On the Connectivity and Giant Component Size of Random K-out Graphs Under Randomly Deleted Nodes

Eray Can Elumar, Mansi Sood, Osman Yagan

Abstract: Random K-out graphs, denoted <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbb{H}(n;K)$</tex> , are generated by each of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> nodes drawing <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K$</tex> out-edges towards <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K$</tex> 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbb{H}(n;K_{n},\gamma_{n})$</tex> , i.e., random K-out graph when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\gamma_{n}$</tex> of its nodes, selected uniformly at random, are deleted. First, we derive conditions for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$K_{n}$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> that ensure, with high probability (whp), the connectivity of the remaining graph when the number of deleted nodes is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\gamma_{n}=\Omega(n)$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\gamma_{n}=o(n)$</tex> , respectively. Next, we derive conditions for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbb{H}(n;K_{n}, \gamma_{n})$</tex> to have a giant component, i.e., a connected subgraph with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\Omega(n)$</tex> nodes, whp. This is also done for different scalings of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\gamma_{n}$</tex> 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.