Spectra of lifted Ramanujan graphs
A random -lift of a base graph is its cover graph on the vertices , where for each edge in there is an independent uniform bijection , and has all edges of the form . A main motivation for studying lifts is understanding Ramanujan graphs, and namely whether typical covers of such a graph are also Ramanujan.
Let be a graph with largest eigenvalue and let be the spectral radius of its universal cover. Friedman (2003) proved that every “new” eigenvalue of a random lift of is with high probability, and conjectured a bound of , which would be tight by results of Lubotzky and Greenberg (1995). Linial and Puder (2008) improved Friedman’s bound to . For -regular graphs, where and , this translates to a bound of , compared to the conjectured .
Here we analyze the spectrum of a random -lift of a -regular graph whose nontrivial eigenvalues are all at most in absolute value. We show that with high probability the absolute value of every nontrivial eigenvalue of the lift is . This result is tight up to a logarithmic factor, and for it substantially improves the above upper bounds of Friedman and of Linial and Puder. In particular, it implies that a typical -lift of a Ramanujan graph is nearly Ramanujan.
Over the last quarter of a century, expander graphs have played a vital role in a remarkable variety of areas, ranging from combinatorics to discrete geometry to theoretical computer science, while exhibiting deep connections to algebra and number theory. Notable applications of expanders, to name just a few, include the design of efficient communication networks, explicit error-correcting codes with efficient encoding and decoding schemes, derandomization of randomized algorithms, compressed sensing and the study of metric embeddings. See the expository article of Sarnak [Sarnak] on these intriguing objects, as well as the comprehensive survey of Hoory, Linial and Wigderson [HLW] demonstrating their many applications.
Informally, an expander is a graph where every small subset of the vertices has a relatively large edge boundary (see Section 2.1 for a formal definition). Most applications utilize -regular sparse expanders ( fixed), where it is well-known that expansion is related to the ratio between and , the second largest eigenvalue in absolute value of the adjacency matrix. The smaller is, the better the graph expansion becomes. As a consequence of the Alon-Boppana bound [Nilli] (see also [Friedman2]) where the -term tends to as the graph size tends to . Graphs for which are in that respect optimal expanders and are called Ramanujan graphs.
A proof that -regular expanders exist for any was given by Pinsker [Pinsker] in the early 70’s via a simple probabilistic argument. However, constructing good expanders explicitly is far more challenging and particularly important in applications (see [RVW] and the references therein), a task that was first achieved by Margulis [Margulis1]. Thereafter Ramanujan graphs were constructed explicitly in the seminal works of Lubotzky-Phillips-Sarnak [LPS] and Margulis [Margulis2], relying on deep number theoretic facts. Till this date Ramanujan graphs remain mysterious: Not only are there very few constructions for such graphs, but for instance it is not even known whether they exist for any . A striking result of Friedman [Friedman08] shows that almost every -regular graph on vertices is nearly Ramanujan — it has (the -term tends to as ). What proportion of these graphs satisfy remains an intriguing open problem.
The useful connection between expanders and the topological notion of covering maps was extensively studied by many authors over the last decade. Various properties of random covers of a given graph were thoroughly examined (see e.g. [AL1, AL2, BL, LP]), motivated in part by the problem of generating good (large) expanders from a given one.
Given two simple graphs and , a covering map is a homomorphism that for every induces a bijection between the edges incident to and those incident to . In the presence of such a covering map we say that is a lift (or a cover) of , or alternatively that is a quotient of . The fiber of is the set , and if is connected then all fibers are of the same cardinality, the covering number.
One well-known connection between covers and expansion is the fact that the universal cover of any -regular graph is the infinite -regular tree , whose spectral radius is , the eigenvalue threshold in Ramanujan graphs. In fact, Greenberg and Lubotzky [Greenberg] (cf. [Lubotzky]*Chapter 4) extended the Alon-Boppana bound to any family of general graphs in terms of the spectral radius of its universal cover (also see [Friedman1]*Theorem 4.1).
