Existence and Size of the Giant Component in Inhomogeneous Random K-Out Graphs

Random K-out graphs are receiving attention as a model to construct sparse yet well-connected topologies in distributed systems including sensor networks, federated learning, and cryptocurrency networks. In response to the growing heterogeneity in emerging real-world networks, where nodes differ in resources and requirements, inhomogeneous random K-out graphs were proposed recently. In this model, first, each of the
$n$
nodes is classified as type-1 (respectively, type-2) with probability
$\mu $
(respectively,
$1-\mu$
) independently from the others, where
$0 < \mu < 1$
. Next, each type-1 (respectively, type-2) node draws 1 arc towards a node (respectively,
$K_{n}$
arcs towards
$K_{n}$
distinct nodes) selected uniformly at random. The orientation of the arcs is ignored yielding the inhomogeneous random K-out graph, denoted by
$\mathbb {H}(n;\mu,K_{n})$
. It was recently established that
$\mathbb {H}(n;\mu,K_{n})$
is connected with high probability (whp) if and only if
$K_{n}=\omega (1)$
. Motivated by practical settings where establishing links is costly and only a bounded choice of
$K_{n}$
is feasible (
$K_{n} = O(1)$
), we study the size of the largest connected sub-network of
$\mathbb {H}(n;\mu,K_{n})$
. We first show that the trivial condition of
$K_{n} \geq 2$
for all
$n$
is sufficient to ensure that
$\mathbb {H}(n;\mu,K_{n})$
contains a giant component of size
$n-O(1)$
whp. Next, to model settings where nodes can fail or get compromised, we investigate the size of the largest connected sub-network in
$\mathbb {H}(n;\mu,K_{n})$
when
$d_{n}$
nodes are selected uniformly at random and removed from the network. We show that if
$d_{n}=O(1)$
, a giant component of size
$n- {O}(1)$
persists for all
$K_{n} \geq 2$
whp. Further, when
$d_{n}=o(n)$
nodes are removed from
$\mathbb {H}(n;\mu,K_{n})$
, the remaining nodes contain a giant component of size
$n(1-o(1))$
whp for all
$K_{n} \geq 2$
. We present numerical results to demonstrate the size of the largest connected component when the number of nodes is finite.