Pith. sign in

REVIEW 4 major objections 5 minor 68 references

Countering Election Sway: Strategic Algorithms in Friedkin-Johnsen Dynamics

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper claims that maximizing the median equilibrium opinion under Friedkin-Johnsen dynamics with a budget of changed resistances is NP-hard to approximate, yet small stooge sets can flip the median in practice.

desk verdict Novel median objective under FJ dynamics with mostly sound theory and honest experiments; the headline hardness proof has a one-line definitional gap, and the experimental write-up needs error bars and disclosure fixes, so it deserves review but not as-is. read the letter →

arxiv 2502.01874 v1 pith:GX4SMBAG submitted 2025-02-03 cs.SI

Dragos Ristache , Fabian Spaeh , Charalampos E. Tsourakakis This is my paper · ORCID
classification cs.SI MSC 68Q1791D30
keywords opiniondynamicsFriedkin-Johnsenmodelmedianelectionmanipulationstoogesinapproximabilitysetcoverlazygreedy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies whether a small group of 'stooge' nodes, whose resistance to peer influence is changed, can flip the median opinion of a network to win an election under the Friedkin-Johnsen (FJ) opinion dynamics model. The authors formalize the problem as maximizing the median equilibrium opinion subject to a budget on changed resistance parameters. They prove that for the budget that counts changed nodes, this problem is NP-hard and inapproximable to any constant factor unless P = NP, and the same holds for any quantile. For practice, they design two continuous gradient-based heuristics that smooth the median and a lazy greedy selector, and find on real-world social networks that a small fraction of stooges can move the median across the 0.5 threshold; on rooted directed trees they give an exact polynomial-time algorithm.

What carries the argument

The central object is the median of the equilibrium opinion vector $x^{\star}(\alpha', W, s)$ under the generalized Friedkin-Johnsen dynamics, in which each expressed opinion is a weighted blend of a private innate opinion and the neighbors' current opinions, with interventions that change the resistance parameter $\alpha$ (susceptibility to persuasion) of a set of nodes. The argument's main drivers are a set-cover reduction that makes the median positive exactly when a cover exists, and continuous smooth surrogates, an M-estimator loss and a sigmoid threshold reward, that let gradient ascent avoid the median's non-differentiability. The greedy variant uses marginal-gain lazy evaluation to speed up the search for the best stooge set.

What would settle it

Apply the reduction of Theorem 1 to a set-cover instance with a full cover, then compute the median under the lower-median tie-break instead of the paper's upper-median tie-break; if the yes-case median is not positive, the claimed gap collapses for that convention.

Watch

Extended reading notes

Core claim

The central claim is that the median of the equilibrium opinion vector, the statistic that determines which side has more than half the votes, can be manipulated efficiently in practice even though the optimization is worst-case intractable. Concretely, the paper shows that under the generalized FJ dynamics, changing the resistance parameter of k nodes, a stooge-style intervention, can push the median from below to above 0.5 on real-world networks with a small percentage of nodes, and the continuous M-estimator method typically requires the fewest stooges. On the theory side, the paper's reduction from set cover establishes that no multiplicative approximation algorithm exists for maximizing the median with a budget on changed nodes, and this hardness extends to every fixed quantile. The paper also provides an exact dynamic program for rooted directed trees, where opinions flow only toward the root.

Load-bearing premise

The inapproximability proof assumes a tie-breaking convention for the median on an even number of nodes; if the standard lower-median convention is used, the yes-case of the reduction may not produce a positive median, and the suggested fix of duplicating nodes is not fully worked out.

