On the Connectivity and Giant Component Size of Random K-out Graphs Under Randomly Deleted Nodes

Random K-out graphs, denoted
$\mathbb{H}(n;K)$
, are generated by each of the
$n$
nodes drawing
$K$
out-edges towards
$K$
distinct nodes selected uniformly at random, and then ignoring the orientation of the arcs. Recently, random K-out graphs have been used in applications as diverse as random (pairwise) key predistribution in ad-hoc networks, anonymous message routing in crypto-currency networks, and differentially-private federated averaging. In many applications, connectivity of the random K-out graph when some of its nodes are dishonest, have failed, or have been captured is of practical interest. We provide a comprehensive set of results on the connectivity and giant component size of
$\mathbb{H}(n;K_{n},\gamma_{n})$
, i.e., random K-out graph when
$\gamma_{n}$
of its nodes, selected uniformly at random, are deleted. First, we derive conditions for
$K_{n}$
and
$n$
that ensure, with high probability (whp), the connectivity of the remaining graph when the number of deleted nodes is
$\gamma_{n}=\Omega(n)$
and
$\gamma_{n}=o(n)$
, respectively. Next, we derive conditions for
$\mathbb{H}(n;K_{n}, \gamma_{n})$
to have a giant component, i.e., a connected subgraph with
$\Omega(n)$
nodes, whp. This is also done for different scalings of
$\gamma_{n}$
and upper bounds are provided for the number of nodes outside the giant component. Simulation results are presented to validate the usefulness of the results in the finite node regime.