[Clips] Odlyzko, et al.: Metcalfe's Law is Wrong
R.A. Hettinga
rah at shipwright.com
Sat Jul 15 04:01:52 PDT 2006
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Date: Sat, 15 Jul 2006 06:58:43 -0400
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From: "R.A. Hettinga" <rah at shipwright.com>
Subject: [Clips] Odlyzko, et al.: Metcalfe's Law is Wrong
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Andrew Odlyzko does it again...
Cheers,
RAH
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<http://spectrum.ieee.org.nyud.net:8080/print/4109>
IEEE Spectrum
Metcalfe's Law is Wrong
By: Bob Briscoe, Andrew Odlyzko, and Benjamin Tilly
ILLUSTRATION: SERGE BLOCH
Of all the popular ideas of the Internet boom, one of the most dangerously
influential was Metcalfe's Law. Simply put, it says that the value of a
communications network is proportional to the square of the number of its
users.
The law is said to be true for any type of communications network, whether
it involves telephones, computers, or users of the World Wide Web. While
the notion of "value" is inevitably somewhat vague, the idea is that a
network is more valuable the more people you can call or write to or the
more Web pages you can link to.
Metcalfe's Law attempts to quantify this increase in value. It is named for
no less a luminary than Robert M. Metcalfe, the inventor of Ethernet.
During the Internet boom, the law was an article of faith with
entrepreneurs, venture capitalists, and engineers, because it seemed to
offer a quantitative explanation for the boom's various now-quaint mantras,
like "network effects," "first-mover advantage," "Internet time," and, most
poignant of all, "build it and they will come."
By seeming to assure that the value of a network would increase
quadratically-proportionately to the square of the number of its
participants-while costs would, at most, grow linearly, Metcalfe's Law gave
an air of credibility to the mad rush for growth and the neglect of
profitability. It may seem a mundane observation today, but it was hot
stuff during the Internet bubble.
Remarkably enough, though the quaint nostrums of the dot-com era are gone,
Metcalfe's Law remains, adding a touch of scientific respectability to a
new wave of investment that is being contemplated, the Bubble 2.0, which
appears to be inspired by the success of Google. That's dangerous because,
as we will demonstrate, the law is wrong. If there is to be a new,
broadband-inspired period of telecommunications growth, it is essential
that the mistakes of the 1990s not be reprised.
The law was named in 1993 by George Gilder, publisher of the influential
Gilder Technology Report. Like Moore's Law, which states that the number of
transistors on a chip will double every 18 to 20 months, Metcalfe's Law is
a rough empirical description, not an immutable physical law. Gilder
proclaimed the law's importance in the development of what came to be
called "the New Economy."
Soon afterward, Reed E. Hundt, then the chairman of the U.S. Federal
Communications Commission, declared that Metcalfe's Law and Moore's Law
"give us the best foundation for understanding the Internet." A few years
later, Marc Andreessen, who created the first popular Web browser and went
on to cofound Netscape, attributed the rapid development of the Web-for
example, the growth in AOL's subscriber base-to Metcalfe's Law.
There was some validity to many of the Internet mantras of the bubble
years. A few very successful dot-coms did exploit the power of the Internet
to provide services that today yield great profits. But when we look beyond
that handful of spectacular successes, we see that, overall, the law's
devotees didn't fare well. For every Yahoo or Google, there were dozens,
even hundreds, of Pets.coms, EToys, and Excite at Homes, each dedicated to
increasing its user base instead of its profits, all the while increasing
expenses without revenue.
Because of the mind-set created, at least in small part, by Metcalfe's Law,
even the stocks of rock-solid companies reached absurd heights before
returning to Earth. The share price of Cisco Systems Inc., San Jose,
Calif., for example, fell 89 percent-a loss of over US $580 billion in the
paper value of its stock-between March 2000 and October 2002. And the rapid
growth of AOL, which Andreessen attributed to Metcalfe's Law, came to a
screeching halt; the company has struggled, to put it mildly, in the last
few years.
Metcalfe's Law was over a dozen years old when Gilder named it. As Metcalfe
himself remembers it, in a private correspondence with one of the authors,
"The original point of my law (a 35mm slide circa 1980, way before George
Gilder named it...) was to establish the existence of a cost-value
crossover point-critical mass-before which networks don't pay. The trick is
to get past that point, to establish critical mass." [See "To the Point," a
reproduction of Metcalfe's historic slide.]