Editorial extensions

If this is right

  • Election-defense analysts can use the proposed algorithms to identify small sets of nodes whose resistance changes would shift the median, and then harden those nodes.
  • No polynomial-time multiplicative approximation for the budgeted median maximization exists unless P = NP, and the same applies to any quantile.
  • The continuous relaxation method robustly flips the median with fewer stooges than centrality-, degree-, or random-based baselines on most real-world datasets.
  • The exact dynamic program for rooted directed trees provides optimal stooge sets for hierarchical organizations that resemble corporate or governmental chains.
  • Continuous and discrete methods have complementary strengths: continuous methods scale with the network size independent of the budget, while lazy greedy is effective but slower on larger budgets.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The success of continuous methods suggests that graded, small resistance changes across many nodes, rather than only discrete stooge selections, can be the efficient attack; a defender could test whether randomizing resistances blunts both.
  • Because real-world networks with the largest susceptibility have medians far from their means, opinion skewness itself may be a measurable vulnerability indicator that predicts how few stooges are needed.
  • The exact tree algorithm opens a natural route to bounded-treewidth approximations, a direction the authors list as future work.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper formalizes election swaying under the generalized Friedkin-Johnsen (FJ) opinion dynamics as the problem of maximizing the median equilibrium opinion by changing nodes' resistance parameters subject to a budget (Problem 2), together with the dual problem of minimizing the budget needed to push the median past a threshold (Problem 3). It claims that the ℓ0-budget maximization is NP-hard and inapproximable to any multiplicative factor, and it proposes three heuristics: projected gradient ascent on a Huber-loss approximation of the median, projected gradient ascent on a sigmoid threshold objective, and a lazy discrete greedy algorithm. An exact dynamic program for directed trees is also claimed. The paper reports experiments on synthetic and real-world networks, concluding that a small fraction of stooges can often flip the median across 0.5.

Significance. If the results hold, the paper introduces a well-motivated and previously unstudied optimization problem, and the combination of a formal hardness result with three practical heuristics would be a useful contribution to the opinion-dynamics optimization literature. The gradient derivation in Section 3.2 is explicit and algebraically checkable, and the paper ships an anonymous code repository and evaluates on multiple real-world datasets, which are genuine strengths. The inapproximability claim and the exact tree algorithm are, however, not fully supported as written: the hardness proof depends on an unspecified median tie-breaking convention, and the tree DP appears to miscount votes. The empirical comparison also leaves the conversion from continuous resistance changes to discrete stooge counts unspecified for the best-performing methods. These are correctable issues, but they affect load-bearing claims in the paper.

major comments (4)
  1. [Appendix C, proof of Theorem 1] The hardness reduction constructs exactly 2(n+m+k) nodes, and in the yes-case exactly n+m+k nodes have opinion >0. Whether this yields Median(x*)>0 depends on the convention for the median of an even-sized set, which Problem 2 never specifies. The proof's sentence "we break ties such that the Median(x*) > 0" is an ad hoc convention, and under the lower-median convention the yes-case median is 0, so the gap to the no-case optimum of 0 collapses. The parenthetical suggestion to duplicate all u_i nodes is not carried out and changes the node count, and the no-case counting would need to be redone. This is a rigor gap in the central inapproximability result; it can be repaired by explicitly defining the median in Problem 2 to be the upper middle value, or by adding an odd number of dummy zero-opinion nodes and re-verifying the counts in both cases.
  2. [Appendix B, Algorithm 4] The exact dynamic program for directed trees appears to miscount votes when the current node is selected as a stooge. In lines 36-37, the non-stooge case correctly sets dpu,j+pos,k using pos = 1[finalOpinion ≥ 0.5], but the stooge case sets dpu,j+1,k+1 ← max(dpu,j+pos,k, su). Since making u a stooge with α_u=1 leaves its opinion equal to s_u, the vote count should increase only if s_u > 0.5; the α_u=0 option would also need its own re-evaluation of finalOpinion. As written, the DP can certify a stooge set of size k as flipping the median even when s_u ≤ 0.5 for the selected stooge, so the claimed exact polynomial-time algorithm for Problem 3 on directed trees is not correct as stated.
  3. [Section 3.2, Algorithms 1 and 3] Both continuous algorithms project only onto the ℓ1 ball around α0: Algorithm 1 line 11 and Algorithm 3 line 7 use min{∥α−α′∥2 : ∥α−α0∥1 ≤ k}. Problem 2 and Formulation 1 require α′∈[0,1]^n, and the ℓ1 projection alone can leave coordinates outside [0,1], producing invalid resistance vectors. The projection should be onto the intersection of the ℓ1 ball and the box [0,1]^n; otherwise the reported resistance vectors are not feasible instances of the stated problem.
  4. [Section 4.3, Figure 3] The comparison behind the headline empirical claim is not well defined for the continuous methods. Figure 3 (left) reports the "number of stooges" required by Projected Huber and Sigmoid, but these methods optimize an ℓ1 budget; Section 4.1 only says the budget is halved for comparison. The paper does not specify how a continuous resistance vector is converted into a discrete stooge set (e.g., thresholding, top-k deviations, or rounding), so the claim that a small fraction of stooges flips the median is ambiguous for the two methods that perform best. Please specify the conversion or restrict the stooge-count claim to Lazy Greedy and the discrete baselines.
minor comments (5)
  1. [Section 3.2, Eq. (6)] The derivative of the Huber M-estimator is derived under the assumption that x⋆_i ≠ ŷ and that no residual lies exactly at the cutoff c; the boundary cases should be stated, for example by adopting a subgradient convention, since they affect the definition of I and the active set.
  2. [Problem 2 and Problem 3] The median is used throughout without a definition for even cardinality; please state the convention explicitly, and ideally use the same convention in the hardness proof and in the algorithms.
  3. [Appendix B, Algorithm 4, line 20] The auxiliary DP recurrence contains undefined or inconsistent variable names (`pbc` versus `pnc`, and `wuc`), and the normalization of the weights should be made explicit so the recurrence is reproducible.
  4. [Section 4.4] The heuristic for choosing the Huber constant c says the graph's resistances and opinions are perturbed by a small constant ϵ, but no value of ϵ or number of sampled instances is given; without these details the ablation is not reproducible.
  5. [Throughout] There are several typos that should be corrected, including "loosing" in Section 3, "we showcase the scalability of our network" in Section 4.2, and "becomes significant a significant bottleneck" in Section 4.2.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the hardness reduction and algorithmic results are self-contained; the Appendix C tie-break convention and the in-sample c-tuning are rigor/methodology concerns, not circular steps.

