## Analysis of ant colony behavior

Ants, it turns out, are extremely good at estimating the concentration of other ants in their vicinity. This ability appears to play a role in several communal activities, particularly in the voting procedure whereby an ant colony selects a new nest.

Biologists have long suspected that ants base their population-density estimates on the frequency with which they — literally — bump into other ants while randomly exploring their environments.

That theory gets new support from a theoretical paper that researchers from MIT’s Computer Science and Artificial Intelligence Laboratory will present at the Association for Computing Machinery’s Symposium on Principles of Distributed Computing conference later this month. The paper shows that observations from random exploration of the environment converge very quickly on an accurate estimate of population density. Indeed, they converge about as quickly as is theoretically possible.

Beyond offering support for biologists’ suppositions, this theoretical framework also applies to the analysis of social networks, of collective decision making among robot swarms, and of communication in ad hoc networks, such as networks of low-cost sensors scattered in forbidding environments.

“It’s intuitive that if a bunch of people are randomly walking around an area, the number of times they bump into each other will be a surrogate of the population density,” says Cameron Musco, an MIT graduate student in electrical engineering and computer science and a co-author on the new paper. “What we’re doing is giving a rigorous analysis behind that intuition, and also saying that the estimate is a very good estimate, rather than some coarse estimate. As a function of time, it gets more and more accurate, and it goes nearly as fast as you would expect you could ever do.”

Random walks

Musco and his coauthors — his advisor, NEC Professor of Software Science and Engineering Nancy Lynch, and Hsin-Hao Su, a postdoc in Lynch’s group — characterize an ant’s environment as a grid, with some number of other ants scattered randomly across it. The ant of interest — call it the explorer — starts at some cell of the grid and, with equal probability, moves to one of the adjacent cells. Then, with equal probability, it moves to one of the cells adjacent to that one, and so on. In statistics, this is referred to as a “random walk.” The explorer counts the number of other ants inhabiting every cell it visits.

In their paper, the researchers compare the random walk to random sampling, in which cells are selected from the grid at random and the number of ants counted. The accuracy of both approaches improves with each additional sample, but remarkably, the random walk converges on the true population density virtually as quickly as random sampling does.

That’s important because in many practical cases, random sampling isn’t an option. Suppose, for instance, that you want to write an algorithm to analyze an online social network — say, to estimate what fraction of the network self-describes as Republican. There’s no publicly available list of the network’s members; the only way to explore it is to pick an individual member and start tracing connections.

Similarly, in ad hoc networks, a given device knows only the locations of the devices in its immediate vicinity; it doesn’t know the layout of the network as a whole. An algorithm that uses random walks to aggregate information from multiple devices would be much easier to implement than one that has to characterize the network as a whole.