Metcalfe was ideally situated to watch and analyze the growth of networks
and their profitability. In the 1970s, first in his Harvard Ph.D. thesis
and then at the legendary Xerox Palo Alto Research Center, Metcalfe
developed the Ethernet protocol, which has come to dominate
telecommunications networks. In the 1980s, he went on to found the highly
successful networking company 3Com Corp., in Marlborough, Mass. In 1990 he
became the publisher of the trade periodical InfoWorld and an influential
high-tech columnist. More recently, he has been a venture capitalist.
The foundation of his eponymous law is the observation that in a
communications network with n members, each can make (n-1) connections with
other participants. If all those connections are equally valuable-and this
is the big "if" as far as we are concerned-the total value of the network
is proportional to n(n-1), that is, roughly, n 2. So if, for example, a
network has 10 members, there are 90 different possible connections that
one member can make to another. If the network doubles in size, to 20, the
number of connections doesn't merely double, to 180, it grows to 380-it
roughly quadruples, in other words.
If Metcalfe's mathematics were right, how can the law be wrong? Metcalfe
was correct that the value of a network grows faster than its size in
linear terms; the question is, how much faster? If there are n members on a
network, Metcalfe said the value grows quadratically as the number of
members grows.
We propose, instead, that the value of a network of size n grows in
proportion to n log(n). Note that these laws are growth laws, which means
they cannot predict the value of a network from its size alone. But if we
already know its valuation at one particular size, we can estimate its
value at any future size, all other factors being equal.
The distinction between these laws might seem to be one that only a
mathematician could appreciate, so let us illustrate it with a simple
dollar example.
ILLUSTRATION: SERGE BLOCH
Imagine a network of 100 000 members that we know brings in $1 million. We
have to know this starting point in advance-none of the laws can help here,
as they tell us only about growth. So if the network doubles its membership
to 200 000, Metcalfe's Law says its value grows by (200 0002/100 0002)
times, quadrupling to $4 million, whereas the n log(n) law says its value
grows by 200 000 log(200 000)/100 000 log(100 000) times to only
$2.1 million. In both cases, the network's growth in value more than
doubles, still outpacing the growth in members, but the one is a much more
modest growth than the other. In our view, much of the difference between
the artificial values of the dot-com era and the genuine value created by
the Internet can be explained by the difference between the Metcalfe-fueled
optimism of n 2 and the more sober reality of n log(n).
This difference will be critical as network investors and managers plan
better for growth. In North America alone, telecommunications carriers are
expected to invest $65 billion this year in expanding their networks,
according to the analytical firm Infonetics Research Inc., in San Jose,
Calif. As we will show, our rule of thumb for estimating value also has
implications for companies in the important business of managing
interconnections between major networks.
The increasing value of a network as its size increases certainly lies
somewhere between linear and exponential growth [see diagram, "Growth
Curves"]. The value of a broadcast network is believed to grow linearly;
it's a relationship called Sarnoff's Law, named for the pioneering RCA
television executive and entrepreneur David Sarnoff. At the other extreme,
exponential-that is, 2n-growth, has been called Reed's Law, in honor of
computer networking and software pioneer David P. Reed. Reed proposed that
the value of networks that allow the formation of groups, such as AOL's
chat rooms or Yahoo's mailing lists, grows proportionally with 2n.
We admit that our n log(n) valuation of a communications network
oversimplifies the complicated question of what creates value in a network;
in particular, it doesn't quantify the factors that subtract from the value
of a growing network, such as an increase in spam e-mail. Our valuation
cannot be proved, in the sense of a deductive argument from first
principles. But if we search for a cogent description of a network's value,
then n log(n) appears to be the best choice. Not only is it supported by
several quantitative arguments, but it fits in with observed developments
in the economy. The n log(n) valuation for a network provides a
rough-and-ready description of the dynamics that led to the disappointingly
slow growth in the value of dot-com companies. On the other hand, because
this growth is faster than the linear growth of Sarnoff's Law, it helps
explain the occasional dot-com successes we have seen.
The fundamental flaw underlying both Metcalfe's and Reed's laws is in the
assignment of equal value to all connections or all groups. The underlying
problem with this assumption was pointed out a century and a half ago by
Henry David Thoreau in relation to the very first large telecommunications
network, then being built in the United States. In his famous book Walden
(1854), he wrote: "We are in great haste to construct a magnetic telegraph
from Maine to Texas; but Maine and Texas, it may be, have nothing important
to communicate."