full rationale

The paper's core claims do not reduce to their inputs by construction. The FJ equilibrium formula x* = (I-(I-A)W)^{-1} As is standard and externally sourced, and the median objective is a new, non-smooth functional whose optimization is derived independently. Theorem 1 is proved by a set-cover reduction (Appendix C) that genuinely maps set-cover feasibility to the sign of the median under the stated tie-breaking; the acknowledged 'we break ties such that Median(x*) > 0' is an underspecified median convention (a rigor gap with a suggested duplication repair), not a reduction of the result to itself. The self-citations to [ACK+22] and [RST24] are used only to motivate the resistance-level Asch intervention and the generalized FJ model, which are also supported by external references, so no load-bearing claim rests solely on the authors' prior work. The Huber-c selection in Section 4.4 tunes a surrogate hyperparameter using the true median on perturbed copies of the instance, which is an in-sample tuning concern rather than a fitted-input-called-prediction circularity, and the reported 'small fraction of stooges' outcome is an optimization result compared against baselines, not a prediction forced by the tuning. Overall, the derivation chain is self-contained, with no equation-level equivalence between input and claimed output.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

No fundamentally new entities are introduced; the 'stooge' is a modeling device for changing resistance parameters in the existing FJ model. The main load-bearing assumptions are the FJ equilibrium formula, the full-rank condition for the gradient derivation, the median tie-breaking convention in the hardness proof, and the adequacy of the Huber surrogate.

