Local Improvement Gives Better Expanders

It has long been known that random regular graphs are with high probability good expanders. This was first established in the 1980s by Bollob'as by directly calculating the probability that a set of vertices has small expansion and then applying the union bound. In this paper we improve on this analysis by relying on a simple high-level observation: if a graph contains a set of vertices with small expansion then it must also contain such a set of vertices that is locally optimal, that is, a set whose expansion cannot be made smaller by exchanging a vertex from the set with one from the set’s complement. We show that the probability that a set of vertices satisfies this additional property is significantly smaller. Thus, after again applying the union bound, we obtain improved lower bounds on the expansion of random $\Delta$-regular graphs for $\Delta\ge 4$. In fact, the gains from this analysis increase as $\Delta$ grows, a fact we explain by extending our technique to general $\Delta$. Thus, in the end we obtain an improvement not only for some small special cases but on the general asymptotic bound on the expansion of $\Delta$-regular graphs given by Bollob'as.

💡 Research Summary

The paper revisits the classic problem of estimating the expansion of random Δ‑regular graphs, a topic that dates back to Bollobás’s seminal work in the 1980s. Bollobás proved that such graphs are, with high probability, good expanders by directly computing the probability that a given vertex set S has a small edge boundary ∂S and then applying a union bound over all possible S. While elegant, this approach treats every set S as equally likely to be “bad” and ignores any structural information that might make some sets far less probable. The authors of the present work introduce a simple yet powerful high‑level observation: if a graph contains any set with small expansion, it must also contain a locally optimal set—one that cannot be improved by swapping a vertex inside the set with a vertex outside. By focusing on locally optimal sets, the authors dramatically reduce the number of candidate sets that need to be considered, because the local optimality condition imposes strong combinatorial constraints on the degree distribution of vertices inside and outside the set.

The paper defines a set S⊂V to be locally optimal if for every pair (u∈S, v∈V∖S) the exchange S′=(S{u})∪{v} does not decrease the size of the edge boundary, i.e., |∂S′|≥|∂S|. This definition guarantees two crucial properties. First, any set with small expansion can be transformed, via a sequence of improving swaps, into a locally optimal set with no larger expansion; thus the existence of a “bad” set implies the existence of a “bad” locally optimal set. Second, the locally optimal condition forces the internal and external degree profiles of the vertices to satisfy a set of inequalities that are highly unlikely under the random stub‑matching model that generates Δ‑regular graphs.

To quantify this unlikelihood, the authors analyze the stub‑matching process: each vertex contributes Δ half‑edges (stubs) which are paired uniformly at random. For a fixed size k, the numbers of internal edges, external edges, and the degree sequence of the cut are jointly distributed according to a multinomial law. Adding the locally optimal constraints translates into additional linear inequalities on these counts. Using Stirling’s approximation for factorials together with Chernoff‑type concentration bounds, the authors derive an exponential‑type upper bound on the probability that a random Δ‑regular graph contains a locally optimal set of size k with expansion at most α. This bound is substantially tighter than the one obtained by Bollobás, which only accounted for the raw multinomial probabilities.

Summing the refined probabilities over all k (the union bound) yields a new lower bound on the edge‑expansion h(G) of a random Δ‑regular graph. For Δ≥4 the authors obtain
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