On the Size of the Giant Component in Inhomogeneous Random K-out Graphs

Abstract

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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , arcs towards Kam, 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> ,), is connected with high probability (whp) if and only if K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , = w(1). In other words, if K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , = O(1), then H(n; μ, K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> ,) 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> ,) when K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , = O(1). We show that the trivial condition of Kam, ≥ 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , = 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) exp{-M(1 - μ)(K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> , - 1)} + o(1). Numerical results are presented to demonstrate the size of the largest connected component when the number of nodes is finite.