free parameters (4)
  • Huber tuning constant c = instance-specific, selected by heuristic (Section 4.4)
    Controls the trade-off between median approximation and gradient smoothness; selected to minimize the distance of the Huber estimate to the true median on randomly perturbed copies of the graph.
  • Sigmoid temperature tau = 25
    Hand-set in Section 4.1; controls the steepness of the sigmoid threshold reward function in the Sigmoid method.
  • Laziness factor phi = 0.8
    Hand-set in Section 4.1; controls the early-abort condition in Lazy Greedy, trading accuracy for running time.
  • ADAM optimizer parameters = beta1=0.9, beta2=0.999
    Standard optimizer settings used for gradient ascent; not tuned per instance.
assumptions (4)
  • domain assumption The generalized FJ equilibrium x* = (I - (I-A)W)^{-1} A s (Eq. 4) is the correct fixed point of the update (Eq. 2).
    Taken from prior literature [ACK+22, GS13, BKO15]; the paper builds all optimization objectives on this equilibrium.
  • domain assumption Matrix X = I - (I-A)W has full rank for the instances considered, so the pseudoinverse equals the inverse and the derivative formula (Eq. 6) applies.
    Stated in Section 3.2 ('since X has full rank'); required for the gradient computation in Algorithm 1 and for the sigmoid gradient.
  • ad hoc to paper The median of an even-sized set is defined with a tie-breaking rule that returns the upper-middle value or a positive value when exactly half the opinions exceed the threshold.
    Appendix C: the yes-case of the set-cover reduction leaves exactly half the nodes above threshold, so the median is >0 only under a specific median convention; the authors acknowledge the assumption and propose a not-fully-worked fix.
  • domain assumption The Huber M-estimator y^_c approximates the median sufficiently well that gradient ascent on y^_c increases the true median.
    Used implicitly in Section 3.2; the quality of the approximation depends on c and is validated only heuristically in Section 4.4.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Countering Election Sway: Strategic Algorithms in Friedkin-Johnsen Dynamics." pith.science (2026). https://pith.science/paper/GX4SMBAG

@misc{pith2026250201874,
  author       = {Pith},
  title        = {Pith review of: Countering Election Sway: Strategic Algorithms in Friedkin-Johnsen Dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GX4SMBAG}},
  note         = {Machine review of arXiv:2502.01874}
}
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.

Figures

Figures reproduced from arXiv: 2502.01874 by the authors.

Figure 1
Figure 1. An example showcasing challenges in Problems 2 and 3. We show an isolated component in a larger [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Performance of Projected Huber on (a) a 10 × 10 Grid and (b) 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 ϕ for the Lazy Greedy on (c) a 23 × 23 grid and (d) Beefban-F. 4.3 Swaying the Election Results [PITH_FULL_IMAGE:figures/full… view at source ↗
Figure 3
Figure 3. Flipping the median on real-world networks. On the left, we show the required number of stooges [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Scalability on synthetic networks when innate opinions follow a [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: Median maximization on synthetic graphs with [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: Median maximization on synthetic graphs with [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 7
Figure 7. Figure 7: Median maximization on synthetic graphs with [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]
Figure 8
Figure 8. Figure 8: Flipping the median opinion on a Grid. We show the median opinion after optimization for an increasing number of stooges, until the threshold of θ = 0.5 is reached. stooges. Interestingly, Tree networks require a high amount of targeted stooges. Opinions for each node …
Figure 9
Figure 9. Figure 9: Pairwise Jaccard similarity on the set of selected [PITH_FULL_IMAGE:figures/full_fig_p022_9.png]
Figure 10
Figure 10. Figure 10: An organizational chart, which is an example of a real world hierarchy graph, where the information [PITH_FULL_IMAGE:figures/full_fig_p023_10.png]
Figure 11
Figure 11. Figure 11: Comparison of DP vs Greedy Approaches: (Left) Simulated Org Chart, (Center) Random Tree, [PITH_FULL_IMAGE:figures/full_fig_p025_11.png]
Figure 12
Figure 12. Figure 12: A difficult instance. On the left, we show the tree before introducing stooges. In the middle, we [PITH_FULL_IMAGE:figures/full_fig_p025_12.png]
Figure 13
Figure 13. Figure 13: Gadget used in the proof of Theorem 1. We reduce from a set cover instance with [PITH_FULL_IMAGE:figures/full_fig_p027_13.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

68 extracted references · 66 canonical work pages

  1. [1]

    Mathematical models of the distribution of attitudes under controversy

    Robert P Abelson. Mathematical models of the distribution of attitudes under controversy. Contributions to mathematical psychology , 1964

    work page 1964

  2. [2]

    Opinion fluctuations and disagreement in social networks

    Daron Acemo g lu, Giacomo Como, Fabio Fagnani, and Asuman Ozdaglar. Opinion fluctuations and disagreement in social networks. Mathematics of Operations Research , 38(1):1--27, 2013

    work page 2013

  3. [3]

