Robustness of the 2-Choices Dynamics to Node Failures

TL;DR

This work analyzes the robustness of the 2-choices consensus dynamics when nodes occasionally fail to follow the update rule, modeling failures with a constant probability $\alpha$. The authors show a sharp phase transition at $\alpha=1/3$: on complete graphs (and, more generally, on well-behaved expanders), the system exhibits metastability with $t_{\text{mix}}$ and $\mathbb{E}[T_{1/2}]$ growing as $\exp(\Theta(n))$ for $\alpha<1/3$, but rapid, $O(\log n)$ behavior for $\alpha>1/3$. They extend these results to a broad class of general graphs characterized by spectral gap and degree-homogeneity conditions, proving analogous exponential vs. logarithmic scaling regimes and providing explicit drift-based criteria via a function $F_{\alpha,L}$. The paper combines spectral graph theory, continuous-time Markov chain analysis, and drift arguments to establish sharp robustness thresholds, offering insights into the persistence of initial majority information under substantial random perturbations and enabling analyses of similar nonlinear dynamics on networks. The results have implications for designing resilient distributed consensus protocols in both dense and sparse networks.

Abstract

In many applications, it becomes necessary for a set of distributed network nodes to agree on a common value or opinion as quickly as possible and with minimal communication overhead. The classical 2-choices rule is a well-known distributed algorithm designed to achieve this goal. Under this rule, each node in a network updates its opinion at random instants by sampling two neighbours uniformly at random and then adopting the common opinion held by these neighbours if they agree. For a sufficiently well-connected network of $n$ nodes and two initial opinions, this simple rule results in the network being absorbed in a consensus state in $O(\log n)$ time (with high probability) and the consensus is obtained on the opinion held by the majority of nodes initially. In this paper, we study the robustness of this algorithm to node failures. In particular, we assume that with a constant probability $α$, a node may fail to update according to the 2-choices rule and erroneously adopt any one of the two opinions uniformly at random. This is a strong form of failure under which the network can no longer be absorbed in a consensus state. However, we show that as long as the error probability $α$ is less than a threshold value, the network is able to retain the majority support of the initially prevailing opinion for an exponentially long time ($Ω(\poly(\exp(n)))$). In contrast, when the error probability is above a threshold value, we show that any opinion quickly ($O(\log n)$ time) loses its majority support and the network reaches a state where (nearly) an equal proportion of nodes support each opinion. We establish the above phase transition in the dynamics for both complete graphs and expander graphs with sufficiently large spectral gaps and sufficiently homogeneous degrees. Our analysis combines spectral graph theory with Markov chain mixing and hitting time analyses.

Figures (4)

• Figure 4.1: Mixing time ($t_{\text{mix}}$) as a function of the network size ($n$) for different values of the failure probability ($\alpha$) for complete graphs. (a)$\alpha=0.2 <1/3$. (b)$\alpha=0.4 > 1/3$. In both cases, to compute $t_{\text{mix}}$, the transient distribution of $A(t)$ is found by solving the forward Kolmogorov equation, and the stationary distribution $\boldsymbol{\pi}$ is calculated by solving the detailed balance equations $\boldsymbol{\pi}(\{a\})q_+(a)=\boldsymbol{\pi}(\{a+1\})q_-(a+1)$ for $a\in \{0,1,\ldots,n-1\}$.
• Figure 4.2: Plot of $f_{n,\alpha}$ as a function of the fraction of nodes with opinion $1$ for different values of $\alpha$. The arrows in both plots indicate the direction of the drift (right for positive drift and left for negative drift). In both plots, we have chosen $n=100$. (a) For $\alpha=0.2 <1/3$ there are three distinct real roots of $f_{n,\alpha}$, which, in the increasing order of their values, are at $r_{n,\alpha}, 1/2,$ and $1-r_{n,\alpha}$. All trajectories are attracted either toward $r_{n,\alpha}$ or toward $1-r_{n,\alpha}$.(b) For $\alpha = 0.4>1/3$ the unique root of $f_{n,\alpha}$ is at $1/2$ and all trajectories are attracted toward this root.
• Figure 5.1: Plots showing $T_{1/2}$ as a function of network size for Erdős--Rényi random graphs with mean degree $2\log(n)$ for failure probabilities both below and above the critical threshold of $1/3$. A number of runs were performed on a fixed instance of the graph starting at state $\mathbf{0}$, and the results were averaged. This was then repeated for a number of different Erdős--Rényi graph realisations with the same parameters and then the results were further averaged across these realisations. (a)$\alpha=0.2$. The number of simulations per graph was 10 and 5 graphs were generated per $n$ value. (b)$\alpha=0.5$. The number of simulations per graph was 30 and 10 graphs were generated per $n$ value.
• Figure 5.2: Plots showing $T_{1/2}$ as a function of network size for random $d$-regular graphs with $d=10$ for different values of the failure probability $\alpha$. A number of runs were performed on a fixed instance of the graph starting at state $\mathbf{0}$, and the results were averaged. This was then repeated for a number of different random $d$-regular graph realisations with the same parameters and then the results were further averaged across these realisations. (a)$\alpha=0.2$. The number of runs per graph was 20 and 5 graph instances were generated per $n$ value. (b)$\alpha=0.5$. The number of runs per graph was 50 and 4 graphs were generated per $n$ value.

Theorems & Definitions (18)

• Lemma 3.1
• Corollary 3.2
• proof
• Theorem 4.1
• Lemma 4.2
• Lemma 4.3
• Lemma 4.4
• Theorem 5.1
• Corollary 5.2
• Lemma 5.3