Characterization of the relationship between redundancy and robustness

Establish a full characterization of the relationship between the $(r,r')$-redundancy condition and the $r$-robustness property for general communication graphs.

Background

The paper introduces $(r,r')$-redundancy as a tractable topological condition supporting fully Byzantine-resilient actor-critic multi-agent reinforcement learning under Byzantine edge attacks. It also recalls the established notion of $r$-robustness, which is used in several existing resilient distributed-learning methods but is computationally difficult to verify in general.

The paper proves that a specific class of graphs constructed according to Proposition 1 is $r$-robust when $n\geq 2r-1$. However, this result applies only to that particular construction and does not determine the general relationship between $(r,r')$-redundancy and $r$-robustness for arbitrary graphs. A complete characterization of their implications and possible equivalence remains unresolved.