It is easy to see that any lift of a -regular base graph is itself -regular and inherits all the original eigenvalues of . One hopes that the lift would also inherit the expansion properties of its base graph, and in particular that almost every cover of a (small) Ramanujan graph will also be Ramanujan.
Since our focus here is on lifts of Ramanujan graphs (regular by definition) we restrict our attention to base graphs that are -regular for .
A random uniform -lift of a base graph (a uniform cover of with covering number ) has the following convenient description: It is the graph on the vertices , where for each edge in there is an independent uniform bijection , and has all edges of the form . The random lift of a complicated base-graph is thus a hybrid between the complex geometry of the quotient and the randomness due to the bijections.
In an important development in the study of the spectrum of random lifts Friedman [Friedman1] showed in 2003 that with high probability (w.h.p.) every “new” eigenvalue of an -lift (one that is not inherited from the base graph) is at most , where is the largest eigenvalue of the base-graph and is the spectral-radius of its universal cover. When the base-graph is -regular, and Friedman’s result implies that in its random -lift the largest absolute value of all nontrivial eigenvalues is w.h.p.
where denotes . Conversely, and by Alon-Boppana it is also at least . This lower bound was conjectured by Friedman [Friedman1] to be tight (for general graphs he conjectured that all new eigenvalues are at most as in the Greenberg-Lubotzky bound).
In a recent paper [LP], Linial and Puder were able to significantly improve Friedman’s bound and show that w.h.p. all the new eigenvalues of are at most . Consequently, an -lift of a -regular w.h.p. satisfies
When is a -regular expander with nontrivial eigenvalues of as is the case for Ramanujan graphs, this translates to .
Our main result in this work is the new near optimal upper bound of when is -regular with all nontrivial eigenvalues at most in absolute value. For it substantially improves the known bounds (1.1),(1.2), and when as in Ramanujan graphs it is tight up to a logarithmic factor, giving .
Let be a -regular graph with all nontrivial eigenvalues at most in absolute value and let be the spectral radius of its universal cover. Let be a random -lift of . For some explicit absolute constant , every nontrivial eigenvalue of is at most in absolute value except with probability .
Let be a -regular Ramanujan graph vertices and let be a random -lift of . With probability every nontrivial eigenvalue of is at most in absolute value, where is an explicit absolute constant.
Note that the above corollary implies that typical random -lifts of Ramanujan graphs are nearly Ramanujan. No attempt was made to optimize the explicit constant in Theorem 1. Finally, the statement of Theorem 1 holds even when the size of the base-graph is allowed to grow with provided that is large enough in comparison (e.g., ).
1.1. Related work
The previous bounds on the spectra of random -lifts of a fixed graph due to Friedman [Friedman1] and Linial and Puder [LP] were both obtained via Wigner’s trace method. The fact that the universal cover of a connected graph is the infinite tree of non-backtracking walks from an arbitrarily chosen vertex makes the trace method particularly useful for relating the new eigenvalues of the lift with , the spectral-radius of .
Even when the geometry of a graph is very well understood, bounding its nontrivial eigenvalues can be extremely challenging. For instance, a line of papers (cf. [BS, FKS, Friedman1, Friedman2]) established various bounds for the second eigenvalue of certain random regular graphs, culminating in the optimal bound for a uniformly chosen -regular graph on vertices, proved by Friedman [Friedman08] using highly sophisticated arguments.
It turns out that this model is essentially the special case of an -lift of a graph comprising a single vertex with self-loops: It is easy to see that for even, the random -regular graph obtained by independent uniform permutations in is equivalent to an -lift of the base-graph that has a single vertex with loops (this model is in fact contiguous to the uniform random -regular graph for , cf. e.g. [Wormald]). Unfortunately, when the base-graph features a complex and rich structure (e.g. the LPS-expanders, whose expansion properties hinge on a deep theorem of Selberg) it becomes significantly harder to control the spectrum of its lifts. Indeed, there are many examples of geometric properties that have been pinpointed precisely for the random regular graph yet remain unknown for arbitrary expanders (see [LS] for a recent such example). Estimating the number of closed walks in lifts of arbitrary Ramanujan graphs thus appears to be a formidable task.
