On the Robustness, Connectivity and Giant Component Size of Random K-out Graphs

TL;DR

This paper analyzes the connectivity, giant component size, and robustness of random K-out graphs, providing formal guarantees and conditions for network resilience against failures and adversaries, with comparisons to Erdős-Rényi graphs.

Contribution

It offers a comprehensive analysis of the resilience properties of random K-out graphs, including connectivity, giant component size, and r-robustness, with formal probabilistic guarantees.

Findings

Conditions for high-probability connectivity and giant component existence.

Derivation of parameters for achieving r-robustness.

Numerical simulations comparing K-out and Erdős-Rényi graphs.

Abstract

Random K-out graphs are garnering interest in designing distributed systems including secure sensor networks, anonymous crypto-currency networks, and differentially-private decentralized learning. In these security-critical applications, it is important to model and analyze the resilience of the network to node failures and adversarial captures. Motivated by this, we analyze how the connectivity properties of random K-out graphs vary with the network parameters $K$, the number of nodes ($n$), and the number of nodes that get failed or compromised ($\gamma_n$). In particular, we study the conditions for achieving \emph{connectivity} {with high probability} and for the existence of a \emph{giant component} with formal guarantees on the size of the largest connected component in terms of the parameters $n,~K$, and $\gamma_n$. Next, we analyze the property of \emph{$r$-robustness} which is a stronger property than connectivity and leads to resilient consensus in the presence of malicious nodes. We derive conditions on $K$ and $n$ under which the random K-out graph achieves r-robustness with high probability. We also provide extensive numerical simulations and compare our results on random K-out graphs with known results on Erd\H{o}s-R\'enyi (ER) graphs.