  Random K-out graphs are garnering interest in designing distributed systems
including secure sensor networks, anonymous crypto-currency networks, and
differentially-private decentralized learning. In these security-critical
applications, it is important to model and analyze the resilience of the
network to node failures and adversarial captures. Motivated by this, we
analyze how the connectivity properties of random K-out graphs vary with the
network parameters $K$, the number of nodes ($n$), and the number of nodes that
get failed or compromised ($\gamma_n$). In particular, we study the conditions
for achieving \emph{connectivity} {with high probability} and for the existence
of a \emph{giant component} with formal guarantees on the size of the largest
connected component in terms of the parameters $n,~K$, and $\gamma_n$. Next, we
analyze the property of \emph{$r$-robustness} which is a stronger property than
connectivity and leads to resilient consensus in the presence of malicious
nodes. We derive conditions on $K$ and $n$ under which the random K-out graph
achieves r-robustness with high probability. We also provide extensive
numerical simulations and compare our results on random K-out graphs with known
results on Erd\H{o}s-R\'enyi (ER) graphs.
