Agreement in Directed Dynamic Networks

We study distributed computation in synchronous dynamic networks where an omniscient adversary controls the unidirectional communication links. Its behavior is modeled as a sequence of directed graphs representing the active (i.e. timely) communication links per round. We prove that consensus is impossible under some natural weak connectivity assumptions, and introduce vertex-stable root components as a means for circumventing this impossibility. Essentially, we assume that there is a short period of time during which an arbitrary part of the network remains strongly connected, while its interconnect topology may keep changing continuously. We present a consensus algorithm that works under this assumption, and prove its correctness. Our algorithm maintains a local estimate of the communication graphs, and applies techniques for detecting stable network properties and univalent system configurations. Our possibility results are complemented by several impossibility results and lower bounds for consensus and other distributed computing problems like leader election, revealing that our algorithm is asymptotically optimal.

💡 Research Summary

The paper investigates the classic consensus problem in a synchronous dynamic network where an omniscient adversary controls the set of directed communication links each round. The network’s evolution is modeled as a sequence of directed graphs G₁, G₂, …, each representing the timely links for a particular round. The adversary may arbitrarily choose the edge set for every round, which makes the model substantially more challenging than traditional static or undirected dynamic models.

The authors first show that weak connectivity assumptions that are common in the literature—such as the existence of a root component (a strongly‑connected subgraph that can reach all other nodes) in every round—are insufficient for achieving consensus. Even if a root component exists each round, the fact that its vertex set may change arbitrarily prevents any information from persisting long enough to force a common decision. They formalize this impossibility by constructing executions where different initial values remain forever separated.

To overcome this barrier, the paper introduces the notion of a vertex‑stable root component (VSRC). A VSRC is a set of nodes R that forms the root component for a contiguous interval of T rounds, i.e., for every round r in