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arxiv: 2502.15029 · v2 · submitted 2025-02-20 · 💻 cs.MA

Voter Model Meets Rumour Spreading: an FPRAS for Consensus Probabilities on Voter Models with Agnostic Nodes

Marcelo Matheus Gauy , Anna Abramishvili , Eduardo Colli , Nicolaus Heuer , Tiago Madeira , Frederik Mallmann-Trenn , Vin\'icius Franco Vasconcelos , David Kohan Marzag\~ao This is my paper

Pith reviewed 2026-05-23 02:29 UTC · model grok-4.3

classification 💻 cs.MA
keywords voter modelrumour spreadingconsensusFPRASMarkov chain Monte Carloagnostic nodesmulti-agent systemsmixing time
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The pith

A Markov chain Monte Carlo sampler yields an FPRAS for consensus probabilities in voter models that include agnostic nodes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper models consensus on graphs where some nodes begin without an opinion by combining the classical voter model with rumour spreading. It constructs a martingale that gives the exact probability a given colour reaches consensus and derives polynomial bounds on the number of steps until absorption. The central algorithmic result is an MCMC procedure whose running time is O(n² log n) on arbitrary graphs and O(n log n) on Erdős-Rényi graphs, producing a fully polynomial-time randomized approximation scheme for any desired accuracy. A reader cares because many distributed protocols and social-dynamics models must predict outcomes when participants start undecided, and the method supplies a practical computational handle on those probabilities.

Core claim

The paper shows that the probability each colour reaches consensus can be estimated to arbitrary precision in polynomial time by running a Markov chain that interleaves voter-model updates with rumour-spreading steps; the chain mixes in O(n² log n) steps on general graphs and faster on random graphs, yielding an FPRAS, while closed-form expressions are available when the number of agnostic nodes is at most logarithmic.

What carries the argument

A Markov chain Monte Carlo process that simulates the combined voter-rumour dynamics to approximate the martingale-derived consensus probabilities.

If this is right

  • Closed formulas for consensus probabilities exist when the number of agnostic nodes is at most logarithmic.
  • Termination time is bounded by known results from rumour spreading and voter models.
  • Fewer Monte Carlo runs suffice for a fixed error level as the number of nodes grows.
  • The scheme works for any fixed number of colours on any connected graph.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same MCMC reduction may apply to other opinion dynamics that mix adoption and transmission steps.
  • The polynomial-time estimator could be used to compare robustness of different update rules when undecided agents are present.
  • If the martingale extends to weighted or directed graphs, the FPRAS would immediately carry over.

Load-bearing premise

The dynamics with agnostic nodes can be accurately captured by a combination of the classical voter model and rumour-spreading processes, so that existing martingale and mixing-time results apply directly.

What would settle it

An explicit family of graphs on which the MCMC chain requires super-polynomial steps to reach constant total-variation distance from stationarity would refute the claimed FPRAS running time.

Figures

Figures reproduced from arXiv: 2502.15029 by Anna Abramishvili, David Kohan Marzag\~ao, Eduardo Colli, Frederik Mallmann-Trenn, Marcelo Matheus Gauy, Nicolaus Heuer, Tiago Madeira, Vin\'icius Franco Vasconcelos.

Figure 1
Figure 1. Figure 1: A motivational example of an undirected graph [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Every configuration 𝑠1𝑖 which can be reached from 𝑆0 in Example 1.1 after one round, i.e. 𝑎𝑖 := P(𝑆1 = 𝑠1𝑖 |𝑆0) > 0. The probability of a red consensus in each case is denoted by 𝑏𝑖 and can be calculated by applying Proposition 2.1. Therefore, the probability of red to win in Example 1.1 is P8 𝑖=1 𝑎𝑖𝑏𝑖 = 5 8 . Observe that 𝑋𝑡+1 = P 𝑣∈𝑉 𝜇(𝑣)P(𝑅(𝑣)|𝑆𝑡+1), and we have, by linear￾ity of expectation: E(𝑋𝑡+1 | 𝑆… view at source ↗
Figure 3
Figure 3. Figure 3: Counterexample for the conjecture that the Mar [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: A comparison of the standard error of the proba [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: A comparison of the standard error of the proba [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 6
Figure 6. Figure 6: A comparison of the cumulative standard error of [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
read the original abstract

Problems of consensus in multi-agent systems are often viewed as a series of independent, simultaneous local decisions made between a limited set of options, all aimed at reaching a global agreement. Key challenges in these protocols include estimating the likelihood of various outcomes and finding bounds for how long it may take to achieve consensus, if it occurs at all. To date, little attention has been given to the case where some agents have no initial opinion. In this paper, we introduce a variant of the consensus problem which includes what we call `agnostic' nodes and frame it as a combination of two known and well-studied processes: voter model and rumour spreading. We show (1) a martingale that describes the probability of consensus for a given colour, (2) bounds on the number of steps for the process to end using results from rumour spreading and voter models, (3) closed formulas for the probability of consensus in a few special cases, along with a polynomial-time algorithm for the case where the number of agnostic vertices is at most logarithmic and (4) that the computational complexity of estimating the probability with a Markov chain Monte Carlo process is $O(n^2 \log n)$ for general graphs and $O(n\log n)$ for Erd\H{o}s-R\'enyi graphs, resulting in a fully polynomial-time randomized approximation scheme (FPRAS) for estimating the probabilities of consensus. Furthermore, we present experimental results suggesting that the number of runs needed for a given standard error decreases when the number of nodes increases.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces 'agnostic' nodes (initially opinionless) into the voter model on graphs and frames the resulting dynamics as a combination of the classical voter model and rumour-spreading processes. It claims (1) a martingale for the probability that a given colour reaches consensus, (2) convergence-time bounds obtained by invoking existing results from the two source literatures, (3) closed-form consensus probabilities for special cases together with a polynomial-time algorithm when the number of agnostic nodes is at most logarithmic, and (4) an FPRAS for estimating consensus probabilities whose MCMC complexity is O(n² log n) on general graphs and O(n log n) on Erdős-Rényi graphs. Experimental results suggesting that the number of Monte-Carlo runs needed for a fixed standard error decreases with network size are also reported.

