Ratio Consensus Algorithm

Updated 7 July 2026

Ratio consensus is a distributed algorithm that computes global averages across directed networks by evolving coupled linear state processes and taking local quotients.
It uses column-stochastic dynamics to ensure mass preservation and properly cancels network-dependent scaling to achieve consensus.
The method adapts to real-world conditions like delays and packet drops, providing finite-time stopping rules and computable convergence rate bounds.

Searching arXiv for recent and foundational papers on ratio consensus and closely related variants. arXiv search query: "ratio consensus push-sum distributed averaging delays convergence rate over-the-air open multi-agent systems" Ratio consensus is a class of distributed algorithms for computing a global ratio, typically an average, over a directed network by evolving two coupled linear state processes and taking their nodewise quotient. In its canonical form, each node maintains a value state and an auxiliary weight state, both propagated by the same nonnegative matrix sequence; the estimate at node $i$ is the ratio $x_n^i/w_n^i$ . This construction is central when the network is directed or unbalanced, because column-stochastic updates preserve total mass but do not, by themselves, make each node’s state converge to the average. Ratio consensus corrects that asymmetry by transporting both numerator and denominator through the same mixing dynamics, so that the common network-dependent scaling cancels in the quotient (Gerencsér et al., 2019, Gerencsér, 2021).

1. Problem setting and canonical objective

In the generalized formulation, each node $i$ starts with a value $x^i\in\mathbb R$ and a nonnegative weight $w^i\ge 0$ , not all weights zero, and the network seeks the global ratio

$\frac{\sum_i x^i}{\sum_i w^i}.$

Average consensus is the special case $w^i=1$ for all $i$ , so the target becomes $\frac{1}{p}\sum_i x^i$ on a network with $p$ nodes (Gerencsér et al., 2019). In the fixed-topology distributed-averaging setting studied for delayed networks, node $x_n^i/w_n^i$ 0 initially holds a scalar value $x_n^i/w_n^i$ 1, and the objective is

$x_n^i/w_n^i$ 2

over a fixed directed interconnection topology with $x_n^i/w_n^i$ 3 agents (Prakash et al., 2017).

This objective is technically straightforward on balanced undirected networks with doubly stochastic averaging, but directed graphs are different. Column-stochastic matrices preserve total mass,

$x_n^i/w_n^i$ 4

yet the value process alone does not converge to the average at each node. Ratio consensus is therefore the directed, asynchronous analogue of average-consensus and gossip methods: it preserves the aggregate through column-stochastic dynamics while recovering the desired normalized quantity through a quotient operation (Gerencsér, 2021).

Historically, the modern formulation is closely associated with the push-sum algorithm of Kempe, Dobra, and Gehrke and its later weighted-gossip extensions by Bénézit et al.; the abstract framework was subsequently broadened to strictly stationary ergodic sequences of nonnegative matrices (Gerencsér et al., 2019).

2. Core recursion and the ratio mechanism

The standard recursion is

$x_n^i/w_n^i$ 5

where each $x_n^i/w_n^i$ 6 is a nonnegative matrix, often column-stochastic in averaging applications (Gerencsér et al., 2019, Gerencsér et al., 27 May 2026). If

$x_n^i/w_n^i$ 7

then

$x_n^i/w_n^i$ 8

and the local estimate at node $x_n^i/w_n^i$ 9 is

$i$ 0

The fixed-graph averaging version makes the mechanism explicit by running a numerator process $i$ 1 and a denominator process $i$ 2, with

$i$ 3

A typical initialization is

$i$ 4

so that the numerator tracks the total initial data while the denominator tracks a uniform mass of one per node (Prakash et al., 2017). When both processes evolve under the same linear dynamics, each node asymptotically sees the same network-dependent scaling in both variables; the ratio cancels that scaling and yields the average. In the column-stochastic case,

$i$ 5

and when $i$ 6, the limit is the usual global average (Gerencsér et al., 2019, Gerencsér et al., 27 May 2026).

A useful decomposition clarifies the error mechanism. Writing $i$ 7, one has

$i$ 8

so the consensus error is controlled by the transverse product $i$ 9 together with the denominator size $x^i\in\mathbb R$ 0 (Gerencsér, 2021). This perspective underlies both asymptotic convergence proofs and rate estimates.

3. Directed graphs, delays, and finite-time termination

A practically important setting is ratio consensus on a fixed directed graph with finite, bounded communication delays. In that case, delayed incoming values are incorporated explicitly into the updates: $x^i\in\mathbb R$ 1

$x^i\in\mathbb R$ 2

where $x^i\in\mathbb R$ 3 is the maximum delay bound and delayed packets are treated as “in-transit” contributions that are injected after the specified delay (Prakash et al., 2017). The essential structural assumptions remain column-stochasticity and sufficient connectivity.

