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6.2. DEVIATIONS FROM POWER LAWS 37
tributions of subsets of the WWW with exponents 2.1 and 2.38 − 2.72 respectively [37, 61, 179],
the in-degree distribution of the African web graph with exponent 1.92 [48], a citation graph with
exponent 3 [237], distributions of website sizes and traffic [4], and many others. Newman [215]
provides a long list of such works.
One may wonder: is every distribution a power law? If not, are there deviations? The answer
is that, yes, there are deviations. In log-log scales, sometimes a parabola fits better, or some more
complicated curves fit better. For example, Pennock et al. [231], and others, have observed deviations
from a pure power-law distribution in several datasets. Common deviations are exponential cutoffs,
the so-called “lognormal” distribution, and the “doubly-Pareto-lognormal” distribution. We briefly
cover them all, next.
6.2.1 EXPONENTIAL CUTOFFS
Sometimes the distribution looks like a power law over the lower range of values along the x-axis,
but decays very quickly for higher values. Often, this decay is exponential, and this is usually called
an exponential cutoff:
y(x = k) ∝ e −k/κ k −γ
(6.3)
where e −k/κ is the exponential cutoff term and k −γ is the power-law term. Amaral et al. [23] find
such behaviors in the electric power-grid graph of Southern California and the network of airports,
the vertices being airports and the links being non-stop connections between them. They offer two
possible explanations for the existence of such cutoffs. One: high-degree nodes might have taken a
long time to acquire all their edges and now might be “aged,” and this might lead them to attract
fewer new edges (for example, older actors might act in fewer movies). Two: high-degree nodes
might end up reaching their “capacity” to handle new edges; this might be the case for airports
where airlines prefer a small number of high-degree hubs for economic reasons, but are constrained
by limited airport capacity.
6.2.2 LOGNORMALS OR THE “DGX” DISTRIBUTION
Pennock et al. [231] recently found while the whole WWW does exhibit power-law degree distributions,
subsets of the WWW (such as university homepages and newspaper homepages) deviate
significantly. They observed unimodal distributions on the log-log scale. Similar distributions were
studied by Bi et al. [46], who found that a discrete truncated lognormal (called the Discrete Gaussian
Exponential or “DGX” by the authors) gives a very good fit. A lognormal is a distribution whose
logarithm is a Gaussian; its pdf (probability density function) looks like a parabola in log-log scales.
The DGX distribution extends the lognormal to discrete distributions (which is what we get in
degree distributions), and can be expressed by the formula:
y(x = k) =
A(μ, σ )
k
(ln k − μ)2
exp −
2σ 2
k = 1, 2,... (6.4)