Significance. If the claimed martingale and mixing-time results are shown to carry over, the work supplies the first FPRAS for consensus probabilities in this extended model and a practical polynomial-time algorithm for the logarithmic-agnostic regime; both would be useful for analysing opinion dynamics with undecided agents. The experimental observation on variance reduction with size is a potentially interesting empirical finding.

major comments (2)
  1. [Abstract and martingale derivation section] Abstract and the section deriving the martingale: the claim that the probability of consensus for each colour is a martingale rests on the assertion that the combined voter-model/rumour-spreading update rule preserves the one-step conditional-expectation property used in the classical voter-model proofs. No explicit verification that the agnostic-node rumour-spreading step leaves this invariance intact is supplied.
  2. [FPRAS and complexity paragraph] The paragraph stating the FPRAS complexity: the O(n² log n) (general graphs) and O(n log n) (ER graphs) MCMC bounds are obtained by directly citing mixing-time results from rumour spreading and voter models. The paper does not demonstrate that the additional dependence introduced by agnostic nodes preserves the conductance or coupling conditions required by those cited bounds; without this step the reduction to an FPRAS does not follow.
minor comments (2)
  1. [Experimental results section] The experimental plots would be clearer if the precise graph-generation parameters (e.g., edge probability for the ER instances) and the number of independent trials per data point were stated explicitly.
  2. [Model definition] The update rule for an agnostic node should be written as a single displayed equation rather than described in prose, to remove any ambiguity about the probability with which each colour is adopted.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive major comments. Both points identify places where the manuscript relies on implicit carry-over from the cited voter-model and rumour-spreading literature without supplying the required verification steps. We address each comment below and will revise the paper to close these gaps.

read point-by-point responses

  1. Referee: [Abstract and martingale derivation section] Abstract and the section deriving the martingale: the claim that the probability of consensus for each colour is a martingale rests on the assertion that the combined voter-model/rumour-spreading update rule preserves the one-step conditional-expectation property used in the classical voter-model proofs. No explicit verification that the agnostic-node rumour-spreading step leaves this invariance intact is supplied.

    Authors: We acknowledge that the current derivation does not contain an explicit one-step calculation for the agnostic-node update. The manuscript treats the combined process as inheriting the martingale property from the classical voter model, but does not verify the conditional-expectation step when an agnostic node adopts an opinion. In the revision we will insert a short lemma that computes the one-step expectation explicitly for both the voter-model edge selection and the rumour-spreading adoption rule, confirming that the invariance is preserved because an agnostic node adopts colour c with probability exactly equal to the fraction of its neighbours that already hold c. revision: yes

  2. Referee: [FPRAS and complexity paragraph] The paragraph stating the FPRAS complexity: the O(n² log n) (general graphs) and O(n log n) (ER graphs) MCMC bounds are obtained by directly citing mixing-time results from rumour spreading and voter models. The paper does not demonstrate that the additional dependence introduced by agnostic nodes preserves the conductance or coupling conditions required by those cited bounds; without this step the reduction to an FPRAS does not follow.

    Authors: We agree that simply invoking the cited mixing-time theorems is insufficient once agnostic nodes are present. The manuscript does not supply an argument that the conductance or coupling arguments remain valid under the additional state variables carried by agnostic nodes. In the revision we will add a paragraph (or short appendix) showing that the process can be coupled to a standard voter model after a logarithmic number of steps (once all agnostic nodes have adopted an opinion) and that the pre-adoption phase contributes only a lower-order term to the mixing time; this will justify reusing the cited O(n² log n) and O(n log n) bounds. revision: yes

Circularity Check

0 steps flagged

No load-bearing circularity; martingale and FPRAS rest on external voter/rumour theorems applied to new dynamics

full rationale

The paper introduces agnostic nodes by combining voter-model updates with rumour-spreading steps, then states a martingale for consensus probability and invokes existing mixing-time and martingale results from the two classical literatures to bound termination and obtain the O(n² log n) MCMC complexity. No equations reduce a claimed prediction to a fitted parameter or self-citation by construction, and the cited theorems are standard external results rather than prior work by the same authors. The special-case closed formulas and logarithmic-agnostic poly-time algorithm are presented as direct derivations, keeping the central FPRAS claim independent of any self-referential loop.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities beyond the modeling choice of agnostic nodes are stated.

invented entities (1)
  • agnostic nodes no independent evidence
    purpose: Model agents that begin without an opinion
    New modeling element introduced to extend the voter model

pith-pipeline@v0.9.0 · 5855 in / 1103 out tokens · 55396 ms · 2026-05-23T02:29:19.672247+00:00 · methodology

discussion (0)

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