As it turns out, Maine did have quite a bit to communicate with Texas-but
not nearly as much as with, say, Boston and New York City. In general,
connections are not all used with the same intensity. In fact, in large
networks, such as the Internet, with millions and millions of potential
connections between individuals, most are not used at all. So assigning
equal value to all of them is not justified. This is our basic objection to
Metcalfe's Law, and it's not a new one: it has been noted by many
observers, including Metcalfe himself.
There are common-sense arguments that suggest Metcalfe's and Reed's laws
are incorrect. For example, Reed's Law says that every new person on a
network doubles its value. Adding 10 people, by this reasoning, increases
its value a thousandfold (210). But that does not even remotely fit our
general expectations of network values-a network with 50 010 people can't
possibly be worth a thousand times as much as a network with 50 000 people.
At some point, adding one person would theoretically increase the network
value by an amount equal to the whole world economy, and adding a few more
people would make us all immeasurably rich. Clearly, this hasn't happened
and is not likely to happen. So Reed's Law cannot be correct, even though
its core insight-that there is value in group formation-is true. And, to be
fair, just as Metcalfe was aware of the limitations of his law, so was Reed
of his law's.
Metcalfe's Law does not lead to conclusions as obviously counterintuitive
as Reed's Law. But it does fly in the face of a great deal of the history
of telecommunications: if Metcalfe's Law were true, it would create
overwhelming incentives for all networks relying on the same technology to
merge, or at least to interconnect. These incentives would make isolated
networks hard to explain.
To see this, consider two networks, each with n members. By Metcalfe's Law,
each one's value is on the order of n 2, so the total value of both of
these separate networks is roughly 2n 2. But suppose these two networks
merge. Then we will effectively have a single network with 2n members,
which, by Metcalfe's Law, will be worth (2n)2 or 4n 2-twice as much as the
combined value of the two separate networks.
Surely it would require a singularly obtuse management, to say nothing of
stunningly inefficient financial markets, to fail to seize this obvious
opportunity to double total network value by simply combining the two. Yet
historically there have been many cases of networks that resisted
interconnection for a long time. For example, a century ago in the United
States, the Bell System and the independent phone companies often competed
in the same neighborhood, with subscribers to one being unable to call
subscribers to the other. Eventually, through a combination of financial
maneuvers and political pressure, such systems connected with one another,
but it took two decades.
Similarly, in the late 1980s and early 1990s, the commercial online
companies such as CompuServe, Prodigy, AOL, and MCIMail provided e-mail to
subscribers, but only within their own systems, and it wasn't until the
mid-1990s that full interconnection was achieved. More recently we have had
(and continue to have) controversies about interconnection of instant
messaging systems and about the free exchange of traffic between Internet
service providers. The behavior of network operators in these examples is
hard to explain if the value of a network grows as fast as Metcalfe's n 2.
There is a further argument to make about interconnecting networks. If
Metcalfe's Law were true, then two networks ought to interconnect
regardless of their relative sizes. But in the real world of business and
networks, only companies of roughly equal size are ever eager to
interconnect. In most cases, the larger network believes it is helping the
smaller one far more than it itself is being helped. Typically in such
cases, the larger network demands some additional compensation before
interconnecting. Our n log(n) assessment of value is consistent with this
real-world behavior of networking companies; Metcalfe's n 2 is not. [See
sidebar, "Making the Connection," for the mathematics behind this argument.]
We have, as well, developed several quantitative justifications for our n
log(n) rule-of-thumb valuation of a general communications network of size
n. The most intuitive one is based on yet another rule of thumb, Zipf's
Law, named for the 20th-century linguist George Kingsley Zipf.
Zipf's Law is one of those empirical rules that characterize a surprising
range of real-world phenomena remarkably well. It says that if we order
some large collection by size or popularity, the second element in the
collection will be about half the measure of the first one, the third one
will be about one-third the measure of the first one, and so on. In
general, in other words, the kth-ranked item will measure about 1/k of the
first one.
