Cutting Corners

Go to just about any college campus or public park and you're bound to see two kinds of trails: "official" paths defined by paved or gravel surfaces and "unofficial" routes trodden into the grass or dirt marking where pedestrians have preferred to walk or runners to jog.

Such collective path breaking is an example of a human social phenomenon that involves some form of self-organization. In 1997, a team of researchers developed a mathematical model of pedestrian motion to explore the evolution of trails in urban green spaces.

In the article "Modelling the evolution of human trail systems," published in Nature, Dirk Helbing, Joachim Keltsch, and Peter Molnar reported that their model was "able to reproduce many of the observed large-scale spatial features of trail systems."

The paths pedestrians forge can be quite complex. "In many cases, the pedestrians' desire to take the shortest way and the specific properties of the terrain are insufficient for an explanation of the trail characteristics," the researchers observed. In other words, the trails generated by walkers don't always reflect the most direct routes between entry and exit points.

Several factors come into play. People usually want to take the most direct route, but they also prefer to walk on worn, existing trails rather than tramp over rougher terrain to blaze new ones. Moreover, the nearer a segment of an existing trail is to a walker, the greater an attraction it holds.

Helbing and his colleagues built these assumptions about human behavior into a set of equations that describes how the routes taken by pedestrians would change as time passes. They could then observe what happens when a crowd of "virtual" walkers with given entry points and destinations venture across the park at random times. Because not every route is used by the same number of walkers, some paths typically start to become more pronounced than others.

The type of trail pattern that emerges depends largely on a parameter the researchers called attractiveness (or comfort of walking). If attractiveness is low, a direct-route system develops. If attractiveness is high, a minimal-route system forms. Otherwise, a compromise between the two extremes appears, as is often observed in real trail systems.

The attractiveness parameter determines whether a direct-route (left) or a minimal-route (right) system of trails forms to connect three park gates.

In one simulation, the researchers started with a square, level, grassy park having a gate at each corner. Initially, randomly oriented walkers tend to use direct paths from gate to gate. However, because frequently used trails become more comfortable, the different routes begin to blend together in such a way that the overall length of the trail system decreases.

The evolution of a path system in a square park. Direct paths are prominent in a new park (left) whereas a compromise path system develops in older parks (right).

"The resulting [trail] system…could serve as a planning guideline," the researchers said. "It provides a suitable compromise between minimal construction costs and maximal comfort. Moreover, it balances the relative detours of all walkers."

Self-organization is evident in a wide variety of human social phenomena, from the evolution of cooperation to the development of traffic patterns and the growth of settlements. Mathematical models help elucidate the factors that influence such processes.

"There is a wealth of possible applications and extensions of such models, not only to human but to animal populations, and to social as well as physical spaces," commented geographer Michael Batty. "These models naturally extend to problems where the goal is to optimize, or design 'best' paths, as is characteristic of many planning problems where the task is to design high-quality urban environments."

May the path be with you!

Originally posted July 28, 1997

See also "Soap Films and Grid Walks."