The Next 700 Impossibility Results in Time-Varying Graphs

We address highly dynamic distributed systems modeled by time-varying graphs (TVGs). We interest in proof of impossibility results that often use informal arguments about convergence. First, we provide a distance among TVGs to define correctly the convergence of TVG sequences. Next, we provide a general framework that formally proves the convergence of the sequence of executions of any deterministic algorithm over TVGs of any convergent sequence of TVGs. Finally, we illustrate the relevance of the above result by proving that no deterministic algorithm exists to compute the underlying graph of any connected-over-time TVG, i.e., any TVG of the weakest class of long-lived TVGs.

💡 Research Summary

The paper tackles the challenge of rigorously proving impossibility results in highly dynamic distributed systems modeled by time‑varying graphs (TVGs). The authors first introduce a metric on the space of TVGs: the distance d_G between two TVGs is defined as 2^‑λ where λ is the supremum of times up to which the two presence functions coincide. This metric is an ultrametric, satisfying symmetry, a strong triangle inequality (max‑type), and, crucially, completeness: every Cauchy sequence of TVGs converges to a limit TVG. An analogous metric d_O is defined on the space of algorithmic outputs (executions) over TVGs, which also forms a complete ultrametric space.

The central theorem (the “convergence framework”) states that for any deterministic algorithm A, if a sequence of TVGs (g_n) converges to a TVG g under d_G, then the corresponding sequence of executions (outputs) of A on the g_n also converges under d_O, and its limit coincides with the execution of A on g. The proof leverages the deterministic nature of A (identical inputs produce identical outputs) and the completeness of both metric spaces.

Using this framework, the authors focus on the weakest long‑lived class of TVGs, called connected‑over‑time (COT). In a COT, for any time t and any pair of processes p, q, there exists a temporal path from p to q after t, but edges may appear only finitely many times (eventual missing edges). The eventual underlying graph U_ω consists of edges that are present infinitely often.

The paper proves that no deterministic algorithm can compute U_ω for all COT TVGs. The proof constructs, for any candidate algorithm A, a Cauchy sequence of TVGs that gradually “hides” a particular edge e: each TVG in the sequence agrees with the limit TVG on longer and longer prefixes, yet e is present infinitely often in the early members and disappears forever in the limit. By the convergence theorem, A’s outputs on the sequence must converge to its output on the limit TVG. However, on each finite member A would have to include e in the computed underlying graph (since e appears infinitely often there), while on the limit TVG e must be excluded. This contradiction shows that such an algorithm cannot exist.

The contribution is twofold. First, the authors provide a mathematically sound notion of convergence for TVGs and their executions, turning informal “limit” arguments into rigorous proofs. Second, they apply this tool to obtain a fundamental impossibility: even in the most permissive dynamic setting (COT), deterministic processes cannot reliably infer which edges are permanently available. This result highlights the inherent limits of deterministic computation in highly dynamic networks and suggests that randomness, additional timing assumptions, or stronger connectivity guarantees are necessary for any algorithm that aims to reconstruct stable topological information. The framework is generic and can be reused to analyze other TVG classes, opening a systematic pathway for proving (im)possibility results in dynamic distributed computing.