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Optimal selection of the most informative nodes for a noisy DeGroot model with stubborn agents

Roberta Raineri, Giacomo Como, Fabio Fagnani

TL;DR

This work addresses how to efficiently select a subset of regular agents to observe in a DeGroot opinion dynamics with stubborn agents and external noise in order to accurately estimate the mean equilibrium opinion $Y$. It derives explicit variance-reduction and MSE formulations, proves that the variance-reduction objective $F(mathcal{K})$ is submodular, and designs a greedy $(1-1/e)$-approximation algorithm with iterative inverse updates to compute gains efficiently. The contributions include a closed-form expression for $F$, a rigorous submodularity proof, and a scalable greedy method demonstrated on simple and real-network cases, with implications for practical polling and social sensing under noise. The results hold without extra assumptions on the noise covariance, enabling robust subset selection in noisy social networks and guiding future work on stochastic stubbornness and noisy network structures.

Abstract

Finding the optimal subset of individuals to observe in order to obtain the best estimate of the average opinion of a society is a crucial problem in a wide range of applications, including policy-making, strategic business decisions, and the analysis of sociological trends. We consider the opinion vector X to be updated according to a DeGroot opinion dynamical model with stubborn agents, subject to perturbations from external random noise, which can be interpreted as transmission errors. The objective function of the optimization problem is the variance reduction achieved by observing the equilibrium opinions of a subset K of agents. We demonstrate that, under this specific setting, the objective function exhibits the property of submodularity. This allows us to effectively design a Greedy Algorithm to solve the problem, significantly reducing its computational complexity. Simple examples are provided to validate our results.

Optimal selection of the most informative nodes for a noisy DeGroot model with stubborn agents

TL;DR

This work addresses how to efficiently select a subset of regular agents to observe in a DeGroot opinion dynamics with stubborn agents and external noise in order to accurately estimate the mean equilibrium opinion . It derives explicit variance-reduction and MSE formulations, proves that the variance-reduction objective is submodular, and designs a greedy -approximation algorithm with iterative inverse updates to compute gains efficiently. The contributions include a closed-form expression for , a rigorous submodularity proof, and a scalable greedy method demonstrated on simple and real-network cases, with implications for practical polling and social sensing under noise. The results hold without extra assumptions on the noise covariance, enabling robust subset selection in noisy social networks and guiding future work on stochastic stubbornness and noisy network structures.

Abstract

Finding the optimal subset of individuals to observe in order to obtain the best estimate of the average opinion of a society is a crucial problem in a wide range of applications, including policy-making, strategic business decisions, and the analysis of sociological trends. We consider the opinion vector X to be updated according to a DeGroot opinion dynamical model with stubborn agents, subject to perturbations from external random noise, which can be interpreted as transmission errors. The objective function of the optimization problem is the variance reduction achieved by observing the equilibrium opinions of a subset K of agents. We demonstrate that, under this specific setting, the objective function exhibits the property of submodularity. This allows us to effectively design a Greedy Algorithm to solve the problem, significantly reducing its computational complexity. Simple examples are provided to validate our results.

Paper Structure

This paper contains 9 sections, 10 theorems, 58 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model and Problem Formulation
  3. Preliminary Results
  4. Main Results
  5. Greedy Algorithm Design
  6. Conclusions
  7. Proof of Proposition \ref{['prop:general covariance']} and Corollary \ref{['lemma: C properties']}
  8. Proof of Proposition \ref{['prop:F formulation']}
  9. Proof of Proposition \ref{['prop:iterative step']}-\ref{['cor:iterative_inverse']}

Key Result

Proposition 1

Let $X$ be the random equilibrium opinion vector of the DeGroot opinion dynamics eqn:dynamical system on an undirected graph with nonempty globally reachable set of stubborn nodes. Then, $X$ has expected value and covariance matrix

Figures (6)

  • Figure 1: Cycle graph with 7 nodes. The blue links represent $\mathcal{E}$, while the red one are associated to $\hat{\mathcal{E}}$.
  • Figure 2: Comparison of centrality measures over a Watts-Strogatz graph. The colormap indicates the normalized Variance Reduction and the normalized Bonacich Centrality, respectively..
  • Figure 3: Comparison between the percentage of residual variance gained through the exact method vs the Greedy Algorithm, as function of the cardinality of the observed set $|\mathcal{K}|$, for a Watts-Strogatz Network.
  • Figure 4: Comparison between the nodes selection through exact method vs Greedy Algorithm, fixed cardinality $s=4$, for a Watts-Strogatz Network.
  • Figure 5: Malawi network used in the case study analized.
  • ...and 1 more figures

Theorems & Definitions (26)

  • Proposition 1
  • proof
  • Corollary 1
  • proof
  • Proposition 2
  • proof
  • Corollary 2
  • Remark 1
  • Example 1
  • Definition 1
  • ...and 16 more