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\title{A refutation of Metcalfe's Law\\
and a better estimate for the value\\
of networks and network interconnections}

%A refutation of Metcalfe's Law\\
%and an alternate valuation of networks}

% \title{The value of networks\\
% and a refutation of Metcalfe's Law}

%  A replacementfalsity of Metcalfe's Law and a proposed replacement}


\titlerunning{Metcalfe's Law}

\author{Andrew Odlyzko\inst{1}
\and Benjamin Tilly\inst{2}}

\authorrunning{Andrew Odlyzko and Ben Tilly}

\institute{Digital Technology Center, University of Minnesota\\
499 Walter Library, 117 Pleasant St. SE\\
Minneapolis, MN 55455, USA\\
\email{odlyzko@umn.edu}\\
\texttt{http://www.dtc.umn.edu/$\sim$odlyzko}\\
\and
\email{ben\_tilly@operamail.com}\\
\texttt{Preliminary version, March 2, 2005}
}


\maketitle

\begin{abstract}

Metcalfe's Law states that the value of a communications
network is proportional to the square of the size of the network.
It is widely accepted and frequently cited.  However, there
are several arguments that this rule is a significant overestimate.
(Therefore Reed's Law is even more of an overestimate,
since it says that the value of a network grows exponentially, in the
mathematical sense, in network size.)
This note presents several quantitative
arguments that suggest the value of a general
communication network of size $n$
grows like $n \log (n)$.  This growth rate is faster than the linear
growth, of order $n$, that, according to Sarnoff's Law,
governs the value of a broadcast network.  On the other hand,
it is much slower
than the quadratic growth of Metcalfe's Law, and helps
explain the failure of the dot-com and telecom booms,
as well as why network interconnection (such as peering
on the Internet) remains a controversial issue.




\end{abstract}



\section{Introduction}

The value of a broadcast network is almost universally
agreed to be proportional to the number of users.  This
rule is called Sarnoff's Law.  For general communication
networks, in which users can freely interact with each
other, it has become widely accepted that Metcalfe's Law
applies, and value is proportional to the square of the
number of users.  This ``law'' (which is just a general
rule-of-thumb, not a physical law) has attained a sufficiently
exalted status that in a recent article it was classed
with Moore's Law as one of the five basic ``rule-of-thumb
'laws' [that] have stood out'' and passed the test
of time \cite{Ross}.  Even as far back as 1996, 
Reed Hundt, the then-chairman of the Federal
Communications Commission, claimed that Metcalfe's Law
and Moore's Law ``give us the best foundation for understanding the
Internet'' \cite{Kaprowski}.  Metcalfe's Law was frequently
cited during the dot-com and telecom booms to justify
business plans that had the infamous ``hockey-stick'' revenue
and profit projections.  The hopes and promises were that once
a service or network attained sufficient size, the
non-linear growth of Metcalfe's Law would kick in, and
network and bandwagon effects would start to operate
to bring great riches to the venture's backers.  This led
to the argument that
there was little need to worry about current disappointing
financial results, and all efforts should be devoted to
growth.

The fundamental insight, that the value of a general
communication network grows faster than linearly, appears
sound.  Although it had not been enunciated explicitly
until a few decades ago, it had motivated decision makers
in the past.  For example, it affected the development
of the phone system a century ago, as shown by the
following quote \cite{NixG}:
\begin{quote}
The president
of AT\&T, Frederick Fish, believed that customers valued access
and that charging a low fee for network membership would maximize
the number of subscribers.  According to Fish, the number of users
was an important determinant of the value of telephony to individual
subscribers.  His desire to maximize network connections led the
firm to adopt a pricing structure in which prices to residential
customers were actually set below the marginal cost of service in
order to encourage subscriptions.  These losses were made up through
increased charges to business customers.
\end{quote}

Serious quantitative modeling of network effects dates back
to the 1974 paper of Jeffrey Rohlfs \cite{Rohlfs1} (see also
\cite{Rohlfs2}, and some of the references in \cite{Rohlfs1} to
earlier work in the area), which was motivated by the lack of success
of the AT\&T PicturePhone{\tm} 
videotelephony service.  (For a survey of economics literature
in this area, see \cite{Economides}.)
Metcalfe's Law itself dates back to a slide that Bob Metcalfe
created around 1980, when he was running 3Com, to sell
the Ethernet standard.  It was dubbed ``Metcalfe's Law''
by George Gilder \cite{Gilder} in the 1990s, and 
citations to it started to proliferate \cite{Kaprowski,Metcalfe1,Metcalfe2}.

