Explicit Expanding Expanders

Deterministic constructions of expander graphs have been an important topic of research in computer science and mathematics, with many well-studied constructions of infinite families of expanders. In some applications, though, an infinite family is not enough: we need expanders which are “close” to each other. We study the following question: Construct an an infinite sequence of expanders G₀, G₁, … , such that for every two consecutive graphs G_i and G_{i + 1}, G_{i + 1} can be obtained from G_i by adding a single vertex and inserting/removing a small number of edges, which we call the expansion cost of transitioning from G_i to G_{i + 1}. This question is very natural, e.g., in the context of datacenter networks, where the vertices represent racks of servers, and the expansion cost captures the amount of rewiring needed when adding another rack to the network. We present an explicit construction of d-regular expanders with expansion cost at most 5d2, for any d≥ 6. Our construction leverages the notion of a “2-lift” of a graph. This operation was first analyzed by Bilu and Linial (Combinatorica 26(5): 495–519, 2006), who repeatedly applied 2-lifts to construct an infinite family of expanders which double in size from one expander to the next. Our construction can be viewed as a way to “interpolate” between Bilu–Linial expanders with low expansion cost while preserving good edge expansion throughout. While our main motivation is centralized (datacenter networks), we also get the best-known distributed expander construction in the “self-healing” model.