Classroom Companion: Business

16.3 · Numerical Analysis of the Long Tail

---

238
16
7
Box 
16.3
 is an example of the application of Zipf ’s law to a completely different 
area, namely, the vulnerability of the Internet and the World Wide Web to targeted 
attacks on the infrastructure. Both the Internet and the Web may be regarded as a 
long tail network. On the Internet, most of the routers are small and connected by 
communications links to few other routers, while the “tail” of the Internet consists 
of rather few big routers connected to thousands of other routers. In the Web, most 
websites are connected to few other websites. These websites make up the “head” 
of the Web. The “tail” of the Web consists of websites connected to very many 
other websites, where search engines are example of websites connected to an enor-
mous number of websites.
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0
0,2
0,4
0,6
0,8
1
1,2
1,4
1,6
1,8
2
R
alpha
. Fig. 16.3 Relative number of books in the tail as a function of alpha. (Authors’ own figure)
 
Chapter 16 · The Long Tail

---

239
16
Box 16.3 The Long Tail and the Vulnerability of Internet
Power-law distributions are not only 
important to determine the economic 
value of the long tail. These distri-
butions are also used to evaluate the 
robustness and vulnerability of the 
Internet and digital services. In 1999, the 
two physicists, Albert-László Barabási 
and Réka Albert, discovered that the 
number of hyperlinks pointing into 
or out of webpages followed a general 
power-law distribution with an expo-
nent approximately equal to 2 (Albert 
& Barabási, 
2001
). The same research-
ers also developed a general theory in 
which they demonstrated that the num-
ber of links connected to the nodes of a 
graph follows a power-law distribution 
if the graph is grown with preference. 
For the Web, “growth with preference” 
means that a new webpage is connected 
to another webpage with a probability 
proportional to the number of existing 
connections to that webpage. This phe-
nomenon is called the “Barabási-Albert 
(BA) random graph model.” Later, it 
was found that the size of Internet rout-
ers measured in terms of the number 
of connections they have with other 
routers also follows the same power law 
(Dorogovtsev & Mendes, 
2001
).
Hence, the Internet and the Web have 
long tails. In these cases, the tail consists 
of the largest and the head consists of 
the smallest routers and webpages. This 
insight directs us toward another obser-
vation, namely, that structures like the 
Internet and the Web are very vulnerable 
to attacks against routers and webpages 
in the long tail. Remove several of the 
biggest Internet routers—there are not 
very many of them—and the Internet 
runs into severe connectivity and capac-
ity problems; remove the search engines 
from the Web, and the Web falls apart. 
The Internet and the Web are vulnerable 
to targeted attacks. On the other hand, 
most of the routers and webpages have 
small connectivity, and a random attack 
on routers and webpages may have little 
overall effect on the Internet or the Web. 
Hence, the Internet and the Web are, 
at the same time, both vulnerable and 
robust against failures and cyberattacks 
(Audestad, 
2007
).