The foundation of Metcalfe's Law is the observation that
in a general communication network with $n$ members, 
there are $n (n-1)/2$
connections that can be made between pairs of participants.
If all those connections are equally valuable (and this is
the big ``if'' that we will discuss in more detail below), the total
value of the network is proportional to $n (n-1)/2$, which, since we
are dealing with rough approximations, grows like $n^2$.

Metcalfe's Law is intuitively appealing, since our personal
estimate of the size of a network is based on the
uptake of that network among friends and family.  Our
derived value also varies directly with that metric.
We therefore see a linear relationship between the
perceived size and value of that network.

Reed's Law \cite{Reed} is based on the further insight
that in a communication network as flexible as the Internet,
in addition to linking pairs of members, one can form groups.
With $n$ participants, there are $2^n$ possible groups, and
if they are all equally valuable, the value of the network
grows like $2^n$.

The fundamental fallacy underlying Metcalfe's and Reed's laws
is in the assumption that all connections or all groups are equally valuable.
The defect in this assumption was pointed out a century and
a half ago by Henry David Thoreau.  In {\em Walden,} he
wrote \cite{Thoreau}:
\begin{quote}
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.
\end{quote}
Now Thoreau was wrong.  Maine and Texas did and do have a lot
to communicate.  Some was (and is) important, and some not,
but all of sufficient value for people to pay for.  Still,
Thoreau's insight is valid, and Maine does not have as much
to communicate with Texas as it does with Massachusetts
or New York, say.

In general, connections are not used with the same intensity
(and most are not used at all in large networks, such as
the Internet), so assigning equal value to them is not
justified.  This is the basic objection to Metcalfe's Law,
and it has been stated explicitly in many places,
for example \cite{Krugman,McAfeeO,Odlyzko7,Rohlfs2,Ross}.  
This observation was apparently first made by Metcalfe himself,
in his first formal publication devoted to Metcalfe's Law
\cite{Metcalfe1}.  Some users (those who generate spam,
worms, and viruses, for example) actually subtract from the value of a 
network.  Even if we disregard the clearly objectionable aspects
of communication, such as spam, many users do complain about
the problem of dealing with too many options.

There are additional scaling arguments that suggest
Metcalfe's and Reed's laws are incorrect.  For example,
Reed's Law is implausible because of its exponential
(in the precise mathematical sense of the term) nature \cite{KilkkiK}.
If a network's value were proportional to $2^n$, then
there would be a threshold value $m$ such that for
$n$ below $m-50$, the value of the network would be
no more than 0.0001\% of the value of the whole economy,
but once $n$ exceeded $m$, the value of the
network would be more than 99.9999\% of the value of all assets.
Beyond that stage, the addition of a single member to 
the network would have the effect of almost doubling the
total economic value of the world.  
This does not fit general
expectations of network values and thus also suggests
that Reed's Law is not correct.

Metcalfe's Law is slightly more plausible than Reed's Law,
as its quadratic growth does not lead to the extreme
threshold effect noted above,
but it is still improbable.  The problem is that Metcalfe's Law
provides irresistible incentives
for all networks relying on
the same technology to merge or at least
interconnect.  To see this, consider two networks,
each with $n$ members.  By Metcalfe's Law, each one
is (disregarding the constant of proportionality)
worth $n^2$, so the total value of both is $2 n^2$.
But suppose these two networks merge, or one
acquires the other, or they come to an agreement
to interconnect.  Then we will effectively have a single network
with $2 n$ members, which, by Metcalfe's Law, will
be worth $4n^2$, or twice as much as the two separate
networks.  Surely it would require a combination of
singularly obtuse management and singularly inefficient
financial markets not to seize this obvious opportunity
to double total network value by a simple combination.
Yet historically there have been many cases of networks that resisted
interconnection for a long time.  For example, a
century ago in the U.S., 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 (see \cite{Odlyzko7}, or
\cite{Mueller} for much more detail).  Eventually
interconnection was achieved (through a combination
of financial maneuvers and political pressure), but
it took two decades.  
% More recently,
In the late 1980s and early 1990s, the commercial
online companies such as CompuServe, Prodigy, AOL, and MCIMail, 
provided email to subscribers, but only within their
own systems, and it was only in the mid-1990s that
full interconnection was achieved.  More recently
yet, AOL for many years resisted interconnecting
its IM system with those of its competitors, and
it is only in 2004 that an agreement, at least in
principle, was reached to carry this out.  In addition,
short messaging services of some wireless carriers
do not interoperate.  And there has been a long series
of controveries about interconnection policies of
ISPs (Internet Service Providers).  Thus the general conclusion
(drawn first in \cite{Odlyzko7}) is that the incentives
to interconnect cannot be too strong, and so Metcalfe's Law 
cannot be valid.

