On the Size of the Giant Component in Inhomogeneous Random K-out Graphs
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Inhomogeneous random K-out graphs were recently introduced to model heterogeneous sensor networks secured by random pairwise key predistribution schemes. First, each of the n nodes is classified as type-1 (respectively, type-2) with probability 0<μ<1 (respectively, 1-μ) independently from each other. 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, and then the orientation of the arcs is ignored. It was recently established that this graph, denoted by ℍ(n;μ,K_n), is connected with high probability (whp) if and only if K_n=ω(1). In other words, if K_n=O(1), then ℍ(n;μ,K_n) has a positive probability of being not connected as n gets large. Here, we study the size of the largest connected subgraph of ℍ(n;μ,K_n) when K_n = O(1). We show that the trivial condition of K_n ≥ 2 for all n is sufficient to ensure that inhomogeneous K-out graph has a connected component of size n-O(1) whp. Put differently, even with K_n =2, all but finitely many nodes will form a connected sub-network in this model under any 0<μ<1. We present an upper bound on the probability that more than M nodes are outside of the largest component, and show that this decays as O(1){-M(1-μ)(K_n-1)} +o(1). Numerical results are presented to demonstrate the size of the largest connected component when the number of nodes is finite.
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