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of number of people affected or MW lost) and its probability of occurrence. Recall
hurricane Katrina, whose initial damage could not be avoided but whose subsequent
destructive impact, many argue, resulted from a lack of preparedness for its magnitude,
duration and secondary destructive influence. It is important to design the infrastructure
to be robust against large failures regardless of whether one is talking about hurricanes,
blackouts, or, indeed, economic recessions. Moreover, it is not just the United States that
has an inverse power-law size-frequency relationship in event size in either megawatts
or customers, or both, but the international blackout data show the same pattern [
10
].
This universality of the blackout data indicates that the inverse power-law relationship
is a reflection of the fundamental structure of the electrical power grid.
It is useful to make some formal observations here. The first is that the probability
of failure in simple statistical phenomena is exponential, as one finds in uncomplicated
manufacturing processes, say in the making of light bulbs. Empirically it is observed
that for a large number of light bulbs there is a constant rate of failure per unit of time,
say
. This constant rate of failure implies that the probability a given fraction of this
large number of light bulbs fails in a time
t
is given by
λ
e
−
λ
t
F
(
t
)
=
,
(1.49)
so that, although one cannot say whether this bulb or that bulb will fail after a time
t
,it
is possible to discuss fairly accurately what will happen to a large number of bulbs in
that time. Consequently, it is useful for the factory owner to know this number in order
to anticipate what the manufacturing cost of the bulbs will be over the next year and
to design the production line to have an acceptable rate of failure. The assumption is
that, although the specifics of the process cannot be predicted or controlled, an overall
parameter such as the average failure rate
λ
can be changed to make the production
process more efficient.
However, this exponential functional form is not observed in blackouts, or, indeed, in
the failure of any other complex network.
It is web complexity that invalidates this exponential distribution of failure and its
simple one-parameter characterization. Quite generally complexity is manifest in the
loss of web characterization by a small number of parameters. In this chapter we have
seen that normal statistics, with its mean and variance, is replaced by inverse power-law
or Pareto statistics in which one or both of these quantities diverge. In subsequent chap-
ters we show that these complex webs have failure probabilities that are also given by
inverse power laws, just like the blackouts. Large blackouts typically involve compli-
cated and cascading chains of events leading to failure that propagate through a power
network by means of a variety of processes. It is useful to point out that each particular
blackout can be explained as a causal chain of events after the fact, but that explanation
cannot be constructed predictively prior to the blackout [
3
]. The same condition exists
for most complex networks, so we do not spend much time on discussing particular
failure mechanisms. We leave that to the subject-matter experts.
But we are getting ahead of ourselves. We need to see a great deal more data before
the ubiquity of the inverse power-law structure of complex webs becomes evident.
Moreover, we need to see the various ways complexity blurs and distorts our traditional
