Fault-tolerant Consensus in Anonymous Dynamic Network

TL;DR

This work investigates fault-tolerant consensus in anonymous dynamic networks where a dynamic adversary selects round-by-round communication graphs. It introduces the $(T,D)$-dynaDegree stability property to capture evolving connectivity and proves tight conditions for crash- and Byzantine-tolerant approximate consensus, along with impossibility results for exact consensus under restrictive dynamic-degree regimes. The authors present two phase-based algorithms, DAC and DBAC, achieving crash- and Byzantine-tolerant approximate consensus under the stated degree conditions, with proven convergence and termination properties. The results illuminate the limits and possibilities of consensus when identities are absent and connectivity fluctuates, offering a structured framework and concrete algorithms for practical, anonymity-preserving distributed systems.

Abstract

This paper studies the feasibility of reaching consensus in an anonymous dynamic network. In our model, $n$ anonymous nodes proceed in synchronous rounds. We adopt a hybrid fault model in which up to $f$ nodes may suffer crash or Byzantine faults, and the dynamic message adversary chooses a communication graph for each round. We introduce a stability property of the dynamic network -- $(T,D)$-dynaDegree for $T \geq 1$ and $n-1 \geq D \geq 1$ -- which requires that for every $T$ consecutive rounds, any fault-free node must have incoming directed links from at least $D$ distinct neighbors. These links might occur in different rounds during a $T$-round interval. $(1,n-1)$-dynaDegree means that the graph is a complete graph in every round. $(1,1)$-dynaDegree means that each node has at least one incoming neighbor in every round, but the set of incoming neighbor(s) at each node may change arbitrarily between rounds. We show that exact consensus is impossible even with $(1,n-2)$-dynaDegree. For an arbitrary $T$, we show that for crash-tolerant approximate consensus, $(T,\lfloor n/2 \rfloor)$-dynaDegree and $n > 2f$ are together necessary and sufficient, whereas for Byzantine approximate consensus, $(T,\lfloor (n+3f)/2 \rfloor)$-dynaDegree and $n > 5f$ are together necessary and sufficient.

Figures (2)

• Figure 1: Illustration of an example message adversary. Figure \ref{['fig:sub1']} shows that during odd rounds, the message adversary removes all the links, whereas Figure \ref{['fig:sub2']} shows that during even rounds, the adversary removes two links $(1,3)$ and $(3,1)$.
• Figure 2: Partition of $\operatorname{interval}(V^{(p)})$ into $V_1,V_2$ and $V_3$.

Theorems & Definitions (32)

• Definition 1: $(T,D)$-dynaDegree
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Definition 7: Convergence Rate
• Lemma 1
• proof
• Claim 2