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, Kn arcs towards Kn distinct nodes) selected uniformly at random. The orientation of the arcs is ignored yielding the inhomogeneous random K out graph, denoted by H(n; <mu>, K-n). It was recently established that 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 H(n; mu, K-n). We first show that the trivial condition of K-n >= 2 for all n is sufficient to ensure that 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 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 >= 2 whp. Further, when d(n) = o(n) nodes are removed from H(n; mu, K-n), the remaining nodes contain a giant component of size n(1 - o(1)) whp for all K-n > 2. We present numerical results to demonstrate the size of the largest connected component when the number of nodes is finite.