If we reject Metcalfe's and Reed's laws, can we replace
them with a more accurate but still simple estimate
of the value of a network?  We propose $n \log (n)$
as an alternate rule-of-thumb valuation of a general
communication network of size $n$.  
Metcalfe's law would hold if the value an individual personally 
gets from a network is directly proportional to the number of 
people in that network.  This doesn't seem to hold, there is 
some law of diminishing returns that applies.  Several arguments 
that we will present suggest that the value that a single user gets
from being in a network of $n$ people scales as $\log (n)$, 
leading to a rule-of-thumb valuation of $n \log (n)$ for a network of size $n$.
This growth rate is faster than the linear
growth of Sarnoff's Law, and so explains why connectivity
and not content is king \cite{Odlyzko8}, and why various
network and bandwagon effects do operate.  On the other
hand, this growth rate is only slightly faster than
linear, and this helps explain why interconnection often
requires time, effort, and in many cases regulatory
pressure to achieve.  If we have two networks, each with
$n ~=~ 2^{20} ~=~ 1,048,576$ members, then (assuming
the logarithm in our formula is to base $2$, and the
constant of proportionality is $1$) each is valued
at $20 n$, for total value of $40 n$.  If these two
networks interconnect, the resulting single network
will have size $2 n ~=~ 2^{21} ~=~ 2,097,152$, and its
value by our suggested formula will be $42 n$, only
a 5\% gain over the $40 n$ total valuation of the two separate
networks.  The modest size of such gains helps explain
why network and bandwagon effects, although definitely
present and non-trivial, are often small, and
therefore why so many dot-com and telecom ventures came
to grief.

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.

Our proposed $n \log (n)$ rule for network valuation
does have the advantage of being
consistent with observed behavior of actual 
communication networks, at least when it comes to
rough implications of scaling.  In sections 3, 4, and 5,
we will present some quantitative heuristics that
also lead to this same $n \log (n)$ valuation of
a network of size $n$, although in both cases one
could modify those heuristics to obtain somewhat
different estimates.  Obviously any rule of this
type cannot be very accurate.  Real networks vary
wildly in their behavior.  It is even conceivable
(as has been proposed in \cite{McAfeeO,Rohlfs2}) that
their value may decrease as they grow (say as a result
of spam).  And there have been communication technologies
that almost disappeared after a meteoric rise (as with
citizens' band radios).  But the overwhelmingly dominant pattern
has been for usage and revenues of general communication
networks to grow, often even when
competing technologies appear \cite{Odlyzko7}.  As just one
example, ordinary mail usage continued growing until just
a few years ago, in spite of complaints about junk mail
and the availability of numerous other services.  Thus
the increase in value of a network with increasing size
appears to be well supported by evidence. 

Over the last decade a rich literature has developed
on the structure of networks.  (A detailed technical
survey is available in \cite{Newman}, an account
aimed at a general audience in \cite{Watts}, and 
many references to recent literature can be found
in \cite{RavidR}.)  But it appears hard to obtain
any clear implications of this work for values of
networks, especially since such values depend not
only on structure, but also on usage.  Furthermore,
there are time dependencies in network valuations,
as people learn to use them more intensively and
in new ways, even in the absence of new members.
Reed's insight about value of group formation plays
a key role here, so it is necessary to study not
just pair interactions, but group ones as well.
(Kilkki
and Kalervo have made a first step towards modeling
such phenomena in \cite{KilkkiK}.)

