerned, Metcalfe’s Law is a metaphor, not a law. It’s not specific enough to be provably right or wrong. What is value? What is the metaphor saying? It’s saying that there’s a network effect, and a network effect is exactly what? Maybe, it’s equivalent to a phase transition, as in magnetism or percolation. There are no phase transitions in one dimension. But once you move to two dimensions, and you have a network of interacting nodes, you can have phase transitions. A sparse network may not experience the phase transition, but increase the density, and bang! the behavior of the network changes dramatically. That’s the nonlinearity.

12. Let me recap:

All complex systems are finite.

“Emotional rewards from novel stimuli are processed by dopamine receptors in the striatum”

Well it’s a little more complex than that. In the last decade it became pretty clear that we are not talking about 10b problem but a 100t one[1], give or take a few.

If we take a look at what is “news”/interesting stuff to the brain in a model kind of way. Then it’s best to model it on a pane as signals in time and over time. Over time is basically our current computing model (seq) or connections between active neurons over time, in time is the parallel signals which are not directly connected. Visual,touch … Add building and learning abstractions/generalizations to the mix and the importance of the parallel not directly connected model becomes clear. But that’s where it is the easiest to detect change or news.

Search is based on the seq. model, that’s why they jump through hoops in ranking calculations, 2-3 changes a day at Google. Question is if we can build a parallel model to calculate the information value for any given user, any web page at any given time. Metcalfeâ€™s Law is purely based on the seq. model. I would not apply it to anything where humans are involved.

13. While I mostly agree, your narco/dopamine analogy fails. In truth, a number of assorted stimuli – from sex to certain drug combinations – don’t necessarily run into a tolerance wall.

Sorry to digress.

Of course, we could lurch in the direction of redirective processes and build networks premised on speeding up the loss of Stone Age genes which may be the limiting factors in your consideration of social networks.

Har! See what happens when I comment right after watching Caprica!