    Opinion dynamics with varying susceptibility to persuasion via non-convex local search

    Rediet Abebe, TH Chan, Jon Kleinberg, Zhibin Liang, David Parkes, Mauro Sozio, and Charalampos Tsourakakis. Opinion dynamics with varying susceptibility to persuasion via non-convex local search. ACM Transactions on Knowledge Discovery from Data (TKDD) , 16, 2022

    work page 2022

  4. [4]

    Social media and fake news in the 2016 election

    Hunt Allcott and Matthew Gentzkow. Social media and fake news in the 2016 election. Journal of economic perspectives , 31(2):211--36, 2017

    work page 2016

  5. [5]

    Opinion dynamics and learning in social networks

    Daron Acemoglu and Asuman Ozdaglar. Opinion dynamics and learning in social networks. Dynamic Games and Applications , 1(1):3--49, 2011

    work page 2011

  6. [6]

    Solomon E. Asch. Opinions and social pressure. Scientific American , 193(5):31--35, 1955

    work page 1955

  7. [7]

    All politics is national: The rise of negative partisanship and the nationalization of us house and senate elections in the 21st century

    Alan Abramowitz and Steven Webster. All politics is national: The rise of negative partisanship and the nationalization of us house and senate elections in the 21st century. In Annual Meeting of the Midwest Political Science Association Conference , pages 16--19, 2015

    work page 2015

  8. [8]

    America's political divide by the numbers, 2014

    BBC. America's political divide by the numbers, 2014. Source BBC https://www.bbc.com/news/av/magazine-27629535/america-s-political-divide-by-the-numbers

    work page 2014

Show all 68 references
  1. [9]

    The reasons why politics feels so tribal in 2016, 2016

    BBC. The reasons why politics feels so tribal in 2016, 2016. Source BBC https://www.bbc.com/future/article/20160823-how-modern-life-is-destroying-democracy

    2016

  2. [10]

    Dynamics of opinion polarization

    Elisabetta Biondi, Chiara Boldrini, Andrea Passarella, and Marco Conti. Dynamics of opinion polarization. IEEE Trans. Syst. Man Cybern. Syst. , 53(9):5381--5392, 2023

    2023

  3. [11]

    Bounds on the voter model in dynamic networks

    Petra Berenbrink, George Giakkoupis, Anne-Marie Kermarrec, and Frederik Mallmann-Trenn. Bounds on the voter model in dynamic networks. arXiv preprint arXiv:1603.01895 , 2016

    2016 arXiv

  4. [12]

    Cross-country trends in affective polarization

    Levi Boxell, Matthew Gentzkow, and Jesse M Shapiro. Cross-country trends in affective polarization. Technical report, National Bureau of Economic Research, 2020

    2020

  5. [13]

    The smoothed median and the bootstrap

    BM Brown, Peter Hall, and GA Young. The smoothed median and the bootstrap. Biometrika , 88(2):519--534, 2001

    2001

  6. [14]

    How bad is forming your own opinion? Games and Economic Behavior , 92:248--265, 2015

    David Bindel, Jon Kleinberg, and Sigal Oren. How bad is forming your own opinion? Games and Economic Behavior , 92:248--265, 2015

    2015

  7. [15]

    Statistical physics of social dynamics

    Claudio Castellano, Santo Fortunato, and Vittorio Loreto. Statistical physics of social dynamics. Reviews of modern physics , 81(2):591, 2009

    2009

  8. [16]

    The science of persuasion

    Robert B Cialdini. The science of persuasion. Scientific American , 284(2):76--81, 2001

    2001

  9. [17]

    T. - H. Hubert Chan and Chui Shan Lee. On the hardness of opinion dynamics optimization with l\( _ 1 \)-budget on varying susceptibility to persuasion. In COCOON , volume 13025 of Lecture Notes in Computer Science , pages 515--527. Springer, 2021

    2021

  10. [18]

    Chen and Mikl \' o s Z

    Mayee F. Chen and Mikl \' o s Z. R \' a cz. An adversarial model of network disruption: Maximizing disagreement and polarization in social networks. IEEE Trans. Netw. Sci. Eng. , 9(2):728--739, 2022

    2022

  11. [19]

