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

TL;DR

This paper analyzes the connectivity and giant component size of random K-out graphs under random node deletion, providing probabilistic conditions for maintaining connectivity and large components, with validation through simulations.

Contribution

It offers new probabilistic thresholds for connectivity and giant component existence in K-out graphs with node failures, extending understanding of their robustness.

Findings

Conditions for connectivity with high probability under node deletion

Thresholds for the emergence of a giant component

Simulation validation of theoretical results

Abstract

Random K-out graphs, denoted $\mathbb{H}(n;K)$, 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 $\mathbb{H}(n;K_n,\gamma_n)$, i.e., random K-out graph when $\gamma_n$ 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 $\gamma_n=\Omega(n)$ and $\gamma_n=o(n)$, respectively. Next, we derive conditions for $\mathbb{H}(n;K_n,\gamma_n)$ to have a giant component, i.e., a connected subgraph with $\Omega(n)$ nodes, whp. This is also done for different scalings of $\gamma_n$ 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.