Consensus Through Knot Discovery in Asynchronous Dynamic Networks

TL;DR

The paper addresses consensus in highly dynamic, asynchronous networks where topology can be intermittently disconnected and adversarial. It proposes the Knot Identification Problem (KI) and proves necessary and sufficient conditions for solvability under a knot-transparent, knot-observation-final, asynchronous, non-oblivious adversary, introducing a simple distributed algorithm KIA that disseminates local observation graphs to identify a globally observable knot. It provides formal definitions (KI, KI agreement/termination), impossibility results for knot-opaque adversaries, and a $O(n)$ complexity bound for KIA, alongside performance evaluations demonstrating practical knot detection in dynamic topologies. The work offers a topology-driven path to consensus in environments with minimal connectivity guarantees and outlines extensions to resource-constrained settings and IoT deployments, with potential to inform broader dynamic-network coordination tasks.

Abstract

We state the Problem of Knot Identification as a way to achieve consensus in dynamic networks. The network adversary is asynchronous and not oblivious. The network may be disconnected throughout the computation. We determine the necessary and sufficient conditions for the existence of a solution to the Knot Identification Problem: the knots must be observable by all processes and the first observed knot must be the same for all processes. We present an algorithm KIA that solves it. We conduct KIA performance evaluation.

Figures (5)

• Figure 1: Knot formation example. Edge labels denote states when the edges are present. Process $e$ observes knot $K_1 = \{b,c,d\}$; process $d$ is the first to observe knot $K_2 = \{a,b,c,d\}$.
• Figure 2: Illustration for the proof of Lemma \ref{['lemNoAsynch']}. In figure a), in computation $\sigma_i$, process $p_1$ observes knot $K_1$ with event $e_1$ in state $s_i$. In figure b), in computation $\sigma_{ij}$, process $p_1$ outputs knot $K_1$ in state $s_{ij}$. In figure c), in the same computation $\sigma_{ij}$, process $p_2$ observes knot $K_2$ in state $s_k$ with event $e_2$. In figure d), in computation $\sigma_{ijkl}$, process $p_2$ outputs $K_2$ in state $s_l$.
• Figure 3: Intermittent Connectivity Topology Example. The underlying topology contains a single knot: the cycle.
• Figure 4: Longest knot output time as a function of the knot size.
• Figure 5: Longest knot output time as a function of the maximum number of edges per state.

Theorems & Definitions (10)

• definition 1: Consensus
• definition 2: Knot Identification
• proposition 1
• proposition 2
• lemma 1
• proof
• lemma 2
• proof
• theorem 1
• proof