[Paper Review] Approximate Consensus in Highly Dynamic Networks.

Bernadette Charron-Bost, Matthias Függer+1 more|arXiv (Cornell University)|Aug 4, 2014

Distributed systems and fault tolerance7 citations

TL;DR

This paper establishes that approximate consensus is achievable in highly dynamic networks—where topology changes unpredictably—provided each round's communication graph contains a rooted spanning tree, even if the coordinator (root) changes arbitrarily. It shows that simple averaging algorithms fully capture solvability, achieving precision ε in O(n^{n+1} log(1/ε)) synchronous rounds, with tighter bounds under bounded message delays.

ABSTRACT

In this paper, we investigate the approximate consensus problem in highly dynamic networks in which topology may change continually and unpredictably. We prove that in both synchronous and partially synchronous systems, approximate consensus is solvable if and only if the communication graph in each round has a rooted spanning tree, i.e., there is a coordinator at each time. The striking point in this result is that the coordinator is not required to be unique and can change arbitrarily from round to round. Interestingly, the particularly simple class of averaging algorithms entirely captures the solvability issue of approximate consensus in that the problem is solvable if and only if it can be solved using any averaging algorithm. We show that approximate consensus can be achieved with precision of $\varepsilon$ in a coordinated network model in $O(n^{n+1} \log\frac{1}{\varepsilon})$ synchronous rounds, and in $O(\Delta n^{n\Delta+1} \log\frac{1}{\varepsilon})$ rounds when the maximum round delay for a message to be delivered is $\Delta$. We investigate various network models in which this exponential bound in the number of nodes reduces to a polynomial bound, and we prove that a general upper bound on the time complexity of averaging algorithms has to be exponential. We apply our results to networked systems with a fixed topology and classical benign fault models, and deduce both known and new results for approximate consensus in these systems. In particular, we show that for solving approximate consensus, a complete network can tolerate up to $2n-3$ arbitrarily located link faults at every round, in contrast with the impossibility result established by Santoro and Widmayer (STACS '89) showing that exact consensus is not solvable with $n-1$ link faults per round originating from the same node.

Motivation & Objective

• To determine the necessary and sufficient conditions for solving approximate consensus in highly dynamic networks with unpredictable topology changes.
• To investigate whether simple averaging algorithms can fully characterize the solvability of approximate consensus in such systems.
• To establish tight bounds on the time complexity of approximate consensus under various network models, including bounded message delays.
• To extend known results on consensus under benign fault models, particularly in fixed-topology networks with link faults.
• To reconcile and refine prior impossibility results for exact consensus by showing that approximate consensus remains feasible under stronger fault models.

Proposed method

• Prove that approximate consensus is solvable in both synchronous and partially synchronous systems if and only if each round’s communication graph has a rooted spanning tree.
• Demonstrate that the solvability of approximate consensus is completely captured by the class of averaging algorithms, regardless of network dynamics.
• Analyze the convergence time of averaging algorithms, deriving an upper bound of O(n^{n+1} log(1/ε)) synchronous rounds for ε-precision.
• Extend the analysis to partially synchronous systems with maximum message delay Δ, yielding a bound of O(Δ n^{nΔ+1} log(1/ε)) rounds.
• Identify network models where the exponential bound reduces to polynomial time, showing that the exponential bound is tight in general.
• Apply the results to fixed-topology networks with benign faults, deriving new feasibility conditions for approximate consensus under link failures.

Experimental results

Research questions

• RQ1 What is the necessary and sufficient condition for approximate consensus in highly dynamic networks with unpredictable topology changes?
• RQ2 Can averaging algorithms fully characterize the solvability of approximate consensus in dynamic networks, regardless of coordinator changes?
• RQ3 What is the tightest possible time complexity bound for achieving ε-precision in approximate consensus under bounded message delays?
• RQ4 How does the solvability of approximate consensus compare to exact consensus under the same fault models, particularly in terms of fault tolerance?
• RQ5 In fixed-topology networks with link faults, what is the maximum number of link faults per round that still allows approximate consensus to be achieved?

Key findings

• Approximate consensus is solvable in highly dynamic networks if and only if each round’s communication graph contains a rooted spanning tree, even when the root (coordinator) changes arbitrarily from round to round.
• The class of averaging algorithms entirely captures the solvability of approximate consensus in dynamic networks, meaning the problem is solvable iff it is solvable via any averaging algorithm.
• The time complexity for achieving ε-precision is O(n^{n+1} log(1/ε)) synchronous rounds in the worst case.
• With bounded message delay Δ, the time complexity increases to O(Δ n^{nΔ+1} log(1/ε)) rounds, showing a dependence on both network size and delay.
• In certain network models, the exponential bound on time complexity reduces to polynomial, but a general upper bound for averaging algorithms must be exponential.
• For fixed-topology networks, a complete network can tolerate up to 2n−3 arbitrarily located link faults per round in approximate consensus, exceeding the n−1 fault limit that breaks exact consensus (as per Santoro and Widmayer, STACS '89).