The general conclusion is that accurate valuation
of networks is complicated, and no simple rule will
apply universally.  But if we have to select a
parsimonious rule, one that is concise and yet
captures many of the key features of communication
networks, then we feel that our $n \log (n)$ formula
fits the available evidence and is supported
by reasonable heuristics.





\section{Value of interconnection}

When Sarnoff's Law holds, and the value of a network grows
linearly in its size, as it does for a broadcast network,
there is no net gain in value from combining two networks.
Mergers and acquisitions in such situations are likely driven by other
factors, such as scale efficiencies in operations, or attempts
to increase bargaining power in purchases of content.  On the
other hand, when the value of a network grows faster than
linearly in its size, then generally (subject to some smoothness
assumptions we will not discuss), there is a net gain from
a merger or interconnection, as the value of the larger network
that results is greater than the sum of the values of the 
two constituent pieces.  We next consider how that gain is distributed among
the customers of the two networks.

Let us assume that Metcalfe's Law holds, and we have two ISPs, call them
$A$ and $B$, with
$m$ and $n$ customers, respectively, and that on average the customers
are comparable.  Then interconnection would provide each of the $m$
customers of $A$ with additional value $n$ (assuming the constant of proportionality
in Metcalfe's Law is $1$), or a total added value of 
\begin{center}
$m n$ 
\end{center}
for all
the customers of $A$.  Similarly, each member of $B$ would gain
$m$ in value, so all the customers of $B$ would 
gain total value of \begin{center}
$n m$ 
\end{center}
from interconnection.  Thus aggregate gains to customers of $A$ and $B$ would
be equal, and the two ISPs should peer, if they are rational.
However, the incentives are different if our $n \log (n)$ rule for network
valuation holds.  In that case, each of the $m$ customers of
$A$ would gain value $\log (m + n) - \log (m)$ from interconnection,
and so all the customers of $A$ would gain in total 
\begin{center}
$m (\log (m + n) - \log (m))$.
\end{center}
On the other hand, the total gain to
to customers of $B$ would now be
\begin{center}
$n (\log (m + n) - \log (n))$.
\end{center}
If $m$ and $n$ are not equal, this would no longer be the same as the total gain to
to customers of $A$.  As a
simple example, if $m ~=~ 2^{20} ~=~ 1,048,576$, and $B$ has 8 times as
many customers as $B$, so $n ~=~ 2^{23} ~=~ 8,388,608$, then
(again taking logarithms to base 2) we find that interconnection
would increase the value of the service to $A$'s customers by about 3,323,907, while
$B$'s customers would gain about 1,425,434.  Thus the smaller network
would gain more than twice as much as the larger one.  This clearly reduces the
incentive for the latter to interconnect without compensation.
This is a very simplistic model of ISP interconnection, of course,
and it does not deal with other important aspects
that enter into actual negotiations, such as geographical
spans of networks, and balance of outgoing and incoming
traffic.  All we are trying to do is show that is that there
may be sound economic reasons for larger networks to demand
payment for interconnection from smaller one, a very common
phenomenon in real life.





\section{Gravity laws and the value of locality}

Thoreau's comments in {\em Walden} about Maine and
Texas not having much to communicate reflect the
phenomenon that most traffic, whether in physical
goods, or in information, has historically been
local.  
Although distance is in many aspects becoming
less important, as described by the ``death of distance''
phrase \cite{Cairncross} and some jobs are being
outsourced halfway around the world, locality is
still important.  As was noted in \cite{Odlyzko7},
an investment bank relocated its high-tech branch
from San Francisco to Menlo Park, to be even closer
to Silicon Valley.  A recent report on New York City's
recovery from the Sept. 11, 2001 disaster noted that
businesses displaced by that tragedy ``tended to relocate
in areas where other businesses in their line of work were
already clustered'' \cite{Krueger}.  
Some careful quantitative studies \cite{CummingsK} (with
an account for a general audience in \cite{Young})
show that researchers working far apart lose productivity.
Thus the value of
geographical locality is still important.  

