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Trevisan’s L^1-style Cheeger inequality

My expander class at ETH continues, and for the moment the course notes are kept updated (and in fact I have some more in the notes than I did in class).

Last week, I presented the inequalities relating the definition of expanders based on the expansion ratio with the “spectral” definition. In fact, I used the random walk definition first, which is only very slightly different for general families of graphs (with bounded valency), and is entirely equivalent for regular graphs. For a connected k-regular graph \Gamma, the inequalities state that
\frac{2}{k}\lambda_1(\Gamma)\leq h(\Gamma)\leq k\sqrt{2\lambda_1(\Gamma)},
where:

  • \lambda_1(\Gamma) is the smallest non-zero eigenvalue of the normalized Laplace operator \Delta, i.e., the operator acting on functions on the vertex set by
    \Delta\varphi(x)=\varphi(x)-\frac{1}{k}\sum_{y\text{ adjacent to x}}{\varphi(y)}.
  • h(\Gamma) is the expansion ratio of \Gamma, given by h(\Gamma)=\min_{1\leq |W|\leq |\Gamma|/2}\frac{|E(W)|}{|W|}, the set E(W) in the numerator being the set of edges in \Gamma with one extremity in W and one outside.

The left-hand inequality is rather simple, once one knows (or notices) the variational characterization
\lambda_1(\Gamma)=\min_{\varphi\perp 1}\frac{\langle \Delta\varphi,\varphi\rangle}{\|\varphi\|^2}
of the spectral gap: basically, for a characteristic function \varphi of a set W of vertices, the right-hand side of this expression reduces (almost) to the corresponding ratio |E(W)|/|W|. The second inequality, which is called the discrete Cheeger inequality, is rather more subtle.

The proof I presented is the one presented by L. Trevisan in his recent lectures on expanders (see, specifically, Lecture 4). Here, the idea is to construct a test set for h(\Gamma) of the form
W_t=\{x\in V_{\Gamma}\,\mid\, \varphi(x)\leq t\},
for a suitable real-valued function \varphi on the group. The only minor difference is that, in order to motivate the distribution of a random “threshold” t that Trevisan uses to construct those test sets, I first stated a general lemma where the threshold is picked according to a fairly general probability measure. Then I first showed what happens for a uniform choice in this interval, to motivate the “better” one that leads to Cheeger’s inequality.

As it turns out, I just found an older blog post of Trevisan explaining his proof, where he also discusses first the computations based on the uniform measure supported on [\min \varphi,\max\varphi], and explains why it is problematic.

But there is some good in this inequality. First, the precise form it takes (which I didn’t find elsewhere, and will therefore name Trevisan’s inequality) is
h(\Gamma)\leq \frac{A}{B}
where, for any non-constant real-valued function \varphi with minimum a, maximum b and median t_0, we have
A=\frac{1}{2}\sum_{x,y\in V}{a(x,y)|\varphi(x)-\varphi(y)|},
(where a(x,y) is the number of edges between x and y) and
B=\sum_{x\in V}{|\varphi(x)-t_0|}.

The interesting fact is that this can be sharper than Cheeger’s inequality. Indeed, this happens at least for the family of graphs \Gamma_n which are the Cayley graphs of \mathfrak{S}_n with respect to the generating set
S_n=\{(1\ 2),(1\ 2\ \cdots\ n)^{\pm 1}\}.
From an analysis of the random walks by Diaconis and Saloff-Coste (see Example 1 in Section 3 of their paper), it is known that
\lambda_1(\Gamma_n)\asymp \frac{1}{n^3},
and Cheeger’s inequality leads to
h(\Gamma_n)\ll n^{-3/2}.
However (unless I completely messed up my computations…), one sees that the test function used by Diaconis and Saloff-Coste, namely the cyclic distance between \sigma^{-1}(1) and \sigma^{-1}(2), satisfies
\frac{A}{B}\ll \frac{1}{n^2},
and therefore we get a better upper-bound
h(\Gamma)\ll \frac{1}{n^2}
using Trevisan’s inequality.

(Incidentally, we see here an easy example of a sequence of graphs which forms an esperantist family, but is not an expander.)

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