Expander graphs and their applications
Shlomo Hoory, Nathan Linial, Avi Wigderson · Bulletin of the American Mathematical Society · 2006
A major consideration we had in writing this survey was to make it accessible to mathematicians as well as computer scientists, since expander graphs, the protagonists of our story come up in numerous and often surprising contexts in both fields. A glossary of some basic terms and facts from computer science can be found at the end of this article. But, perhaps, we should start with a few words about graphs in general. They are, of course, one of the prime objects of study in Discrete Mathematics. However, graphs are among the most ubiquitous models of both natural and human-made structures. In the natural and social sciences they model relations among species, societies, companies, etc. In computer science, they represent networks of communication, data organization, computational devices as well as the flow of computation, and more. In Mathematics, Cayley graphs are useful in Group Theory. Graphs carry a natural metric and are therefore useful in Geometry, and though they are “just ” one-dimensional complexes they are useful in certain parts of Topology, e.g. Knot Theory. In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems. The study of these models calls, then, for the comprehension of the significant structural properties of the relevant graphs. But are there nontrivial structural properties which are universally important? The property of being an expander seems significant in many of these mathematical, computational and physical contexts. It is not surprising that expanders are useful in the design and analysis of communication networks. What is less obvious is that expanders