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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.

Figure 2. Computational models of physical phenomena . . .Click to Enlarge Image

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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