Voter Model Meets Rumour Spreading: A Study of Consensus Protocols on Graphs with Agnostic Nodes [Extended Version]

TL;DR

This work introduces a variant of consensus with agnostic nodes, integrating the classical voter model with rumour-spreading dynamics. It develops a martingale for consensus probability under reversibility, derives convergence-time bounds using rumour-spreading results, and provides an efficient Markov chain Monte Carlo estimator for the red-consensus probability with worst-case complexity $O(n^2\log n)$ and Erdős–Rényi efficiency $O(n\log n)$. The authors validate the approach with experiments showing that only a modest number of runs yields accurate estimates, with better performance as the graph grows. Collectively, the work enables fast, reliable probabilistic assessment of consensus outcomes in networks containing undecided agents across general and random graph topologies.

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, 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 Erdős-Rényi graphs, which makes it an efficient method for estimating 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.

Figures (7)

• Figure 1: A motivational example of an undirected graph with an initial configuration $S_0 = s_0$ consisting of one blue node ($v_1$), one red node ($v_3$), and two agnostic nodes ($v_2$ and $v_4$). Transition probabilities are uniform, i.e., $v_3$ has $\frac{1}{3}$ chance of choosing a given neighbour, whereas $v_4$ chooses $v_3$ and becomes red with probability $1$. What are the probabilities of consensus in this case?
• Figure 2: Every configuration $s_{1i}$ which can be reached from $S_0$ in Example \ref{['exm:motivation']} after one round, i.e. $a_i:=\mathbb{P}(S_1=s_{1i}|S_0)>0$. The probability of a red consensus in each case is denoted by $b_i$ and can be calculated by applying Proposition \ref{["prop:nicola's-result"]}. Therefore, the probability of red to win in Example \ref{['exm:motivation']} is $\sum_{i=1}^8a_ib_i=\frac{5}{8}$.
• Figure 3: Counterexample for the conjecture that the Martingale property (Theorem \ref{['thm:martingale']}) is valid for non-reversible chains. Note that edge weights were omitted from Figures \ref{['fig:counterexample1']},\ref{['fig:counterexample2']}, \ref{['fig:counterexample3']}, and \ref{['fig:counterexample4']} for readability.
• Figure 4: A comparison of the cumulative standard error of the probability of red consensus after all nodes are gnostic and the actual consensus until the simulation finishes. Each simulation ran 400 times on cliques, cycles and a connected subgraph of the Pokec social network with 1001 nodes (5% red, 5% blue, 90% agnostic). For cycles, the initial configuration has all red nodes side by side follow by all blue nodes side by side. For the Pokec subgraph, red and blue nodes were assigned at random, with the rest being agnostic.
• Figure 5: A comparison of the standard error of the probability of red consensus running the simulation 400 times for cliques and cycles of different sizes starting with different proportions of gnostic nodes (5%, 50%). Again, the initial configuration for cycles has all red nodes side by side follow by all blue nodes side by side.

Theorems & Definitions (13)

• proposition 1: cooper2016linear cooper2016linear
• definition 1: Voter Model with agnostic Nodes
• Remark
• theorem 1: Martingale Property
• corollary 1: Solution for Complete Graph
• lemma 1: Time bounds for consensus
• proposition 2: rumour spreading bounds for general graphs
• proposition 3: rumour spreading bounds for random graphs
• proposition 4: rumour spreading bounds for agnostic
• proposition 5: Rumour spreading bounds for random graphs - pull version