In this work, the bounds obtained for the spectra of lifts of arbitrary expanders rely on an approach introduced by Kahn and Szemerédi [FKS], which is quite different from Wigner’s trace method. This approach was originally used to control the spectrum of a random regular graph, and several new ideas are required to adapt it to the more complicated geometry of the lifts considered here.
Another related problem in the study of spectra of lifts, yet of a rather different nature, considers the -lift of a base-graph (rather than -lifts of a small fixed graph). Bilu and Linial [BL] showed that for any -regular graph there exists a -lift with all new eigenvalues at most . This was shown by means of the Lovász Local Lemma, combined with the crucial observation of [BL] whereby the new eigenvalues correspond precisely to the eigenvalues of a signing of the adjacency matrix of (the matrix obtained by replacing a subset of its entries by ). In the absence of such a characterization when the covering number is large, different tools are needed for the problem studied here, where we seek a bound that holds for almost every -lift with sufficiently large.
1.2. The distribution of the second eigenvalue
As stated above, while a random -regular graph has second eigenvalue w.h.p. (the -term tending to as ), the probability that is Ramanujan is unknown. See [HLW, Sarnak] for some experimental results suggesting that this probability is bounded away from and . As this is essentially the simplest special case of a random lift (the quotient being a single vertex with self-loops), it is natural to conjecture the following:
For any Ramanujan graph there exist some such that its random -lift satisfies , where the -term tends to as .
Note that the limiting constant in the above conjecture depends on the base-graph , as it is plausible that its structure may affect the probability of being Ramanujan. For instance, a random cover of a complete graph on vertices might behave quite differently compared to lifts of a sparse -regular Ramanujan graph. However, as we next elaborate, experimental results lead us to suspect that up to normalization this is not the case.
Figure 2(a) shows the cumulative distribution function (c.d.f.) of where is the -lift of 3 different -regular Ramanujan base-graphs: (complete graph on vertices), the -vertex Petersen graph and the -vertex Dodecahedral graph. Each curve was evaluated from random lifts. In these simulations, the probability of a random lift being Ramanujan for each of these three base-graphs was bounded between and .
Somewhat surprisingly, aligning the number of vertices of the graph cover to be the same (via -lifts of the Dodecahedral graph, -lifts of the Petersen graph and -lifts of , giving -vertex covers for each graph) resulted in the curves of the individual c.d.f.’s coinciding fairly accurately. This is demonstrated in Figure 2(b), in light of which we speculate that the following stronger version of the statement of Conjecture 3 holds.
First, it seems plausible that for any integer the limiting distribution of the second eigenvalue of the random cover is independent of the base-graph. Namely, there exists a distribution on such that for any -regular Ramanujan graph on vertices, the distribution of for its random -lift converges to as . Second, the strong fit between the curves after aligning the total graph sizes suggests that even the rate of convergence to depends on rather than on the geometry of the base-graph or even its relative size (in case is allowed to depend to ). Of-course, one clearly needs some level of “burn-in” for the covering number compared to since the cover starts as Ramanujan at . For example, it may be that for any the total-variation distance between the distribution of and decays as a function of alone, namely that where depends only on and tends to as .
2. Preliminaries and outline of the proof
2.1. Combinatorial vs. algebraic expanders
The base-graph from Theorem 1 corresponds to the algebraic definition of an expander known as an -graph. An alternative closely-related criterion is the traditional definition of an expander graph in terms of its combinatorial edge or vertex expansion. Let be a -regular graph on vertices. The Cheeger constant of (also referred to as the edge isoperimetric constant) is defined as
where denotes and is the set of edges with exactly one endpoint in . We say that is a -edge-expander for some fixed if it satisfies . Similarly, one defines a -vertex-expander by replacing with the vertex boundary.
