An Alon-Boppana theorem for powered graphs and generalized Ramanujan graphs

TL;DR

This paper generalizes the Alon-Boppana theorem to powered graphs, introduces a new notion of Ramanujan graphs for these, and demonstrates their robustness in random and adversarial settings, with applications in community detection.

Contribution

It extends spectral bounds to powered graphs, proposes a more robust Ramanujan graph definition, and applies these results to community testing under adversarial conditions.

Findings

Powered graphs can be almost Ramanujan even with local irregularities.

The powering operator provides a more robust notion of Ramanujan graphs than nonbacktracking operators.

Robust community testing is possible in adversarial block models using powered graph properties.

Abstract

The r-th power of a graph modifies a graph by connecting every vertex pair within distance r. This paper gives a generalization of the Alon-Boppana Theorem for the r-th power of graphs, including irregular graphs. This leads to a generalized notion of Ramanujan graphs, those for which the powered graph has a spectral gap matching the derived Alon-Boppana bound. In particular, we show that certain graphs that are not good expanders due to local irregularities, such as Erdos-Renyi random graphs, become almost Ramanujan once powered. A different generalization of Ramanujan graphs can also be obtained from the nonbacktracking operator. We next argue that the powering operator gives a more robust notion than the latter: Sparse Erdos-Renyi random graphs with an adversary modifying a subgraph of log(n)^c$ vertices are still almost Ramanujan in the powered sense, but not in the nonbacktracking sense. As an application, this gives robust community testing for different block models.