The delayed formulation matters for two reasons. First, it preserves the mass-conservation logic of ratio consensus even though the system now has memory. Second, it changes the stopping problem. Classical ratio consensus is asymptotic: the iterates approach the average but do not intrinsically indicate when computation may stop. The delayed setting makes naive stopping especially unsafe because local stability of recent iterates does not rule out future changes induced by delayed arrivals.

To address this, a distributed finite-time stopping criterion was developed for delayed ratio consensus on fixed directed topologies (Prakash et al., 2017). The stopping logic is based on the fact that convergence within a prescribed tolerance can be certified from the behavior of ratio histories over a window related to the delay bound and network propagation characteristics. A representative global condition is

$x^i\in\mathbb R$ 4

while the distributed implementation uses local surrogates such as

$x^i\in\mathbb R$ 5

for an appropriate horizon $x^i\in\mathbb R$ 6. The crucial guarantee is that the stopping rule is finite-time and correct: all nodes stop after finitely many iterations, and when a node stops, its returned estimate is within the prescribed tolerance of the exact average (Prakash et al., 2017).

4. General theory and convergence-rate analysis

The modern theory of ratio consensus places the algorithm in the framework of products of random nonnegative matrices. Under strict stationarity, ergodicity, nonnegativity, allowability, and sequential primitivity, the ratios

$x^i\in\mathbb R$ 7

converge almost surely to a common limit, and the normalized trajectories

$x^i\in\mathbb R$ 8

become exponentially close in total variation at a rate governed by the spectral gap $x^i\in\mathbb R$ 9, where $w^i\ge 0$ 0 are the Lyapunov exponents of the matrix product (Gerencsér et al., 2019).

In that framework, the almost sure exponential convergence rate is sharp. For almost all positive initial pairs, the worst nodewise error decays exactly at rate

$w^i\ge 0$ 1

and in the column-stochastic case, where $w^i\ge 0$ 2, the rate becomes $w^i\ge 0$ 3 (Gerencsér et al., 2019, Gerencsér et al., 27 May 2026). This sharpness is one of the main structural results of the theory: the spectral gap is not merely an upper bound but the exact asymptotic decay rate in generic settings.

A complementary line of work addresses computability. Gerencsér and Gerencsér identified the exact almost sure rate in terms of the second Lyapunov exponent, but that quantity is generally not computable. A later result gives a computable upper bound for i.i.d. column-stochastic matrix sequences: $w^i\ge 0$ 4 with

$w^i\ge 0$ 5

and more generally

$w^i\ge 0$ 6

for higher tensor powers (Gerencsér, 2021). The importance of this bound is practical rather than asymptotic: it replaces an exact but inaccessible Lyapunov quantity with a finite-dimensional spectral-radius certificate that can be evaluated from the update distribution itself.

A further extension relaxes the structural hypotheses by replacing allowability with column-allowability and sequential primitivity with weak sequential primitivity. This admits augmented-network formulations in which auxiliary or virtual nodes may have zero rows infinitely often, while preserving the same Lyapunov-gap rate formula (Gerencsér et al., 27 May 2026).

5. Robust and application-specific variants

Ratio consensus has been adapted to several communication models in which the main technical task is to preserve an effective column-stochastic, mass-conserving evolution despite drops, delays, changing membership, or nonstandard physical channels.

Variant	Network feature	Key mechanism
Delayed fixed-topology RC (Prakash et al., 2017)	Bounded communication delays	Delay-aware updates and finite-time stopping
ARQ-based RC (Makridis et al., 2022)	Retransmissions and packet drops	Augmented digraph with virtual delay and buffer nodes
OPENRC (Makridis et al., 2024)	Agent arrivals and departures	Acknowledgement-based weight normalization and residual-mass transfer
Over-the-Air RC (Charalambous et al., 30 Jul 2025)	Wireless MAC superposition	Local normalization by aggregate channel gain

In ARQ-based average consensus over directed unreliable links, the protocol exploits ACK/NACK feedback to acquire out-degree information, detect packet arrival, and obtain local upper bounds on retransmission-induced delays. The convergence proof proceeds by building an augmented digraph with virtual delay nodes and a virtual buffer node for packet drops, yielding a random column-stochastic matrix $w^i\ge 0$ 7 on the enlarged state space: $w^i\ge 0$ 8 Because the augmented evolution is column-stochastic, mass is conserved across actual and virtual nodes, and the ratio still converges to the exact average (Makridis et al., 2022).

