Communications of the ACM

moment of collapse. In the sand pile, for example, most new grains lodge firmly into a place on the pile but occasionally one sets off an avalanche that changes the structure. In the Internet, malware can quickly travel via a hub to many nodes and cause a large-scale avalanche of disruption. In an economy, a new technology can suddenly trigger an avalanche that sweeps away an old structure of jobs and professions and establishes a new order, leaving many people stranded. Complexity theory tells us we frequently encounter systems that transition between stability and randomness.

Punctuated equilibrium appears differently in different systems because self-organization manifests in different ways. In the Internet, it may be the vulnerability to the failure of highly connected hubs. In a national highway system, it may be the collapse of maintenance as more roads are added, bringing new traffic that deteriorates old roads faster. In geology, it may be the sudden earthquake that shatters a stable fault and produces a cascade of aftershocks. In a social system, it may be the outbreak of protests when people get “fed up.”

Explanations but Not Predictions

What can we learn from all this? Many
systems have a strong social compo-
nent, which leads to forms of preferen-
tial attachment and power-laws govern-
ing the degrees of connectivity in the
demonstrated the essence of complex-
ity theory. 4 They observed a sand pile
as it formed by dropping grains of sand
on a flat surface. Most of the time, each
new grain would settle into a stable posi-
tion on the growing cone of sand. But at
unpredictable moments a grain would
set off an avalanche of unpredictable
size that cascaded down the side of the
sand pile. The researchers measured the
time intervals between avalanche starts
and the sizes of avalanches. To their sur-
prise, these two random variables did
not fit any classical probability distribu-
tion such as the Normal or Poisson dis-
tributions. Instead, their distributions
followed a “power law,” meaning the
probability of a sample of length x is pro-
portional to x–k, where k a fixed param-
eter of the random process. Power law
distributions have a finite mean only if
k> 2 and variance only if k> 3. This means
a power law with k≤ 2 has no mean or
variance. Its future is unpredictable.
When 2<k≤ 3, the mean is finite but not
the confidence interval. Bak et al. had
discovered something different—a ran-
dom process whose future could not be
predicted with any confidence.

This was not an isolated finding. Most of the random processes tied to chaotic situations obey a power law with k< 3. For example, the appearance of new connections among Web pages is chaotic. The number of Web pages with x connections to other pages is proportional to 1/ x2—the random process of accumulating links produces 1/4 as many pages with 2x connections as with x connections. This was taken as both bad and good news for the Internet. The bad news is that because there are a very few “hubs”— servers hosting a very large number of connections—an attacker could shatter the network into isolated pieces by bringing down the hubs. The good news is the vast majority of servers host few connections and thus random server failures are unlikely to shatter the network. What makes this happen is “preferential attachment”—when a new Web page joins the network, it tends to connect with the most highly connected nodes already in the network. Startup company founders try to plot strategies to bring about rapid adoption of their technologies and transform their new services into hubs.

Hundreds of processes in science and
engineering follow power laws and their
key variables are unpredictable. Innova-
tion experts believe innovations follow a
power law—the number of innovations
adopted by communities of size x is pro-
portional to x– 2—not good news for start-
up companies hoping to predict their in-
novations will take over the market.

Later Bak1 developed a theory of unpredictability that has subsequently been copied by popular writers like Nassim Nicholas Taleb and others. 6 Bak called it punctuated equilibrium, a concept first proposed by Stephen Jay Gould and Niles Eldredge in 1972.3 The idea is that new members can join a complex system by fitting in to the existing structure; but occasionally, the structure passes a critical point and collapses and the process starts over. The community order that has worked for a long time can become brittle. Avalanche is an apt term for the

Hundreds of
processes in science
and engineering
follow power laws
and their key
variables
are unpredictable.

Log-log plot of the exceedance versus intervals between terror attacks follows a straight line. Exceedance is the probability that an interval is greater than x (a tail of the distribution). A straight line on log-log plot is the signature of a power law; here the slope is – 1. 4, telling us the tails of the distribution are a power law y=x– 1. 4. Because 1. 4 is less than 2, this distribution has no finite mean or standard deviation: the time to next terror attack is unpredictable.