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Table 2.2.
A list of empirical laws for complex webs is given according to their accepted names in the
disciplines in which they were first developed. The power-law index
α
is different for each of the laws
and is intended to be generic; the value listed is from the original reference and need not be the
accepted value today.
Discipline
Law's name
Form of law
Physics
1918 [
70
] /
f
noise
Spectrum(
f
)
∝
1
/
f
t
2
.
14
2002 [
25
]
Solar flares
Pr(time between flares
t
)
1
/
∝
Pr(time between events
t
)
∝
1
/
t
2
.
14
2004 [
69
]
Temperature anomalies
Physiology
1959 [
61
]
Neurons;
d
1
.
5
0
=
d
1
.
5
1
+
d
1
.
5
2
Rall
1963 [
76
] Mammalian vascular network Veins and arteries;
d
2
.
7
0
=
d
2
.
7
1
+
d
2
.
7
2
d
0
=
d
1
+
d
2
1963 [
90
]
Bronchial tree
Metabolic rate(body mass
M
)
∝
M
0
.
75
1973 [
48
] McMahon
Pr(isotope expelled in time
t
)
∝
1
/
t
α
1975 [
103
] Radioactive clearance
/
n
1
.
25
1987 [
93
] West-Goldberger
Airway diameter(generation
n
)
∝
1
Surface area
∝
volume
0
.
90
1991 [
30
] Mammalian brain
No. of breaths(interbreath time
t
)
∝
1
/
t
2
.
16
1992 [
77
]
Interbreath variability
1993 [
58
] Heartbeat variability
Power spectrum(frequency
f
)
∝
f
/
t
1
.
61
2007 [
23
]
EEG
Pr(time between EEG events)
∝
1
/
k
α
2007 [
13
] Motivation and addiction
Pr(
k
behavior connections)
1
∝
Psychology
1957 [
75
]
x
α
Psychophysics
Perceived response(stimulus intensity
x
)
∝
Reaction time(trial
N
)
∝
1
/
N
0
.
91
1963 [
71
]
Trial and error
utility(delay time
t
)
∝
1
/
t
α
1961 [
29
] Decision making
Percentage correct recall(time
t
)
∝
1
/
t
α
1991 [
3
]
Forgetting
Response spectrum(frequency
f
)
∝
1
/
f
α
2001 [
20
]
Cognition
Pr(phase-locked interval
<τ
)
∝
1
/τ
α
2009 [
37
] Neurophysiology
Sociology
1926 [
41
]
/
r
2
Lotka
Pr(No. of papers published rank
r
)
1
∝
1949 [
104
] Zipf
Pr(word has rank
r
)
1
/
r
∝
Pr(citation rank
r
)
∝
1
/
r
3
1963 [
16
]
Price
Population density(radius
R
)
∝
1
/
R
α
1994 [
8
]
Urban growth
Pr(
k
connections)
∝
1
/
k
2
.
3
1998 [
88
] Actors
think about these various things. What we hope is apparent to the reader is that when
phenomena become complex their distributions become suspiciously similar.
2.1.2
Physiologic trees
Physiology is one area where the classical scaling approach has been applied, specifi-
cally to the branching of the bronchial tree. The physiologist Rohrer [
66
] assumed that
the bronchial tree is homogeneous and that the volume of air in the
n
th generation of
a bronchial tube, denoted by
V
n
, is divided equally between the two daughter tubes in
