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On final opinions of the Friedkin-Johnsen model over random graphs with partially stubborn community

Lingfei Wang, Yu Xing, Karl H. Johansson

TL;DR

The paper address the problem of how final opinions in the Friedkin-Johnsen model with a partially stubborn community behave on random graphs. It develops a probabilistic framework that bounds the distance between the final opinions on a realized graph and those on the expected graph by leveraging a lower bound on $\lambda_{\min}(M)$ and matrix concentration inequalities, under both mixed and all-stubborn regimes. A key contribution is the explicit bound $\varepsilon_n$ (and its all-stubborn variant $\varepsilon_n'$) that scales with the network size and graph parameters, showing concentration around the expected dynamics when stubborn agents are well-connected. The results are supported by SBM-based examples and simulations demonstrating the substantial influence of cross-community link probabilities and stubbornness on the opinion distance, with practical implications for predicting opinion formation in large random networks.

Abstract

This paper studies the formation of final opinions for the Friedkin-Johnsen (FJ) model with a community of partially stubborn agents. The underlying network of the FJ model is symmetric and generated from a random graph model, in which each link is added independently from a Bernoulli distribution. It is shown that the final opinions of the FJ model will concentrate around those of an FJ model over the expected graph as the network size grows, on the condition that the stubborn agents are well connected to other agents. Probability bounds are proposed for the distance between these two final opinion vectors, respectively for the cases where there exist non-stubborn agents or not. Numerical experiments are provided to illustrate the theoretical findings. The simulation shows that, in presence of non-stubborn agents, the link probability between the stubborn and the non-stubborn communities affect the distance between the two final opinion vectors significantly. Additionally, if all agents are stubborn, the opinion distance decreases with the agent stubbornness.

On final opinions of the Friedkin-Johnsen model over random graphs with partially stubborn community

TL;DR

The paper address the problem of how final opinions in the Friedkin-Johnsen model with a partially stubborn community behave on random graphs. It develops a probabilistic framework that bounds the distance between the final opinions on a realized graph and those on the expected graph by leveraging a lower bound on and matrix concentration inequalities, under both mixed and all-stubborn regimes. A key contribution is the explicit bound (and its all-stubborn variant ) that scales with the network size and graph parameters, showing concentration around the expected dynamics when stubborn agents are well-connected. The results are supported by SBM-based examples and simulations demonstrating the substantial influence of cross-community link probabilities and stubbornness on the opinion distance, with practical implications for predicting opinion formation in large random networks.

Abstract

This paper studies the formation of final opinions for the Friedkin-Johnsen (FJ) model with a community of partially stubborn agents. The underlying network of the FJ model is symmetric and generated from a random graph model, in which each link is added independently from a Bernoulli distribution. It is shown that the final opinions of the FJ model will concentrate around those of an FJ model over the expected graph as the network size grows, on the condition that the stubborn agents are well connected to other agents. Probability bounds are proposed for the distance between these two final opinion vectors, respectively for the cases where there exist non-stubborn agents or not. Numerical experiments are provided to illustrate the theoretical findings. The simulation shows that, in presence of non-stubborn agents, the link probability between the stubborn and the non-stubborn communities affect the distance between the two final opinion vectors significantly. Additionally, if all agents are stubborn, the opinion distance decreases with the agent stubbornness.
Paper Structure (11 sections, 9 theorems, 53 equations, 3 figures)

This paper contains 11 sections, 9 theorems, 53 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Graphs and random graph model
  4. FJ model
  5. Concentration inequalities
  6. Problem formulation
  7. Main results
  8. Lower bound for $\lambda_{\min}(M)$
  9. probability bound for $\|\mathbf{x}(\infty)-\Bar{\mathbf{x}}(\infty)\|$
  10. Examples
  11. Conclusion

Key Result

Lemma 1

For the FJ model eq:FJ, if for each agent $i$, either $\theta_i>0$ or $i$ is connected to some agent $j$ with $\theta_j>0$, the opinion vector $\mathbf{x}(t)$ will converge, and where $P$ is a stochastic matrix.

Figures (3)

  • Figure 1: The $\log$-$\log$ plot of $\mathrm{Dist}$ w.r.t the network size $n$ for Example \ref{['ex:bd_net_size']}. The red shade represents the range of all $\mathrm{Dist}$s generated in the simulation. The gray curve is the change of the medians of $\mathrm{Dist}$ for each $n$. The two dashed curves are drawn for comparison, with slopes $-\frac{1}{2}$ and $-\frac{3}{5}$, respectively.
  • Figure 2: The scatter plot of the medians w.r.t. the degree parameter triplet $(p_s,p_r,p_{sr})$ for Example \ref{['ex:bd_degree']}. The node color is increasing with the corresponding median.
  • Figure 3: Plot of $\mathrm{Dist}$ w.r.t. the agent stubbornness $\theta$ for Example \ref{['ex:bd_stub']}. The red shade represents the range of all $\mathrm{Dist}$s generated in the simulation.

Theorems & Definitions (24)

  • Definition 1: Random graph model
  • Definition 2: SBM
  • Lemma 1
  • Lemma 2: Bernstein inequality
  • Lemma 3: Chernoff inequality
  • Lemma 4
  • proof
  • Remark 1
  • Lemma 5
  • proof
  • ...and 14 more