For as above the eigenvalues of the corresponding adjacency matrix are
by Perron-Frobenius. We say that is an -graph if for all . This notion was introduced by Alon in the 1980’s, motivated by the fact that when is much smaller than such graphs exhibit strong pseudo-random properties, resembling a random graph with edge density . A notable example of this is captured by the Expander Mixing Lemma: if are (not necessarily disjoint) subsets of vertices of an -graph then
where ([AS]*Chapter 9).
Relating the above two notions of expansion is the following well-known discrete analogue of Cheeger’s inequality bounding the first eigenvalue of a Riemannian manifold (Alon [Alon], Alon-Milman [AM], Dodziuk [Dodziuk], Jerrum-Sinclair [JS]):
See the survey [HLW] for further information on expanders.
2.2. Outline of the proof
We begin by describing the Kahn-Szemerédi [FKS] approach for obtaining an bound for random -regular graphs. Following Broder and Shamir [BS], the actual random graph model studied by [FKS] is the -regular graph obtained from the union of permutations, contiguous to the lift of a single vertex with loops.
Let be the random graph in mention and let denote its adjacency matrix. By the Rayleigh quotient principle, the second (in absolute value) eigenvalue of the graph can be written as
where is the trivial eigenvector of . To bound , the authors of [FKS] analyzed the maximal possible value of separating the contribution of the pairs to the bilinear form into two cases:
Heavy pairs: the contribution from those pairs where is suitably large. Here it is shown that w.h.p. the total contribution to by any pair of unit vectors is at most .
Light pairs: the remaining pairs . Here it was shown that two fixed vectors are unlikely to contribute more than to the bilinear form, and an -net argument was used to extend this result to any unit vectors .
Adapting this method to lifts of general graphs requires several additional ingredients. Even in the simpler setting of [FKS], some of the arguments are only sketched and might prove difficult to complete in detail. More crucially, in our case we have little knowledge of the base-graph , hence the study of both the “heavy” and “light” parts becomes significantly more involved.
First, our only input on is the magnitude of its second eigenvalue, which turns the analysis of the heavy part into a delicate optimization problem, requiring two levels of dyadic expansions of the potential contributions to the final bilinear form.
Second, the consideration of the light part relies on a non-trivial martingale argument which may be useful in other applications: In the absence of sufficient control over the expectation (due to the unknown contribution of the heavy part) we resort to an analysis of the increments in the corresponding Doob’s martingale and apply a Bernstein-Kolmogorov type large deviation inequality due to Freedman.
Throughout the paper we use to denote the base-graph, a -regular graph on vertices, and let denote its random -lift. The asymptotic notation is used under the assumption that .
For the sake of clarity, when addressing a vertex in we will typically denote it either by or by using indices and . Whenever are vertices in some graph whose identity is clear from the context, the abbreviation will denote that these two vertices are adjacent. For example, will usually stand for , which in turn implies that in by the definition of the lift.
Unless stated otherwise, all logarithms are using base and denotes the norm in the appropriate Euclidean space.
3. Heavy pairs and large cuts
Let be an -graph (that is, a -regular graph where all nontrivial eigenvalues are at most in absolute value) with adjacency matrix , and be a random -lift of with adjacency matrix . As mentioned before, the largest nontrivial eigenvalue of in absolute value is precisely
where denotes the trivial eigenvector.
3.1. Heavy pairs
We first analyze the typical contribution to from pairs with fairly large products. More precisely, we say that a pair is heavy if , and otherwise it is light. For , define to be the random variable
The next theorem estimates the contribution of the heavy pairs along the edges of . The exponent of in the requirement was selected to simplify the exposition and can be replaced by for any .
Let be an -graph with and let be a random -lift of for . Then with probability at least every with and satisfy and moreover .
