Capacity of Byzantine Consensus with Capacity-Limited Point-to-Point Links

We consider the problem of maximizing the throughput of Byzantine consensus, when communication links have finite capacity. Byzantine consensus is a classical problem in distributed computing. In existing literature, the communication links are implicitly assumed to have infinite capacity. The problem changes significantly when the capacity of links is finite. We define the throughput and capacity of consensus, and identify upper bound of achievable consensus throughput. We propose an algorithm that achieves consensus capacity in complete four-node networks with at most 1 failure with arbitrary distribution of link capacities.

💡 Research Summary

The paper investigates the fundamental limits of Byzantine consensus when communication links have finite capacities, a scenario largely ignored in prior work that typically assumes unlimited bandwidth. The authors first formalize a notion of “consensus throughput” by treating each node’s input as an infinite bitstream and defining throughput as the asymptotic rate at which all fault‑free nodes agree on bits (lim b(t)/t where b(t) is the number of bits agreed upon by time t). This moves the analysis from round‑based complexity to a continuous‑flow perspective more appropriate for real networks.

The network model is a synchronous, fully‑connected directed graph G(V,E) with n nodes and n(n‑1) directed links. Each link e_{ij} has a capacity c_{ij} (bits per unit time); a capacity of zero simply means the link does not exist. An omniscient adversary may corrupt up to f < n/3 nodes, controlling all their behavior (message forging, collusion, crashes, etc.). Despite this powerful adversary, the classic Byzantine properties—termination, consistency, and validity—must still hold.

The central theoretical contribution is Theorem 1, which establishes an upper bound on the achievable consensus capacity C_con(G). For any subset S of nodes with |S| ≤ f, define Γ_S as the collection of subsets γ ⊂ V \ S of size n − |S| − f. For each γ, let I_S(γ) = Σ_{j∈γ, i∈S} c_{ji}, the total incoming capacity to S from γ. Let I*S = min{γ∈Γ_S} I_S(γ). The theorem proves that C_con(G) ≤ min_{|S|≤f} I*_S. The proof proceeds by contradiction: assuming a protocol exceeds this bound, the authors construct two distinct input scenarios that are indistinguishable to a set of fault‑free nodes because they receive the same limited amount of information (≤ t·I*_S bits). This violates the validity condition, yielding a contradiction.

Focusing on the smallest non‑trivial case, the paper then examines complete four‑node networks (all directed links have positive capacity) with at most one faulty node (f = 1). In this setting the bound simplifies to the minimum sum of any two incoming link capacities to a node, denoted I*. Theorem 2 and its corollary show that for any transmission rate R < I*, at least three pairs of bidirectional links have combined capacity exceeding R. This structural property guarantees enough bandwidth to support the forthcoming algorithm.

The authors design a capacity‑achieving consensus protocol that operates in “generations”. Each generation processes R bits (organized as R packets of size c bits) and proceeds through five operational modes: Undetected 2=, Undetected 1=1, Undetected 2, Detected, and Identified. The first three modes are used while no failure has been detected; they differ in how many nodes appear to have identical inputs. The protocol relies on linear network coding: each node creates coded packets such that any R of them are linearly independent. Nodes exchange packets either directly (if the bidirectional link capacity exceeds R) or via a third node that forwards as many packets as possible. After receiving enough packets, a node checks “consistency” by solving for a unique solution; the result (consistent or not) is broadcast using a 1‑bit reliable Byzantine broadcast primitive.

If inconsistency is detected, a full‑scale diagnostic broadcast (similar to the one used in prior broadcast‑only work) isolates the faulty node to either a pair of candidates (Detected mode) or a single identified node (Identified mode). Once the faulty node is narrowed down, the remaining fault‑free nodes continue consensus using the appropriate mode, guaranteeing that all decisions are identical and equal to the original inputs of the fault‑free nodes.

Crucially, each round’s total transmitted bits never exceed the minimal incoming capacity I*; therefore, for any R arbitrarily close to I*, the protocol can sustain that rate by pipelining generations. Consequently, the algorithm attains the upper bound of Theorem 1, proving that the bound is tight for complete four‑node networks with one possible Byzantine failure.

In summary, the paper extends Byzantine consensus theory to realistic, capacity‑constrained networks, provides a tight information‑theoretic upper bound on consensus throughput, and constructs an explicit protocol that meets this bound in a canonical four‑node topology. This work bridges a gap between abstract consensus models and practical network limitations, offering both theoretical insight and a concrete algorithmic framework for high‑throughput Byzantine‑resilient systems.