Introduction to expander graphs Michael A. Nielsen1, ∗ 1School of Physical Sciences, The University of Queensland, Brisbane, Queensland 4072, Australia (Dated: June 22, 2005) I. INTRODUCTION TO EXPANDERS applied. I’m not learning about expanders with any spe- cific intended application in mind, but rather because Expander graphs are one of the deepest tools of theoret- they seem to behind some of the deepest insights we’ve ical computer science and discrete mathematics, popping had in recent years into information and computation. up in all sorts of contexts since their introduction in the What is an expander graph? Informally, it’s a graph 1970s. Here’s a list of some of the things that expander G = (V, E) in which every subset S of vertices expands graphs can be used to do. Don’t worry if not all the items quickly, in the sense that it is connected to many vertices on the list make sense: the main thing to take away is the in the set S of complementary vertices. Making this def- sheer range of areas in which expanders can be applied. inition precise is the main goal of the remainder of this section. • Reduce the need for randomness: That is, ex- Suppose G = (V, E) has n vertices. For a subset S of panders can be used to reduce the number of ran- V we define the edge boundary of S, ∂S, to be the set dom bits needed to make a probabilistic algorithm of edges connecting S to its complement, S. That is, ∂S work with some desired probability. consists of all those edges (v, w) such that v ∈ S and w 6∈ S. The expansion parameter for G is defined by • Find good error-correcting codes: Expanders can be used to construct error-correcting codes for pro- |∂S| h(G) ≡ min , (1) tecting information against noise. Most astonish- S:|S|≤n/2 |S| ingly for information theorists, expanders can be used to find error-correcting codes which are effi- where |X| denotes the size of a set X. ciently encodable and decodable, with a non-zero One standard condition to impose on expander graphs rate of transmission. This is astonishing because is that they be d-regular graphs, for some constant d, i.e., finding codes with these properties was one of the they are graphs in which every vertex has the same de- holy grails of coding theory for decades after Shan- gree, d. I must admit that I’m not entirely sure why this non’s pioneering work on coding and information d-regularity condition is imposed. One possible reason is theory back in the 1940s. that doing this simplifies a remarkable result which we’ll discuss later, relating the expansion parameter h(G) to • A new proof of PCP: One of the deepest results the eigenvalues of the adjacency matrix of G. (If you in computer science is the PCP theorem, which don’t know what the adjacency matrix is, we’ll give a tells us that for all languages L in NP there is a definition later.) randomized polyonomial-time proof verifier which Example: Suppose G is the complete graph on n ver- need only check a constant number of bits in a pur- tices, i.e., the graph in which every vertex is connected ported proof that x ∈ L or x 6∈ L, in order to de- to every other vertex. Then for any vertex in S, each termine (with high probability of success) whether vertex in S is connected to all the vertices in S, and the proof is correct or not. This result, originally thus |∂S| = |S| × |S| = |S|(n − |S|). It follows that the established in the earlier 1990s, has recently been expansion parameter is given by given a new proof based on expanders. lnm h(G) = min n − |S| = . (2) What’s remarkable is that none of the topics on this S:|S|≤n/2 2 list appear to be related, a priori, to any of the other topics, nor do they appear to be related to graph the- For reasons I don’t entirely understand, computer sci- ory. Expander graphs are one of these powerful unifying entists are most interested in the case when the degree, tools, surprisingly common in science, that can be used d, is a small constant, like d = 2, 3 or 4, not d = n − 1, to gain insight into an an astonishing range of apparently as is the case for the complete graph. Here’s an example disparate phenomena. with constant degree. I’m not an expert on expanders. I’m writing these Example: Suppose G is an n × n square lattice in 2 notes to help myself (and hopefully others) to under- dimensions, with periodic boundary conditions (so as to stand a little bit about expanders and how they can be make the graph 4-regular). Then if we consider a large connected subset of the vertices, S, it ought to be plausi- ble that that the edge boundary set ∂S contains roughly one edge for each vertex on the perimeter of the region S. p ∗[email protected] and www.qinfo.org/people/nielsen We expect there to be roughly |S| such vertices, since 2 we are in two dimensions, and so |∂S|/|S| ≈ 1/p|S|. Example: In this example the family of graphs is in- Since the graph can contain regions S with up to O(n2) dexed by a prime number, p. The set of vertices for vertices, we expect the graph Gp is just the set of points in Zp, the field of integers modulo p. We construct a 3-regular graph by 1 −1 h(G) = O (3) connecting each vertex x 6= 0 to x − 1, x + 1 and x . n The vertex x = 0 is connected to p − 1, 0 and 1. Ac- cording to the lecture notes by Linial and Wigderson, for this graph. I do not know the exact result, but am this was proved to be a family of expanders by Lubotsky, confident that this expression is correct, up to constant Phillips and Sarnak in 1988, but I don’t know a lower factors and higher-order corrections. It’d be a good ex- bound on the expansion parameter. Note that even for ercise to figure out exactly what h(G) is. Note that as p = O(2n) we can do basic operations with this graph the lattice size is increased, the expansion parameter de- (e.g., random walking along its vertices), using compu- creases, tending toward 0 as n → ∞. tational resources that are only polynomial in time and Example: Consider a random d-regular graph, in space. This makes this graph potentially far more use- which each of n vertices is connected to d other vertices, ful in applications than the random graphs considered chosen at random. Let S be a subset of at most n/2 earlier. vertices. Then a typical vertex in S will be connected Example: A similar but slightly more complex exam- to roughly d × |S|/n vertices in S, and thus we expect ple is as follows. The vertex set is Z × Z , where m |∂S| ≈ d × |S||S|/n, and so m m is some positive integer, and Zm is the additive group of |∂S| |S| integers modulo m. The degree is 4, and the vertex (x, y) ≈ d . (4) has edges to (x ± y, y), and (x, x ± y), where all addition |S| n is done modulo m. Something which concerns me a lit- Since |S| has its minimum at approximately n/2 it follows tle about this definition, but which I haven’t resolved, is that h(G) ≈ d/2, independent of the size n. what happens when m is even and we choose y = m/2 Exercise: Show that a disconnected graph always has so that, e.g., the vertices (x + y, y) and (x − y, y) coin- expansion parameter 0. cide with one another. We would expect this duplication In each of our examples, we haven’t constructed just to have some effect on the expansion parameter, but I a single graph, but rather an entire family of graphs, haven’t thought through exactly what. indexed by some parameter n, with the property that as n gets larger, so too does the number of vertices in the graph. Having access to an entire family in this way turns III. GRAPHS AND THEIR ADJACENCY out to be much more useful than having just a single MATRICES graph, a fact which motivates the definition of expander graphs, which we now give. How can we prove that a family of graphs is an ex- Suppose we have a family Gj = (Vj,Ej) of d-regular pander? Stated another way, how does the expansion graphs, indexed by j, and such that |Vj| = nj for some parameter h(G) vary as the graph G is varied over all increasing function nj. Then we say that the family {Gj} graphs in the family? is a family of expander graphs if the expansion parame- One way of tackling the problem of computing h(G) is ter is bounded strictly away from 0, i.e., there is some to do a brute force calculation of the ratio |∂S|/|S| for small constant c such that h(Gj) ≥ c > 0 for all Gj in every subset S of vertices containing no more than half the family. We’ll often abuse nomenclature slightly, and the vertices in the graph. Doing this is a time-consuming just refer to the expander {Gj}, or even just G, omitting task, since if there are n vertices in the graph, then there explicit mention of the entire family of graphs.