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Network value

Apr 08, 2019

Despite widespread endorsement of competition law theories, the western world has become very monopolistic when it comes to anything that is connected to the Internet. We still have two global communications networks – the Internet and the Public Switched Telephone Network, but the latter has become subservient to the former and is in the process of sun-setting. Regarding social networks, there is only one large recreational network (with 2.3 billion users), one large business-oriented one (with 610 million users), and a plethora of small specialized ones. There is basically one huge online retailer ($230 billion annual revenue), and one online b2c and c2c shopping platform (1.2 billion listings, 95 billion volume). There is one Open Source smartphone operating system, and one proprietary one. And don’t believe anyone who tells you that there is more than one search engine (according to Google Trends, the verb “to google” appears seven times more frequently in Google searches than “to search”!).

In this blog post I’d like to explain how this came about.

A short history of the PSTN

Everyone knows that Alexander Graham Bell invented the telephone; but how many know that Theodore Newton Vail invented the telephone network? Bell’s business model was to sell pairs of phones to customers, and leave them to wire and power them up. Vail, the first president of Bell Telephone Company and former General Superintendent of the US Railway Mail Service, organized telephony as a service. Indeed, the term Vailism is used to mean the philosophy that public services should be run as closed centralized monopolies for the public good.

Vail’s contributions to the Bell System, to the PSTN, and to communications in general, are prodigious. Vail defended Bell’s patent for the telephone from challenges from Edison and Elisha Gray. He funded the development of metallurgy techniques to produce copper wire for telephone lines. He introduced Bell’s innovation of twisted pairs to combat cross-talk. He orchestrated the formation of AT&T as a subsidiary of Bell Telephone tasked with building and operating a long distance network. Despite coining the slogan “One Policy, One System, Universal Service” he oversaw the Kingsbury Commitment that led to connection openness. And he pioneered the procedure of funneling a percentage of operating profits into R&D, thus spawning Bell Labs, which is responsible for so many of the scientific and technological breakthroughs upon which the modern world is based.

Over and beyond all of these, Vail seems to have been the first to grasp the idea that the value of a network far exceeds the sum of its parts.

How much is your network worth?

Networks have intrinsic value. I bundle into the term network such diverse things as the Internet, corporate networks, personal networks of friends and colleagues, web-based social networks, even neural networks used in machine learning. We take for granted that we are all connected to single telephony and email networks – but that was not always the case. If instead you had to travel to physically meet someone, how much would that cost you? If you need to speak with someone and have a close friend who can introduce you, how much is that introduction worth? If someone were to steal your list of contacts with their phone numbers, email addresses, and other communications addresses, how much would you pay to get them back?

The first to attempt quantification of a network’s value was Robert Metcalfe, the inventor of Ethernet. Metcalf reasoned that a member of a network with N members (computers, telephones, friends, participants, etc.) might wish to contact any of the other N-1 members. So, the value to each member is N (ignoring the “-1” in “N-1” as compared to “N” for large N). This makes sense; the ability to contact twice as many members is worth twice as much! The value of the entire network is the sum of the value of the network to all N members, which is N*N = N2.

We have thus shown that

Metcalfe’s Law   V = N2

Basically Metcalfe’s law says that the value of a network is equal to the number of pairings of its members – N*(N-1) ≈ N2. The implicit assumption being that as N grows all the new possible pairings remain valuable. In real life, as a network grows, its geographic extent and variability increases, reducing the value to each new member. So, we need to appreciate Metcalfe’s law as a rule of thumb, not as a physical necessity. 

Metcalfe’s rule is not the most optimistic valuation for a network of N participants. David Reed (the designer of UDP) realized that Metcalfe’s law limits communications to one-on-one calls. What about conference calls with three participants? Four participants? All N network members? Reed’s law states that the value of a network of N members equals the number of all possible subsets (the size of the power set), leading to

Reed’s Law  V = 2N

There are also “laws” with N dependence that are weaker than Metcalfe’s. Many years ago I joined LinkedIn in order to keep track of people I know in the high-tech world. High-tech people have a tendency to change jobs (and hence email addresses) frequently, and my box of business cards was hopelessly out of date. When I signed up for LinkedIn I carefully monitored how my network (defined as the number of friends, friends of friends, and friends of friends of friends) grew, how useful it was (e.g., did I even recognize the names of friends of friends?) and how much time and effort I needed to expend to maintain that network (e.g., how many requests for introductions did I receive?). I was particularly interested in the question of when a network became too large (in my case, once the network reaches about 200 contacts, the added value of a typical new contact was less than the added maintenance cost). I came to the conclusion that

Stein’s Law V = N4/3 

This result means that networks are less valuable than Metcalfe’s law implies (and much much less valuable than Reed’s estimation).