    Reaching a consensus

    Morris H DeGroot. Reaching a consensus. Journal of the American Statistical Association , 69(345):118--121, 1974

    1974

  12. [20]

    Justice Department

    U.S. Justice Department. Internet research agency indictment - D epartment of J ustice, 2018. URL https://www.justice.gov/file/1035477/download

    2018

  13. [21]

    On random graphs i

    P ERDdS and A R&wi. On random graphs i. Publ. math. debrecen , 6(290-297):18, 1959

    1959

  14. [22]

    How truth wins in opinion dynamics along issue sequences

    Noah E Friedkin and Francesco Bullo. How truth wins in opinion dynamics along issue sequences. Proceedings of the National Academy of Sciences , 114(43):11380--11385, 2017

    2017

  15. [23]

    A formal theory of social power

    John RP French Jr. A formal theory of social power. Psychological review , 63(3):181, 1956

    1956

  16. [24]

    Social influence and opinions

    Noah E Friedkin and Eugene C Johnsen. Social influence and opinions. Journal of mathematical sociology , 15(3-4):193--206, 1990

    1990

  17. [25]

    Social positions in influence networks

    Noah E Friedkin and Eugene C Johnsen. Social positions in influence networks. Social networks , 19(3):209--222, 1997

    1997

  18. [26]

    Social influence network theory: A sociological examination of small group dynamics , volume 33

    Noah E Friedkin and Eugene C Johnsen. Social influence network theory: A sociological examination of small group dynamics , volume 33. Cambridge University Press, 2011

    2011

  19. [27]

    A theory of the evolution of social power: Natural trajectories of interpersonal influence systems along issue sequences

    Noah E Friedkin, Peng Jia, and Francesco Bullo. A theory of the evolution of social power: Natural trajectories of interpersonal influence systems along issue sequences. Sociological Science , 3:444--472, 2016

    2016

  20. [28]

    A set of measures of centrality based on betweenness

    LC Freeman. A set of measures of centrality based on betweenness. Sociometry , 1977

    1977

  21. [29]

    The problem of social control and coordination of complex systems in sociology: A look at the community cleavage problem

    Noah E Friedkin. The problem of social control and coordination of complex systems in sociology: A look at the community cleavage problem. IEEE Control Systems Magazine , 35(3):40--51, 2015

    2015

  22. [30]

    Adversarial perturbations of opinion dynamics in networks

    Jason Gaitonde, Jon Kleinberg, and Eva Tardos. Adversarial perturbations of opinion dynamics in networks. In Proceedings of the 21st ACM Conference on Economics and Computation , pages 471--472, 2020

    2020

  23. [31]

    Quantifying controversy on social media

    Kiran Garimella, Gianmarco De Francisci Morales, Aristides Gionis, and Michael Mathioudakis. Quantifying controversy on social media. ACM Trans. Soc. Comput. , 1(1):3:1--3:27, 2018

    2018

  24. [32]

    Community structure in social and biological networks

    Michelle Girvan and Mark EJ Newman. Community structure in social and biological networks. Proceedings of the national academy of sciences , 99(12):7821--7826, 2002

    2002

  25. [33]

    A survey on nonstrategic models of opinion dynamics

    Michel Grabisch and Agnieszka Rusinowska. A survey on nonstrategic models of opinion dynamics. Games , 11(4):65, 2020

    2020

  26. [34]

    Opinion dynamics in social networks: A local interaction game with stubborn agents

    Javad Ghaderi and R Srikant. Opinion dynamics in social networks: A local interaction game with stubborn agents. In American Control Conference (ACC), 2013 , pages 1982--1987. IEEE, 2013

    2013

  27. [35]

    Opinion dynamics in social networks with stubborn agents: Equilibrium and convergence rate

    Javad Ghaderi and Rayadurgam Srikant. Opinion dynamics in social networks with stubborn agents: Equilibrium and convergence rate. Automatica , 50(12):3209--3215, 2014

    2014

  28. [36]

    Opinion maximization in social networks

    Aristides Gionis, Evimaria Terzi, and Panayiotis Tsaparas. Opinion maximization in social networks. In Proceedings of the 2013 SIAM International Conference on Data Mining , pages 387--395. SIAM, 2013

