Information Technology Reference
In-Depth Information
Table 4.1.
The occurrence of wars. The rows are
x
, the number of outbreaks of war
per year;
y
, the frequency of years with number of outbreaks
x
;and
z
, the frequency
from a Poisson distribution with mean
λ
≈
0
.
69
years. The table is reproduced with
permission from Richardson [
39
] using the data collected by Wright [
46
].
x
0
1
2
3
4
>
4
Total
y
223
142
48
15
4
0
432
z
216
.
2
149
.
7
51
.
8
12
2
.
1
0
.
3
432
.
1
Finally, the last factor in this expression can be written as an exponential,
(
p
λ
N
−
m
)
λ/
p
1
/
e
−
λ
,
(
1
−
p
)
≈
(
1
−
p
=
1
−
p
)
≈
(4.17)
and the resulting form is the Poisson distribution
m
)
=
λ
e
−
λ
.
p
(
m
;
λ)
=
lim
P
N
(
m
(4.18)
m
!
N
→∞
,
p
→
0
The mean number of “deadly quarrels” can be calculated using this expression to be
N
m
=
mp
(
m
;
λ)
=
λ,
(4.19)
m
=
0
so that the Poisson distribution is parameterized in terms of the average value.
In Table
4.1
the “deadly quarrels” over the past 500 years are recorded in a way
that emphasizes their historical pattern. Richardson [
39
] was interested in the causes
of wars between nations, but he found that, independently of the moral position of a
country, its economy or any of the other reasons that are generally offered to justify
war, the number of wars initiated globally in any given interval of time follows a Poisson
distribution.
Wright [
46
] published information on wars extending from 1482 to 1940 including
278 wars, with the start and end dates, the names of treaties of peace, participating states
and so on. Richardson [
39
] took these data and counted the numbers of years in which
no war began, one war began, two wars began, and so on, as listed in the second row
of the table. In the final row is the fit to the data using a Poisson distribution. It is clear
from Table
4.1
that this quantitative discussion captures an essential aspect of the nature
of war; not that we can predict the onset of a particular event, but that we can determine
a useful property of the aggregate, that is, of the collection of events.
The Poisson distribution also emerges in the elementary discussion of the number of
connections formed among the
N
elements of a web. The elements could represent the
airports distributed in cities around the world, the power stations located outside urban
areas in the United States, neurons in the human brain, scientific collaborators, or the
elements in any of a growing number of complex webs. For the purpose of the present
discussion it does not matter what the elements are, only that they are to be connected in
