Robustness of Random K-out Graphs

We consider a graph property known as r-robustness of the random K-out graphs. Random K-out graphs, denoted as $\mathbb{H}(n;K)$, are constructed as follows. Each of the n nodes select K distinct nodes uniformly at random, and then an edge is formed between these nodes. The orientation of the edges is ignored, resulting in an undirected graph. Random K-out graphs have been used in many applications including random (pairwise) key predistribution in wireless sensor networks, anonymous message routing in crypto-currency networks, and differentially-private federated averaging. r-robustness is an important metric in many applications where robustness of networks to disruptions is of practical interest, and r-robustness is especially useful in analyzing consensus dynamics. It was previously shown that consensus can be reached in an r-robust network for sufficiently large r even in the presence of some adversarial nodes. r-robustness is also useful for resilience against adversarial attacks or node failures since it is a stronger property than r-connectivity and thus can provide guarantees on the connectivity of the graph when up to r – 1 nodes in the graph are removed. In this paper, we provide a set of conditions for Kn and n that ensure, with high probability (whp), the r-robustness of the random K-out graph.