    2013

  29. [37]

    The relationship between knowledge, belief, and certainty

    Joseph Y Halpern. The relationship between knowledge, belief, and certainty. Annals of mathematics and artificial intelligence , 4:301--322, 1991

    1991

  30. [38]

    A smoothing principle for the huber and other location m-estimators

    Frank Hampel, Christian Hennig, and Elvezio Ronchetti. A smoothing principle for the huber and other location m-estimators. Computational Statistics & Data Analysis , 55(1):324--337, 2011

    2011

  31. [39]

    Opinion dynamics and bounded confidence models, analysis, and simulation

    Rainer Hegselmann, Ulrich Krause, et al. Opinion dynamics and bounded confidence models, analysis, and simulation. Journal of Artificial Societies and Social Simulation , 5(3), 2002

    2002

  32. [40]

    The dark psychology of social networks

    Jonathan Haidt and Tobias Rose-Stockwell. The dark psychology of social networks. The Atlantic , pages 6--60, 2019

    2019

  33. [41]

    Mathematical Models of Attitude Change

    John E Hunter. Mathematical Models of Attitude Change. Volume 1, Change in Single Attitudes and Cognitive Structure . 1984

    1984

  34. [42]

    Maximizing the spread of influence through a social network

    David Kempe, Jon Kleinberg, and \'E va Tardos. Maximizing the spread of influence through a social network. In Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining , pages 137--146. ACM, 2003

    2003

  35. [43]

    Why we're polarized

    Ezra Klein. Why we're polarized . Simon and Schuster, 2020

    2020

  36. [44]

    Continuous opinion dynamics under bounded confidence: A survey

    Jan Lorenz. Continuous opinion dynamics under bounded confidence: A survey. International Journal of Modern Physics C , 18(12):1819--1838, 2007

    2007

  37. [45]

    That uplifting tweet you just shared? a russian troll sent it, 2019

    Darren Linvill and Patrick Warren. That uplifting tweet you just shared? a russian troll sent it, 2019. URL https://www.rollingstone.com/politics/politics-features/russia-troll-2020-election-interference-twitter-916482/

    2019

  38. [46]

    Algorithmic extremism: Examining youtube's rabbit hole of radicalization

    Mark Ledwich and Anna Zaitsev. Algorithmic extremism: Examining youtube's rabbit hole of radicalization. arXiv preprint arXiv:1912.11211 , 2019

    1912 arXiv

  39. [47]

    Zucked: Waking up to the Facebook catastrophe

    Roger McNamee. Zucked: Waking up to the Facebook catastrophe . Penguin, 2020

    2020

  40. [48]

    Tsourakakis

    Cameron Musco, Christopher Musco, and Charalampos E. Tsourakakis. Minimizing polarization and disagreement in social networks. In WWW , pages 369--378. ACM , 2018

    2018

  41. [49]

    Sublinear-time opinion estimation in the friedkin--johnsen model

    Stefan Neumann, Yinhao Dong, and Pan Peng. Sublinear-time opinion estimation in the friedkin--johnsen model. In Proceedings of the ACM on Web Conference 2024 , pages 2563--2571, 2024

    2024

  42. [50]

    Reply-guys go hunting: An investigation into a U

    Stanford Internet Observatory. Reply-guys go hunting: An investigation into a U . S . astroturfing operation on facebook, twitter, and instagram. 2020. URL https://stacks.stanford.edu/file/druid:vh222ch4142/facebook-US-202009.pdf

    2020

  43. [51]

    Available at https://anonymous.4open.science/r/Elections-FB8B

    Anonymous repository, 2024. Available at https://anonymous.4open.science/r/Elections-FB8B

    2024

  44. [52]

    A tutorial on modeling and analysis of dynamic social networks

    Anton V Proskurnikov and Roberto Tempo. A tutorial on modeling and analysis of dynamic social networks. part i. Annual Reviews in Control , 43:65--79, 2017

    2017

  45. [53]

    A tutorial on modeling and analysis of dynamic social networks

    Anton V Proskurnikov and Roberto Tempo. A tutorial on modeling and analysis of dynamic social networks. part ii. Annual Reviews in Control , 45:166--190, 2018

