Classroom Companion: Business

9.6 · Estimating the Value of Networks

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140
9
be calculated as V(n) = 5.7 × 10
−9
× n
2
~ n
2
. However, this conclusion depends on 
how value is defined and how the number of users is counted.
The network effect (or feedback term) of Metcalfe’s law is F
Metcalfe
(n)~n. This is 
the most commonly used value for network effects in economic research. The value 
was, in fact, used long before Metcalfe suggested it in 1980. This feedback value is 
the basis for the diffusion model of Frank Bass from 1969 (see 
7
Chap. 
18
).
Andrew Odlyzko and his coworkers have argued that Metcalfe’s law is incor-
rect—in particular, concerning social media networks—since Metcalfe’s law 
assumes that all possible transactions have equal value. In a large population, 
Metcalfe’s law gives an overestimation of the value of the network since each indi-
vidual will only interact with a small number of other individuals, and not all inter-
actions between them will be equally strong. Based on this argument, Odlyzko and 
Tilly proposed the alternative law:
V
n
n
n
O T

 
~ ln .
This law is called Odlyzko-Tilly’s law. The network effect is now reduced to
F
O − T
(n) ~ ln n.
Box 9.4 Derivation of Odlyzko-Tilly’s Law
Odlyzko-Tilly’s law is derived apply-
ing Zipf ’s principle to the frequency 
of interactions an individual has with 
other individuals. Zipf ’s principle is 
an empirical law based on the obser-
vation that several sequences in nature 
and society (e.g., frequency of words, 
size of cities, and length of rivers) fol-
low a rank distribution (called a 
“Pareto distribution” in statistics) in 
which the most frequent or largest 
item is twice as frequent or large as 
the second item in the sequence, three 
times as frequent or large as the third 
item, and so on.
By applying this ranking principle 
on interactions between people, the 
total number of transactions T between 
one individual and all other individuals 
is:
T
c
c
c
c
n
c
n
n
 
 

2 3

~ ln ~ ln .
Applying Zipf’s ranking principle, the 
assumption is that an individual has c 
transactions with the person who is clos-
est to him, only half as many with the 
next closest person, one-third as many 
with the third closest, and so on. The 
value of the network is then 
V
O − T
(n) = nT~n ln n.
Another way to derive Odlyzko-
Tilly’s law is based on the connectivity 
of random graphs. First, assume that 
individuals are nodes in a random graph 
and that the interactions between the 
individuals are the links in the graph; 
that is, two individuals interacting with 
each other are connected by a link, while 
there is no link between two individuals 
who are not interacting with each other. 
The simplest random graph is the Erdös-
Rényi (ER) graph. In ER graphs, the 
probability, p, that a link exists between 
any two nodes is the same for all pairs of 
nodes in the network.