Furthermore,
even in cyberspace, one can use geographical modeling
to represent other types of distances that exist.
Although we are all supposedly connected through
a chain of at most five acquaintances with each
other \cite{Matthews}, most of our interactions
are with smaller, well-defined groups.
A telephone-based language translation service
that AT\&T had started had disappointing results,
since it was primarily emergency and legal service
professionals who used it.  What was discovered is
that people speaking Tibetan, say, had more interest
in speaking with others who spoke Tibetan, even when
those lived far away, and less
with their next-door neighbors who spoke Samoan, say.

The studies of the effects of distance on interaction
have found that the intensity of traffic tends
to follow so-called ``gravity laws'':  If two
cities at distance $d$ apart have populations $A$
and $B$, traffic between them is usually proportional
to $A B / d^{\alpha}$, where $\alpha$ is a constant,
typically between $1$ and $2$.  This observation,
often ascribed to Zipf \cite{Zipf}, actually goes back to the
19th century.
(For extensive references, see \cite{CoffmanO1,Odlyzko7}.)
The degree of locality is not constant in time.  Regular
mail in many instances evolved towards increasing
locality.  On the other hand, in telephony, long distance calling
has grown much faster than local (see \cite{CoffmanO3} for
some growth rate estimates), although local calling
still dominates by far.   Although this is still controversial, 
it appears that traffic on the Internet is also becoming
more local, as had been predicted a long time ago.  

The ``gravity law'' rule is not just an artifact of
distance-sensitive pricing, since it also applies to
regular mail.  There is no fully satisfactory explanation
for why it holds, but it is valid for a remarkably
wide range of interactions, and is often used in
transportation and communication facility planning.
If we assume that on average the value of being able
to communicate with someone at distance $d$ does drop
off as $1 / d^{\alpha}$, then, for a uniform distribution
of populations in a large disk of radius $r$, we find
(after some calculus that we suppress) total value grows
like $r^{3 - \alpha}$ for $1 < \alpha < 2$, and like
$r^2 \log (r)$ for $\alpha ~=~ 2$.  Since the population
is proportional to $r^2$, the area of the disk, we 
deduce that a population of size $n$ with this type 
of spacial distribution,
the value is proportional to
$n \log (n)$ for $\alpha ~=~ 2$ and to $n^{2 - \alpha / 2}$
for $1 < \alpha < 2$.  (For $\alpha > 2$, representing
extreme locality, value is proportional to $n$.)

What value of $\alpha$ is most appropriate?  Different
values apply for different types of traffic.  Also, we
are interested in traffic value (as well as the option value of
being able to engage in a transaction), and that is not the same as
intensity of traffic.  As one example of the contrast between
intensity and value (or, to be more precise, the price
paid), in 1910, only about 3\% of Bell System calls were toll (and 
even the vast majority of those were to nearby locations),
but they brought in 27\% of the revenues.  Thus one can argue
for many different values of $\alpha$.  However, $\alpha ~=~ 2$
is the only nice round value, occurs (at least approximately)
very frequently, and produces a result that is simple to state, so
we adopt it, and the resulting value of $n \log (n)$ for the
valuation of a network.







\section{Zipf's Law}

Zipf's Law \cite{Zipf2} (which again is just a rough empirical rule)
says that if we order some large collection by size or
popularity, the 2nd one will be about half of the first
one in the measure we are using, the 3rd one will be
about one third of the first one, and in general the
one ranked $k$-th will be about $1/k$ of the first one.
This rule has been found to
apply in such widely varying places as the wealth
of individuals, the size of cities, and the amount
of traffic on webservers.  (As Clay Shirky \cite{Shirky} has pointed
out, it can be seen in operation today, producing inequality
in very egalitarian settings, in the weblog and other
spheres, even when there are no economic or political
forces impelling this.)

Zipf's Law is behind phenomena such
as ``content is not king'' \cite{Odlyzko8}, and ``long tails'' \cite{Anderson},
which argue that it is the huge volumes of small items or
interactions, not the few huge hits, that produce the
most value.  It even helps explain why 
both the public and decision makers so often
are preoccupied with the ``hits,'' since, especially
when the total number of items available is relatively
small, they can dominate.  By Zipf's Law, if value follows
popularity, then the value of a collection of $n$ items
is proportional to $\log (n)$.  If we have a billion items,
then the most popular one thousand will contribute a third
of the total value, the next million another third, and
the remaining almost a billion the remaining third.  But
if we have online music stores such as Rhapsody or iTunes
that carry 735,000 titles while the traditional brick-and-mortar
record store carries 20,000 titles, then
the additional value of the ``long tails'' of the download services
is only about 33\% larger than that of record stores.