One network to rule them all

All of these values are superlinear, that is, they all predict that when a network doubles in size its value increases to more than twice the previous value. In Metcalfe’s case, a network of twice the size is worth four times as much. Superlinear value has strong implications on network size. For example, contrast the case of two networks with N members each, with a single network with all 2N members. In the first case Metcalfe’s value is N2 + N2 = 2N2, while in the latter (2N)2 = 4 N2, i.e., twice as much!

This means that if we do have two approximately equal-sized networks, there will be strong pressure for them to merge, allowing members of one network to contact those of the other. If a merger doesn’t take place, there will be pressure for members to leave one network (usually the slightly smaller one) and join the other (the slightly larger one). Over time the slightly larger network will become larger and larger, until the smaller network disappears.

As similar force applies to networked services. For example, the value of an online store can be directly quantified as the savings as compared to purchasing the same items at a local store, minus the constant network connection cost, shipping costs, and importantly the cost of time spent searching for the needed goods. The broader the range of goods offered by the same online store, the lower the search cost. Vailism in its purest form.

The argument for weakly superlinear dependency

On the other hand, for most social networks the value contributed by friends, i.e. first degree connections, scales linearly in N, since these contacts are chosen by the user, and are thus valuable to at least some degree. However, the value of second degree connections is already much less, as your friend’s brother, who is a hair-stylist and lives half-way around the world, is of little interest to you.

In 2005 Andy Odlyzko and Ben Tilly suspected that Metcalfe’s law was too powerful, i.e., implied too much pressure for the emergence of a single network. This feeling was in part motivated by the fact that there were still two large global networks – the Public Switched Telephone Network and the Internet, and various other networks (frame relay, ATM, etc.). These authors reasoned that a network’s value’s dependence on N must be superlinear, but not too strongly so. They proposed a weaker

Odlyzko’s law V = N log N

A dependence similar to the computational complexity of the FFT. The simplest argument for N log N scaling is to assume that randomly chosen connections can be described by Zipf ’s law. One way of explaining Zipf’s law is that for many naturally occurring sets of elements, if we sort the set by decreasing value, the 2nd element will be approximately half as valuable as the first, the 3rd element approximately one third as valuable, and the value of the kth element will only be about 1/k of the first. Since the harmonic sum diverges logarithmically, the value of the network to each participant scales as log N, and the total value as V = N log N.

Exactly as the validity of Metcalfe’s law rests on the equal importance of all possible network members, the validity of this argument rests on strong decline in their importance, i.e., potent localism. This localism need not be geographic – important contacts may be geographically remote but close in interests, job history, or relatives - but Odlyzko’s law banks on each network member only desiring to connect with a small number of other members.

Weak dependency no longer makes sense

Today, the concerns of Odlyzko and Tilly seem misplaced. In a world dominated by a single Internet, world-wide web, Google, Facebook, Amazon, WhatsApp, and eBay it is difficult to believe that the size dependence of any networked service is not strongly superlinear. The same logic may apply to other dominating services or technologies that we don’t usually think of as being networked. A prime example is the English language that so dominates the Internet.

One may object that all of the above is correct only for the west, and that there are rivals to all of these dominating players in China (Baidu instead of Google, Weibo instead of Facebook, Alibaba instead of Amazon, WeChat instead of WhatsApp, ... and of course Chinese instead of English). Although these flourish on the same global Internet as their western counterparts, they don’t really contradict all that we have said above. The Chinese Internet is basically a world of its own, due in equal parts to the Great Firewall of China and language barriers. Such barriers significantly slow down the emergence of a single network or networked service, but do not eliminate the basic pressure that leads to unification.

Because, as Vail realized over a century ago, the value of a single network far exceeds the sum of its parts.


 

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