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Countering Election Sway: Strategic Algorithms in Friedkin-Johnsen Dynamics

Dragos Ristache, Fabian Spaeh, Charalampos E. Tsourakakis

TL;DR

This work studies the vulnerability of elections to adversarial influence under Friedkin-Johnsen opinion dynamics, focusing on shifting the equilibrium median through Asch-like interventions that adjust node susceptibilities. It formalizes the problem as maximizing or flipping the median of the equilibrium under an $\ell_p$ budget, proves NP-hardness and inapproximability for median/quantile objectives, and offers two continuous (Huber M-estimator, sigmoid threshold) and one discrete (lazy greedy) algorithms, with an exact polynomial-time DP on rooted directed trees. Experiments on synthetic and real-world networks show that a small fraction of strategically chosen stooges can significantly sway the median, underscoring the need for defense mechanisms in social-influence systems. The results provide practical algorithms and insights for safeguarding elections and for understanding the limits of manipulation in continuous opinion dynamics.

Abstract

Social influence profoundly impacts individual choices and collective behaviors in politics. In this work, driven by the goal of protecting elections from improper influence, we consider the following scenario: an individual, who has vested interests in political party $Y$, is aware through reliable surveys that parties $X$ and $Y$ are likely to get 50.1\% and 49.9\% of the vote, respectively. Could this individual employ strategies to alter public opinions and consequently invert these polling numbers in favor of party $Y$? We address this question by employing: (i) the Friedkin-Johnsen (FJ) opinion dynamics model, which is mathematically sophisticated and effectively captures the way individual biases and social interactions shape opinions, making it crucial for examining social influence, and (ii) interventions similar to those in Asch's experiments, which involve selecting a group of stooges within the network to spread a specific opinion. We mathematically formalize the aforementioned motivation as an optimization framework and establish that it is NP-hard and inapproximable within any constant factor. We introduce three efficient polynomial-time algorithms. The first two utilize a continuous approach: one employs gradient descent with Huber's estimator to approximate the median, and the other uses a sigmoid threshold influence function. The third utilizes a combinatorial greedy algorithm for targeted interventions. Through comparative analysis against various natural baselines and using real-world data, our results demonstrate that in numerous cases a small fraction of nodes chosen as stooges can significantly sway election outcomes under the Friedkin-Johnsen model.

Countering Election Sway: Strategic Algorithms in Friedkin-Johnsen Dynamics

TL;DR

This work studies the vulnerability of elections to adversarial influence under Friedkin-Johnsen opinion dynamics, focusing on shifting the equilibrium median through Asch-like interventions that adjust node susceptibilities. It formalizes the problem as maximizing or flipping the median of the equilibrium under an budget, proves NP-hardness and inapproximability for median/quantile objectives, and offers two continuous (Huber M-estimator, sigmoid threshold) and one discrete (lazy greedy) algorithms, with an exact polynomial-time DP on rooted directed trees. Experiments on synthetic and real-world networks show that a small fraction of strategically chosen stooges can significantly sway the median, underscoring the need for defense mechanisms in social-influence systems. The results provide practical algorithms and insights for safeguarding elections and for understanding the limits of manipulation in continuous opinion dynamics.

Abstract

Social influence profoundly impacts individual choices and collective behaviors in politics. In this work, driven by the goal of protecting elections from improper influence, we consider the following scenario: an individual, who has vested interests in political party , is aware through reliable surveys that parties and are likely to get 50.1\% and 49.9\% of the vote, respectively. Could this individual employ strategies to alter public opinions and consequently invert these polling numbers in favor of party ? We address this question by employing: (i) the Friedkin-Johnsen (FJ) opinion dynamics model, which is mathematically sophisticated and effectively captures the way individual biases and social interactions shape opinions, making it crucial for examining social influence, and (ii) interventions similar to those in Asch's experiments, which involve selecting a group of stooges within the network to spread a specific opinion. We mathematically formalize the aforementioned motivation as an optimization framework and establish that it is NP-hard and inapproximable within any constant factor. We introduce three efficient polynomial-time algorithms. The first two utilize a continuous approach: one employs gradient descent with Huber's estimator to approximate the median, and the other uses a sigmoid threshold influence function. The third utilizes a combinatorial greedy algorithm for targeted interventions. Through comparative analysis against various natural baselines and using real-world data, our results demonstrate that in numerous cases a small fraction of nodes chosen as stooges can significantly sway election outcomes under the Friedkin-Johnsen model.

Paper Structure

This paper contains 33 sections, 2 theorems, 24 equations, 13 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Opinion Dynamics and the Friedkin-Johnsen Model.
  4. Optimizing objectives within the Friedkin-Johnsen model
  5. Proposed methods
  6. Hardness of Problem \ref{['prob:max-median']}
  7. Huber M-Estimator
  8. Gradient Formulation
  9. Algorithm and complexity
  10. Selecting the Huber tuning constant $c$
  11. Sigmoid Threshold Influence Method
  12. Discrete Lazy Greedy
  13. Experimental Evaluation
  14. Experimental Setup
  15. Datasets.
  16. ...and 18 more sections

Key Result

Theorem 1

Problem prob:max-median for $p=0$ is inapproximable to any multiplicative factor unless $\mathsf{P} = \mathsf{NP}$.

Figures (13)

  • Figure 1: An example showcasing challenges in Problems \ref{['prob:max-median']} and \ref{['prob:elections']}. We show an isolated component in a larger graph, where the black vertices on the left have innate innate opinion $0$, but are only connected to the white vertex on the right with innate opinion $0.5$ but resistance $0$.
  • Figure 2: Performance of $\textsf{Projected Huber}$ on (a) a $10\times 10$Grid and (b) $\textsf{Beefban-F}$ with a budget of $k=50$ stooges, for various values of $c$. The plots display the instance-specific value of $c$ determined by our heuristic strategy to identify an optimal value. Median and runtime as a function of parameter $\phi$ for the $\textsf{Lazy Greedy}$ on (c) a $23 \times 23$ grid and (d) Beefban-F.
  • Figure 3: Flipping the median on real-world networks. On the left, we show the required number of stooges to flip the median across the threshold of $\theta=0.5$, as a percentage of the number of vertices in the graph. On the right, we show the running times, where the budget is the least budget to move the median opinion above the threshold, corresponding to the values on the left.
  • Figure 4: Scalability on synthetic networks when innate opinions follow a $\textsf{Normal}$ distribution.
  • Figure 5: Median maximization on synthetic graphs with $\textsf{Normal}$ innate opinions. We show the number of stooges required to move the median over the threshold $\theta = 0.5$, as a percentage of the number of nodes in the graph. We report average and standard deviation over 10 runs.
  • ...and 8 more figures

Theorems & Definitions (4)

  • Theorem 1
  • Corollary 1
  • proof
  • proof