To take one example, in a typical large body of English-language text, the
most popular word, "the," usually accounts for nearly 7 percent of all word
occurrences. The second-place word, "of," makes up 3.5 percent of such
occurrences, and the third-place word, "and," accounts for 2.8 percent. In
other words, the sequence of percentages (7.0, 3.5, 2.8, and so on)
corresponds closely with the 1/k sequence (1/1, 1/2, 1/3
). Although Zipf
originally formulated his law to apply just to this phenomenon of word
frequencies, scientists find that it describes a surprisingly wide range of
statistical distributions, such as individual wealth and income,
populations of cities, and even the readership of blogs.
To understand how Zipf's Law leads to our n log(n) law, consider the
relative value of a network near and dear to you-the members of your e-mail
list. Obeying, as they usually do, Zipf's Law, the members of such networks
can be ranked in the same sort of way that Zipf ranked words-by the number
of e-mail messages that are in your in-box. Each person's e-mails will
contribute 1/k to the total "value" of your in-box, where k is the person's
rank.
The person ranked No. 1 in volume of correspondence with you thus has a
value arbitrarily set to 1/1, or 1. (This person corresponds to the word
"the" in the linguistic example.) The person ranked No. 2 will be assumed
to contribute half as much, or 1/2. And the person ranked kth will, by
Zipf's Law, add about 1/k to the total value you assign to this network of
correspondents.
That total value to you will be the sum of the decreasing 1/k values of all
the other members of the network. So if your network has n members, this
value will be proportional to 1 + 1/2 + 1/3 +
+ 1/(n-1), which approaches
log(n). More precisely, it will almost equal the sum of log(n) plus a
constant value. Of course, there are n-1 other members who derive similar
value from the network, so the value to all n of you increases as n log(n).
Zipf's Law can also describe in quantitative terms a currently popular
thesis called The Long Tail. Consider the items in a collection, such as
the books for sale at Amazon, ranked by popularity. A popularity graph
would slope downward, with the few dozen most popular books in the upper
left-hand corner. The graph would trail off to the lower right, and the
long tail would list the hundreds of thousands of books that sell only one
or two copies each year. The long tail of the English language-the original
application of Zipf's Law-would be the several hundred thousand words that
you hardly ever encounter, such as "floriferous" or "refulgent."
Taking popularity as a rough measure of value (at least to booksellers like
Amazon), then the value of each individual item is given by Zipf's Law.
That is, if we have a million items, then the most popular 100 will
contribute a third of the total value, the next 10 000 another third, and
the remaining 989 900 the final third. The value of the collection of n
items is proportional to log(n).
Incidentally, this mathematics indicates why online stores are the only
place to shop if your tastes in books, music, and movies are esoteric.
Let's say an online music store like Rhapsody or iTunes carries 735 000
titles, while a traditional brick-and-mortar store will carry 10 000 to 20
000. The law of long tails says that two-thirds of the online store's
revenue will come from just the titles that its physical rival carries. In
other words, a very respectable chunk of revenue-a third-will come from the
720 000 or so titles that hardly anyone ever buys. And, unlike the cost to
a brick-and-mortar store, the cost to an online store of holding all that
inventory is minimal. So it makes good sense for them to stock all those
incredibly slow-selling titles.
At a time when telecommunications is the key infrastructure for the global
economy, providers need to make fundamental decisions about whether they
will be pure providers of connectivity or make their money by selling or
reselling content, such as television and movies. It is essential that they
value their enterprises correctly-neither overvaluing the business of
providing content nor overvaluing, as Metcalfe's Law does, the business of
providing connectivity. Their futures are filled with risks and
opportunities. We believe if they value the growth in their networks as n
log(n), they will be better equipped to navigate the choppy waters that lie
ahead.
About the Authors
BOB BRISCOE is chief researcher at Networks Research Centre, BT (formerly
British Telecom), in Ipswich, England. ANDREW ODLYZKO is a professor of
mathematics and the director of the Digital Technology Center at the
University of Minnesota, in Minneapolis. BENJAMIN TILLY is a senior
programmer at Rent.com, a dot-com company that actually made money, in
Santa Monica, Calif.
To Probe Further
David P. Reed argues for his law in "The Sneaky Exponential" on his Web
site at http://www.reed.com/Papers/GFN/reedslaw.html.
Several additional quantitative arguments are made for the n log(n) value
for Metcalfe's Law on the authors' Web sites at
http://www.cs.ucl.ac.uk/staff/B.Briscoe and http://www.dtc.umn.edu/~odlyzko.
