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in the predictability of the properties of such networks, in particular their failure char-
acteristics. No longer does the survival probability of an element or a network have
the relatively rapid decline of an exponential, but instead failure has a heavy tail that
extends far from the central region, making it both rarer and less reliably predictable.
The why of this observation comes later.
We can pocket another piece of wisdom from the discussion in this chapter. While it is
true that no experimental output is exactly predictable, it is not true that the variability in
the experimental data is necessarily normal, or that the average is the best representation
of those data. It is safe to conclude that, rather than normal statistics, it is hyperbolic
statistics that apply to complex webs, supporting the observation that normalcy is a
myth.
1.5
Problems
1.1 The normal distribution
Verify that the convolution of two Gaussian distributions is also a Gaussian distribution
and thereby verify Equations (
1.22
)-(
1.24
).
1.2 Maximum information
Consider one state with the probability of being occupied
p
and a second state with
a probability of being occupied 1
p
. Consider the information contained in the
two states and show geometrically that the information obtained is maximum when
p
−
=
1
/
2.
1.3 Resolving a contradiction
The argument of Huxley implicitly addresses the criticism of Calder since the over-
all coefficient in the allometric relation becomes a constant of integration. Express the
coefficient of the allometric equation
α
in terms of the initial values
W
a
(
0
)
and
W
b
(
0
)
and use the constant of integration to refute Calder's objection.
1.4 The divine proportion
Another treatment of scaling dates back to the Greeks and is a consequence of a simple
geometric construction. Draw a line of unit length and divide it into two segments of
length
α
and
β
such that the ratio of the original line (
α
+
β
) to the longer segment
α
is
the same as the ratio of
α
to the shorter segment
β
,
α
+
β
α
=
α
β
,
and define the ratio of the two lengths as
? This scaling
proportion was called the golden section or golden mean by the ancient Greeks and
Kepler called it the “divine proportion.” Search the web to find structures that scale
according to this ratio.
φ
=
α/β.
What is the value of
φ