Now let us suppose that the incremental value that
a person gets from other people being part of a network varies 
as Zipf's Law predicts.  Let's
further assume that for most people their most valuable communications
are with friends and family, and the value of those communications is
relatively fixed - it is set by the medium and our makeup as social
beings.  Then each member of a network with $n$ participants derives
value proportional to $\log (n)$, for $n \log (n)$ total value.





\section{Information locality}

The arguments in the previous section were based on
geographic locality.  But cyberspace has other forms of
locality, for instance locality of interests.  Academia
has long been structured this way, and we believe that
the structure of cyberspace will be at least somewhat
similar.

One can tell something about the structure of academic
communication by the distribution of research articles
of interest to a given researcher.  This distribution is
described by Bradford's Law of scattering.  In its
original form \cite{Bradford} it said that for a typical scientist,
there was a core collection $A$ of journals that contained
about a third of all articles that were of interest,
another collection $B$ of journals that was about $\beta$ 
times as large as $A$ that contained another third, and a
final collection $C$ of about ${\beta}^2$ journals that contained
the remaining third, with $\beta$ depending on the scientist.
Since then librarians have realized that if the search
continued on through sets of ${\beta}^3$, ${\beta}^4$, ${\beta}^5$, etc.,
times as many journals, we would continue producing
articles of interest at about equal numbers per set.  So
in the $m$ most interesting journals, the number of
articles of interest tends to scale like $\log (m)$, with
the constant of proportionality the inverse of 
$\log (\beta)$.
It seems reasonable to assume that the number of journals
scales linearly with the number of researchers, so the
total amount of potentially interesting communications
among $n$ researchers scales like $n \log (n)$.

There are many possible objections to this reasoning:
\begin{itemize}
\item
It is obviously impossible to search all journals
for all articles of possible interest.  With
perhaps 200,000 serials, of which about 20,000 are
peer reviewed, the task is far beyond what is
humanly possible.  But we suspect that researchers
come reasonably close to ideal results through a
combination of research tools like keyword
searches, citation indexes, review articles, and
peer recommendations.  (Conversely the potential
value of such tools can be estimated by comparing a
feasible direct search versus what is likely to be
out there.)
\item
We ignored the fact that different researchers have
different constants $\beta$.  We believe that
individual differences average out so this does not
matter.
\item
Bradford's Law of scattering is meant to be a rule
of thumb, and not a precise quantitative law.  True,
but then again our $n \log (n)$ estimate is likewise
meant as more of a rule of thumb than a precise
quantitative law.  Furthermore simple power laws are
ubiquitous in self-organized systems (including
social networks), and $1/n$ scaling factors are
particularly common.  In Bradford's law the the number of
interesting articles in the $n$-th journal averages $1/n$ of 
those in the best journal, consistent with Zipf's Law.
\item
Not all communications of interest are of equal
value.  This is both obvious and difficult to
measure.  
% However see the next consideration.
\item
Researchers cannot read an unlimited number of
interesting articles.  However, articles do not
have to be read to be valuable, as their
availability allows quick pursuit of various
research threads as they are encountered.
Moreover, as the universe of articles available
expands, the most valuable ones, the ones that
do get read, likely grow more valuable.
\item
This calculation depends on the journals being
ordered from most valuable to least.  That is not
exactly how networks grow.  Communication
networks do not grow independently of social
relations.  When people are added, they induce
those close to them to join.  Therefore in a mature
network, those who are most important to people
already in the network are likely to also be members.
So additional growth is likely to occur at the
boundaries of what existing people care about.
\end{itemize}

Therefore the structure of academia suggests that an
$n \log (n)$ scaling rule is at least plausible for communities
that self-organize on the basis of interest.



\section{Conclusions}

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.



% \clearpage


% \section*{Acknowledgments}






% \clearpage


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\end{thebibliography}




\end{document}