Time Complexity of Broadcast and Consensus for Randomized Oblivious Message Adversaries

TL;DR

The paper investigates the time complexity of Broadcast and Consensus in dynamic networks under a stochastic, randomized oblivious adversary. It introduces Uniformly Random Trees and extensions with limited adversarial control (randomly chosen edges, Byzantine nodes) and analyzes information spread using binomial growth, stochastic dominance, and rooted-tree counting, yielding tight high-probability bounds. Key results include Broadcast completing in $O(32 c \ln n)$ rounds on URT and $O(k+\ln n)$ rounds under adversarial edges (with $k \le \frac{2}{3}n-1$ for consensus), with extensions to All-to-All Broadcast and Byzantine Consensus; the analysis further extends to directed Erdős–Rényi graphs, linking sparse random-graph dynamics to robust broadcasting capabilities. The findings demonstrate that stochastic dynamics can dramatically accelerate dissemination compared to worst-case adversaries, and they quantify how adversarial control degrades performance across regimes, providing practical insights for reliable broadcast and consensus in evolving networks.

Abstract

Broadcast and consensus are most fundamental tasks in distributed computing. These tasks are particularly challenging in dynamic networks where communication across the network links may be unreliable, e.g., due to mobility or failures. Indeed, over the last years, researchers have derived several impossibility results and high time complexity lower bounds (i.e., linear in the number of nodes $n$) for these tasks, even for oblivious message adversaries where communication networks are rooted trees. However, such deterministic adversarial models may be overly conservative, as many processes in real-world settings are stochastic in nature rather than worst case. This paper initiates the study of broadcast and consensus on stochastic dynamic networks, introducing a randomized oblivious message adversary. Our model is reminiscent of the SI model in epidemics, however, revolving around trees (which renders the analysis harder due to the apparent lack of independence). In particular, we show that if information dissemination occurs along random rooted trees, broadcast and consensus complete fast with high probability, namely in logarithmic time. Our analysis proves the independence of a key variable, which enables a formal understanding of the dissemination process. More formally, for a network with $n$ nodes, we first consider the completely random case where in each round the communication network is chosen uniformly at random among rooted trees. We then introduce the notion of randomized oblivious message adversary, where in each round, an adversary can choose $k$ edges to appear in the communication network, and then a rooted tree is chosen uniformly at random among the set of all rooted trees that include these edges. We show that broadcast completes in $O(k+\log n)$ rounds, and that this it is also the case for consensus as long as $k \le 0.1n$.

Figures (8)

• Figure 1: Our main results, where $c >0$ is any constant and $q$ is the failure probability.
• Figure 2: The best strategy for the adversary $A$, with $k=6$. Shaded nodes are informed nodes. In the top example, nodes $5, 6, 7, 8, 9$ and $10$ are safe from being informed, whereas node $1$ can still be informed. In the bottom example, nodes $5, 6, 7, 8,$ and $9$ are safe, whereas node $1$ can still be informed. However, node $1$ is safe from being informed by node $10$.
• Figure 3: Shaded nodes are informed nodes. Left: A tree $U$ that is information increasing. Right: A tree $U'$ that is a correction of $U$.
• Figure 4: Illustration for the proof of Lemma \ref{['lem:iinformed']}, Case 1. Shaded nodes are informed nodes. Left: Solid lines represent $E$. Dotted lines are a suitable choice of $a$. Right: Solid lines represent $F$ associated to $E$. Dotted lines represent $b(a)$. Any tree rooted at $9$ on the right yields a suitable choice of $a$ on the left.
• Figure 5: Merging examples

Theorems & Definitions (191)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.3
• Corollary 1.3
• Theorem 2.1
• Lemma 3.1
• proof
• Definition 3.2
• Lemma 3.2