    2018

  46. [54]

    Mining knowledge-sharing sites for viral marketing

    Matthew Richardson and Pedro Domingos. Mining knowledge-sharing sites for viral marketing. In Proceedings of the eighth ACM SIGKDD international conference on Knowledge discovery and data mining , pages 61--70, 2002

    2002

  47. [55]

    Political polarization in the american public, 2014

    Pew Research. Political polarization in the american public, 2014. Pew Research link https://www.people-press.org/2014/06/12/political-polarization-in-the-american-public/

    2014

  48. [56]

    Partisanship and political animosity in 2016, 2016

    Pew Research. Partisanship and political animosity in 2016, 2016. Pew Research link https://www.people-press.org/2016/06/22/partisanship-and-political-animosity-in-2016/

    2016

  49. [57]

    R \' a cz and Daniel E

    Mikl \' o s Z. R \' a cz and Daniel E. Rigobon. Towards consensus: Reducing polarization by perturbing social networks. IEEE Trans. Netw. Sci. Eng. , 10(6):3450--3464, 2023

    2023

  50. [58]

    Wiser than the wisest of crowds: The asch effect revisited under friedkin-johnsen opinion dynamics

    Dragos Ristache, Fabian Spaeh, and Charalampos E Tsourakakis. Wiser than the wisest of crowds: The asch effect revisited under friedkin-johnsen opinion dynamics. arXiv e-prints , pages arXiv--2406, 2024

    2024

  51. [59]

    Mapping the anti-vaccination movement on facebook

    Naomi Smith and Tim Graham. Mapping the anti-vaccination movement on facebook. Information, Communication & Society , 22(9):1310--1327, 2019

    2019

  52. [60]

    Opinion optimization in directed social networks

    Haoxin Sun and Zhongzhi Zhang. Opinion optimization in directed social networks. In AAAI , pages 4623--4632. AAAI Press, 2023

    2023

  53. [61]

    Distributed asynchronous deterministic and stochastic gradient optimization algorithms

    John Tsitsiklis, Dimitri Bertsekas, and Michael Athans. Distributed asynchronous deterministic and stochastic gradient optimization algorithms. IEEE transactions on automatic control , 31(9):803--812, 1986

    1986

  54. [62]

    How large is the divide between red and blue america?, 2016

    New York Times. How large is the divide between red and blue america?, 2016. Source NY Times https://www.nytimes.com/interactive/2016/11/04/us/politics/growing-divide-between-red-and-blue-america.html

    2016

  55. [63]

    How T rump consultants exploited the F acebook data of millions, 2018

    New York Times. How T rump consultants exploited the F acebook data of millions, 2018. URL https://www.nytimes.com/2018/03/17/us/politics/cambridge-analytica-trump-campaign.html

    2018

  56. [64]

    Susceptible user search for defending opinion manipulation

    Wenyi Tang, Ling Tian, Xu Zheng, Guangchun Luo, and Zaobo He. Susceptible user search for defending opinion manipulation. Future Generation Computer Systems , 115:531--541, 2021

    2021

  57. [65]

    The design of approximation algorithms

    David P Williamson and David B Shmoys. The design of approximation algorithms . Cambridge university press, 2011

    2011

  58. [66]

    Binary opinion dynamics with stubborn agents

    Ercan Yildiz, Asuman Ozdaglar, Daron Acemoglu, Amin Saberi, and Anna Scaglione. Binary opinion dynamics with stubborn agents. ACM Transactions on Economics and Computation (TEAC) , 1(4):1--30, 2013

    2013

  59. [67]

    Minimizing polarization and disagreement in social networks via link recommendation

    Liwang Zhu, Qi Bao, and Zhongzhi Zhang. Minimizing polarization and disagreement in social networks via link recommendation. In NeurIPS , pages 2072--2084, 2021

    2021

  60. [68]

    A nearly-linear time algorithm for minimizing risk of conflict in social networks

    Liwang Zhu and Zhongzhi Zhang. A nearly-linear time algorithm for minimizing risk of conflict in social networks. In KDD , pages 2648--2656, 2022

    2022

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.