Chris Anderson's article "The Long Tail" was featured in the October 2004
issue of Wired. Anderson now has an entire Web site devoted to the topic at
http://www.thelongtail.com.
George Gilder dubbed Metcalfe's observation a law in his "Metcalfe's Law
and Legacy," an article that was published in the 13 September 1993 issue
of Forbes ASAP.
An article in the December 2003 issue of IEEE Spectrum, "5 Commandments,"
which can be found at http://www.spectrum.ieee.org/dec03/5com, discusses
Moore's and Metcalfe's laws, as well as three others: Rock's Law ("the cost
of semiconductor tools doubles every four years"); Machrone's Law ("the PC
you want to buy will always be $5000"); and Wirth's Law ("software is
slowing faster than hardware is accelerating").
Figure 1
GROWTH CURVES: How much more valuable does a network become as it grows?
That depends on what kind of network it is. Let's say you have n users. The
value of a television or some other broadcast network grows linearly, as
shown by Sarnoff's Law [green] here. At the other extreme, according to
Reed's Law [purple], growth in value of certain networks-those that can
form groups, such as e-mail lists-is exponential (2n). In between are
ordinary, member-based networks such as AOL, or all of AT&T's phone
customers. Metcalfe's Law [red] says they grow in value quadratically (n
2). We argue, instead, that such growth increases logarithmically as
n log(n) [blue], a much slower rate.
ILLUSTRATION: SERGE BLOCH
Figure 2
TO THE POINT: Robert Metcalfe's original circa-1980 slide, reproduced here,
argued that because the value of a network increased quadratically, it
would quickly surpass its costs, which grew linearly.
ILLUSTRATION: SERGE BLOCH
Sidebar 1
Making the Connection
By: B.B., A.O. & B.T.
A network grows not only by the addition of single members. It can jump in
size by interconnecting or merging with another network. If two networks
are of similar size, each would see roughly the same increase in value if
they combined. You would expect them to interconnect, and indeed, networks
of comparable size do so routinely. But when one network is much bigger
than the other, the larger one usually resists interconnecting.
If Metcalfe's Law were true, no matter what the relative sizes of two
networks, both would gain the same amount by uniting, making the observed
behavior seem irrational [see table below, "The Value of Interconnecting"].
If our n log(n) law holds (or other laws with growth rates falling between
ours and Metcalfe's), then, as the table shows, the smaller network would
gain more than the larger one. For example, if the larger network is eight
times as large as the smaller one, its gain would be less than half that of
the smaller one. This clearly reduces the incentive for the owners of the
larger network to interconnect without compensation.
This model of network interconnection is simplistic, of course, and it does
not deal with other important aspects that enter into actual negotiations,
such as a network's geographical span, its balance of outgoing and incoming
traffic, and any number of additional factors. All we are trying to show is
that there may be sound economic reasons, besides raw market power, for
larger networks to demand payment for interconnection with smaller ones. In
fact, that's a common phenomenon in real life.
Our n log(n) law describes best the increase in value of a single network
as it grows through acquisition of individual members. But our law should
not be applied directly to evaluate the effects of connecting separate
networks. Another important consideration is the degree to which groups
that value each other highly are already contained within the networks
being combined, a factor called clustering.
When clustering is weak, the people you tend to communicate with the
most-family members, work colleagues, fellow hobbyists, and so on-are not
on the same network you are. In these cases, the value of connecting
separate networks can be higher than our n log(n) law predicts.
Nonetheless, given that most networks grow organically, with people drawing
in the people closest to them, the majority of networks are strongly
clustered. Therefore, in most networks, n log(n) appears to be the best
simple description of network value in terms of the network's size.
--
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R. A. Hettinga <mailto: rah at ibuc.com>
The Internet Bearer Underwriting Corporation <http://www.ibuc.com/>
44 Farquhar Street, Boston, MA 02131 USA
"... however it may deserve respect for its usefulness and antiquity,
[predicting the end of the world] has not been found agreeable to
experience." -- Edward Gibbon, 'Decline and Fall of the Roman Empire'
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R. A. Hettinga <mailto: rah at ibuc.com>
The Internet Bearer Underwriting Corporation <http://www.ibuc.com/>
44 Farquhar Street, Boston, MA 02131 USA
"... however it may deserve respect for its usefulness and antiquity,
[predicting the end of the world] has not been found agreeable to
experience." -- Edward Gibbon, 'Decline and Fall of the Roman Empire'
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