In open multi-agent systems, OpenRC extends ratio consensus to time-varying directed networks with arrivals and departures. The target average itself changes with the active set, and the algorithm maintains mass preservation by having remaining agents redistribute their current state, departing agents transfer only their residual relative to their stored joining mass, and arriving agents inject their joining masses. Under the paper’s assumptions, the total active $w^i\ge 0$ 9-mass and $\frac{\sum_i x^i}{\sum_i w^i}.$ 0-mass equal the total joining masses of active nodes at all times; if membership eventually stops changing, the method reduces to standard ratio consensus on the resulting fixed graph and converges to the exact active-set average (Makridis et al., 2024).

In wireless multi-agent systems with over-the-air aggregation, simultaneous transmissions superpose physically at the receiver,

$\frac{\sum_i x^i}{\sum_i w^i}.$ 1

The over-the-air ratio consensus method compensates for arbitrary positive real reciprocal channel gains by dividing the received aggregate by a normalization factor $\frac{\sum_i x^i}{\sum_i w^i}.$ 2, either fixed in the time-invariant case or recomputed at every time step in the time-varying case. This normalization restores an effective column-stochastic update, so the ratio converges asymptotically to the average when receiver noise is negligible and the stated connectivity and reciprocity assumptions hold (Charalambous et al., 30 Jul 2025).

Ratio consensus is closely related to other consensus formulations, but the algorithmic mechanism is specific. The heat-kernel-Pagerank method for consensus computation is conceptually similar in that it recovers a weighted average or follower equilibrium from graph-local structure and invariance properties, yet it is not a ratio-consensus protocol. Rather than iterating two coupled linear recursions and taking nodewise quotients, it reformulates consensus as a Laplacian-based linear system and approximates the solution via heat kernel pagerank and heat-kernel integral identities (Chung et al., 2015).

The phrase “consensus” also appears in problems that are mathematically distinct from distributed averaging. “Consensus Division in an Arbitrary Ratio” studies partitioning a line segment into two subsets so that multiple measures assign a prescribed ratio $\frac{\sum_i x^i}{\sum_i w^i}.$ 3 to one side of the partition; it concerns cut complexity and classes such as $\frac{\sum_i x^i}{\sum_i w^i}.$ 4, $\frac{\sum_i x^i}{\sum_i w^i}.$ 5, and $\frac{\sum_i x^i}{\sum_i w^i}.$ 6, not distributed ratio iterations on directed graphs (Goldberg et al., 2022). Likewise, “Rational Fair Consensus in the GOSSIP Model” studies randomized fair selection of a winning color under coalitional deviations, with fairness defined by proportional winning probabilities, rather than mass-preserving average computation through auxiliary weights (Clementi et al., 2017).

A common misconception is therefore to equate any normalized consensus value with ratio consensus. The defining feature of ratio consensus is narrower: two parallel linear processes driven by the same network evolution, with the quotient compensating for directional bias or nonuniform transport (Gerencsér et al., 2019, Gerencsér, 2021).

7. Significance and current scope

The significance of ratio consensus lies in its ability to solve distributed averaging on directed, unbalanced, delayed, asynchronous, unreliable, or otherwise non-ideal networks without requiring doubly stochastic mixing. Its technical core is the preservation of aggregate mass under column-stochastic or effectively column-stochastic dynamics, together with quotient-based cancellation of the network-dependent scaling. This structure explains why the method extends naturally to augmented-state models with delayed packets, retransmission buffers, virtual nodes, and changing active sets (Prakash et al., 2017, Makridis et al., 2022, Makridis et al., 2024, Gerencsér et al., 27 May 2026).

The mature theory now covers both asymptotic correctness and convergence speed. On one side are sharp spectral-gap results, which identify the exact almost sure exponential rate in broad random-matrix settings (Gerencsér et al., 2019, Gerencsér et al., 27 May 2026). On the other are computable bounds such as

$\frac{\sum_i x^i}{\sum_i w^i}.$ 7

which provide tractable performance certificates for i.i.d. protocols (Gerencsér, 2021). On the algorithmic side, finite-time stopping rules for delayed fixed networks and communication-aware variants for ARQ channels, open systems, and over-the-air wireless aggregation show that ratio consensus is not a single protocol but a family of mass-preserving quotient schemes adapted to directed network constraints (Prakash et al., 2017, Makridis et al., 2022, Makridis et al., 2024, Charalambous et al., 30 Jul 2025).