Contributed by Nancy A. Lynch, August 14, 2017 (sent for review April 18, 2017; reviewed by Yehuda Afek, Ziv Bar-Joseph, and Amos Korman)
Highly complex distributed algorithms are ubiquitous in nature: from the behavior of social insect colonies and bird flocks, to cellular differentiation in embryonic development, to neural information processing. In our research, we study biological computation theoretically, combining a scientific perspective, which seeks to better understand the systems being studied, with an engineering perspective, which takes inspiration from these systems to improve algorithm design. In this work, we focus on the problem of population density estimation in ant colonies, demonstrating that extremely simple algorithms, similar to those used by ants, solve the problem with strong theoretical guarantees and have a number of interesting computational applications.
Many ant species use distributed population density estimation in applications ranging from quorum sensing, to task allocation, to appraisal of enemy colony strength. It has been shown that ants estimate local population density by tracking encounter rates: The higher the density, the more often the ants bump into each other. We study distributed density estimation from a theoretical perspective. We prove that a group of anonymous agents randomly walking on a grid are able to estimate their density within a small multiplicative error in few steps by measuring their rates of encounter with other agents. Despite dependencies inherent in the fact that nearby agents may collide repeatedly (and, worse, cannot recognize when this happens), our bound nearly matches what would be required to estimate density by independently sampling grid locations. From a biological perspective, our work helps shed light on how ants and other social insects can obtain relatively accurate density estimates via encounter rates. From a technical perspective, our analysis provides tools for understanding complex dependencies in the collision probabilities of multiple random walks. We bound the strength of these dependencies using local mixing properties of the underlying graph. Our results extend beyond the grid to more general graphs, and we discuss applications to size estimation for social networks, density estimation for robot swarms, and random walk-based sampling for sensor networks.
↵1C.M., H.S., and N.L. contributed equally to this work.
This contribution is part of the special series of Inaugural Articles by members of the National Academy of Sciences elected in 2016.
An extended abstract of this work has been previously published (1).
Author contributions: C.M., H.-H.S., and N.A.L. designed research; C.M., H.-H.S., and N.A.L. performed research; C.M. and H.-H.S. contributed new reagents/analytic tools; and C.M., H.-H.S., and N.A.L. wrote the paper.
Reviewers: Y.A., Tel-Aviv University; Z.B.-J., Carnegie Mellon University; and A.K., French National Center for Scientific Research.
The authors declare no conflict of interest.
See QnAs on page 10512.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1706439114/-/DCSupplemental.
