On the Connectivity and Giant Component Size of Random K-out Graphs Under Randomly Deleted Nodes
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Random K-out graphs, denoted 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 ℍ(n;K_n,γ_n), i.e., random K-out graph when γ_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 γ_n=Ω(n) and γ_n=o(n), respectively. Next, we derive conditions for ℍ(n;K_n,γ_n) to have a giant component, i.e., a connected subgraph with Ω(n) nodes, whp. This is also done for different scalings of γ_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.
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