Spectral Expanding Expanders.

Dinitz, Schapira, and Valadarsky [5] introduced the intriguing notion of expanding expanders - a family of expander graphs with the property that every two consecutive graphs in the family differ only on a small number of edges. Such a family allows one to add and remove vertices with only few edge updates, making them useful in dynamic settings such as for datacenter network topologies and for the design of distributed algorithms for self-healing expanders. [5] constructed explicit expanding-expanders based on the Bilu-Linial construction of spectral expanders [3]. The construction of expanding expanders, however, ends up being of edge expanders, thus, an open problem raised by [5] is to construct spectral expanding expanders (SEE). In this work, we resolve this question by constructing SEE with spectral expansion which, like [3], is optimal up to a poly-logarithmic factor, and the number of edge updates is optimal up to a constant. We further give a simple proof for the existence of SEE that are close to Ramanujan up to a small additive term. As in [5], our construction is based on interpolating between a graph and its lift. However, to establish spectral expansion, we carefully weigh the interpolated graphs, dubbed partial lifts, in a way that enables us to conduct a delicate analysis of their spectrum. In particular, at a crucial point in the analysis, we consider the eigenvectors structure of the partial lifts.