The main ingredient in the proof of Theorem 3.1 is the following lemma, which provides an upper bound on the number of edges in a cut between subsets of vertices in .
Let be a random -lift of an -graph with and . Then except with probability , every two subsets of vertices with satisfy
Proof of Theorem 3.1.
Consider the following dyadic expansion of :
and identify any element in or with the vertex in . These definitions, together with the assumption on and , implies that
Furthermore, if and then a necessary condition for is that , and so
where we is the number of edges between the two subsets of vertices in corresponding to and .
To prove that , we set
and analyze the sum in (3.3) according to whether or not .
First, consider . As is -regular, trivially , yielding
where the second inequality used (3.2). Similarly,
and since occurs if and only if , altogether in this case
We now focus on . Consider the case where . Repeating the above argument, we now get
By symmetry, the same argument holds for the case , and we infer that
It remains to treat with . This will be achieved with the help of Proposition 3.2, which estimates the size of the cut between two subsets in case . Indeed, for we have
and therefore . Thus, Proposition 3.2 gives that with probability ,
For the first expression in the right-hand-side of (3.6), note that there are at most integers such that (here we used the fact that ). For each such value, we can combine (3.2) with Cauchy-Schwartz to get that
and summing over it follows that
For the second expression in (3.6), again recall that , and so
and the same applies to the analogous quantity for .
Altogether, these two estimates for (3.6) sum up to
Recalling (3.3), we deduce that
To obtain the statement on first note that since is -regular with probability for any two unit vectors . We have already established that, except with probability , every pair of vectors with norm at most satisfies . Hence,
where the last inequality holds for any sufficiently large . Reapplying the result on (along with the triangle inequality) now completes the proof of the theorem (with room to spare). ∎
3.2. Proof of Proposition 3.2
Write and where . Our assumption on then translates into
We aim to show that, except with probability , for any such we have , or in terms of , that
Define the following partition of the fibers according to a dyadic expansion of their proportion that is included in .
Notice that by these definitions, is the number of fibers with about vertices from , and so . In other words, there are about vertices of in fibers of type , and more precisely,
Similarly, we perform an analogous dyadic expansion for :
and again have that
and our bound on will follow from an analysis of the number of edges between the various types of ’s and ’s.
First, consider the case . Here we have and . Since there are edges in between any pair of fibers that correspond to adjacent vertices in , the Expander Mixing Lemma (see (2.1)) applied to the base-graph gives that
Recalling that (see (3.7)) it follows that
where the last inequality used the fact that and so .
Next, consider in case . There is a total of fibers in , thus by definition of the -lift of a -regular graph there are at most fibers, where may have vertices that contribute to . Since has at most vertices in each fiber, and each vertex has neighbors in , we deduce that
where the last inequality followed from (3.8). Similarly, we have
and conclude that
It remains to treat the case for all , where the required bound will only hold w.h.p.
Consider a prescribed set of pairs of vertices () in . We wish to bound the probability that by . By the independence of the different pairs of fibers in the lift, it clearly suffices to show this when all the ’s are on one fiber and all the ’s are on another, i.e., for some and all we have and . When then it is straightforward that this probability is indeed at most . To see this, expose the pairings of one by one, and note that for , the probability to match to , given that so far we succeeded in matching all the previous pairs, is .
Further note that, when considering potential edges between and , the case can only arise when , otherwise no two fibers have more than points of and respectively. Since we excluded the case , the above estimate holds for any of our sets .
and recall that and are contained in the fibers corresponding to and resp., and have at most and vertices on each of these respective fibers. Suppose first that the identity of the fibers and are given (we will account for these later). In this case, the number of configurations of the vertices of on the fibers can be upper bounded by if and by if , here using the well-known inequality . Similarly, the number of configuration of on the fibers is at most . For each such configuration of the vertices of there are at most pairs which may potentially be connected in . From the above estimate on the probability of pairs being adjacent in , it now follows that for any and choice of and ,
we then get