COMPUTING SCIENCE
Undisciplined Science
Brian Hayes
Social Phase Transitions
How does it happen that a sensible and sober-minded physicist strays
into such dangerous neighborhoods as economics, sociology or
political science? Well, one thing leads to another. The road to
ruin may be long and twisting, but each step along the way is easy
enough to trace.
Here's an example. Physics has a long-standing interest in the
phases of matter and the transitions between those phases. This
topic includes not only the familiar solid-liquid-vapor phases but
also related phenomena such as the onset of magnetization in iron.
One strategy for studying phase transitions is to sweep aside all
the intricacy of atomic or molecular structure and build the
simplest model that exhibits the behavior of interest. In the case
of magnetism, the iron atom with its halo of 56 spinning electrons
can be replaced by a single abstract "spin"—which is
merely an arrow that points either up or down and has no other
properties. The spins are arranged on a geometrical grid or lattice,
a cartoon version of the crystal structure of the metal. Quantum
interactions between iron atoms are modeled by a simple tendency for
nearby spins to line up parallel to one another, but this orderly
state can be disrupted by thermal agitation. If this rudimentary
model is a success, then at some temperature most of the spins
should suddenly fall into alignment, mimicking the spontaneous
magnetization of a real magnet.
Having created this model to represent a specific physical system,
you might now discover that the model itself is an interesting
object of study. Variations suggest themselves, with different
lattice geometries or rules of interaction; the variants may or may
not have anything to do with magnetic materials. In some cases the
behavior of the model can be worked out mathematically in full
detail, but more often the only way to understand how the array of
spins evolves is by computer simulation.
Now comes the next step down the path leading out of the Garden of
Physics. After spending some time exploring the universe of abstract
models, you may begin to notice that the lattice of spins could be
given a variety of interpretations; the spins could represent many
things other than magnetic moments of atoms. In particular,
up and down spins might be mapped onto
pro and contra opinions held by people in some
social context. In this new view of the model, the interactions that
were once seen as magnetic couplings now represent the tendency of
people to influence (and be influenced by) their neighbors'
opinions. The phase transition in which the spins all line up
pointing the same way corresponds to the sudden emergence of a
consensus within the population. And thus a physicist becomes a
social scientist.
For another example, consider the process of percolation, where a
fluid trickles through the mazelike passages of a porous medium. Can
the fluid penetrate the entire region, or will it be blocked by
dead-end passages? Again the essentials can be captured in a lattice
model. Each link between adjacent nodes of the lattice is open to
fluid flow with some fixed probability p or is blocked with
probability 1-p. At low values of p, most links
are blocked, and the lattice consists of many small, isolated
clusters of connected nodes. As p increases, there is a
threshold value where a giant connected cluster suddenly appears,
allowing a fluid to infiltrate the entire lattice.
Like the lattice spin system, the percolation model has many
variationsÑand many interpretations distant from the physical
process that inspired it. The idea of something spreading
probabilistically through a network can also model the transmission
of rumors, or the progress of a forest fire or the spread of an
infectious disease. Indeed, maybe the percolation model could model
itself, documenting its own spread from one discipline to the next.
These are a few of the paths radiating from physics to other areas.
But the landscape of science is criss-crossed with trails
going in other directions as well. A mathematician studying random
graphs—structures formed when you start with a set of isolated
nodes and then add links between them at random—would also
discover an abrupt transition where a giant connected component
spontaneously emerges. This sudden change in the structure of the
graphs has all the characteristics of a phase transition, and so the
mathematician wanders onto turf usually claimed by physicists.
A computer scientist could have a similar experience. The
computational problem known as satisfiability concerns Boolean
formulasÑlogical statements such as ((p OR
q) and ((not q) OR r)), where each of the
variables p, q and r has a value of either true or
false. The question is: Can you find an assignment of values that
makes the overall proposition true? For the example given here it's
easy to answer this question by trial and error, but large formulas
are challenging. In the 1980s computer scientists detected an
interesting pattern: As a certain parameter measuring the complexity
of the formula increases, there is a sudden transition. Below the
threshold, almost all satisfiability problems are solvable, but
above it almost none are. The resemblance to phase transitions is
obvious, and so computer scientists found themselves doing physics,
and physicists took up work on the satisfiability problem.
One more example from farther afield: In 1971 Thomas C. Schelling
published a lattice model of racial segregation. Black and white
residents, initially scattered at random over the nodes of the
lattice, were assumed to prefer living among neighbors of the same
race; those who were unhappy with their current surroundings could
move. Schelling's most provocative finding was that it doesn't take
vicious bigotry to produce a sharply segregated housing pattern;
even the mildest preference for neighbors of the same race leads to
a phase separation. Schelling's diagrams look very much like
simulations of a lattice model of magnetic materials, but the paper
makes no reference to the physics literature. (Indeed, it predates
much of it.) Schelling is an economist and political scientist.
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