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arxiv: 2606.26601 · v1 · pith:247XFGXGnew · submitted 2026-06-25 · 💻 cs.SI

Fast Computation and Optimization for Opinion-Based Quantities of Friedkin-Johnsen Model

Haoxin Sun , Yubo Sun , Xiaotian Zhou , Zhongzhi Zhang This is my paper

Pith reviewed 2026-06-26 02:26 UTC · model grok-4.3

classification 💻 cs.SI
keywords friedkin-johnsen modelopinion dynamicspartial rooted forestssublinear algorithmsopinion minimizationpolarizationsampling methodssocial networks
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The pith

Partial rooted forest sampling reduces computation of Friedkin-Johnsen opinion quantities and optimizations from linear to sublinear time.

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

The paper introduces partial rooted forests as a structure that encodes opinion propagation in the Friedkin-Johnsen model. It uses random sampling over these forests to approximate opinion-based quantities that previously required linear-time matrix operations. The same sampling technique yields fast algorithms for two optimization tasks: minimizing the total shift in opinions and minimizing a combination of polarization and disagreement. This drops the running time from linear in the network size to sublinear while preserving high-probability accuracy guarantees. Experiments on real networks confirm that the approximations match exact values closely enough to outperform prior methods and handle graphs too large for earlier approaches.

Core claim

Quantities in the Friedkin-Johnsen model, including expected opinions under stubborn agents and measures of polarization and disagreement, admit exact expressions in terms of partial rooted forests; therefore Monte Carlo sampling over these forests produces unbiased estimators whose variance can be controlled to yield sublinear-time algorithms. The same estimators turn the Opinion Minimization Problem and the Polarization and Disagreement Minimization Problem into tractable sampling-based optimization tasks whose solutions remain accurate on large networks.

What carries the argument

Partial rooted forests, which represent the reachable influence paths from stubborn agents and allow sampling to approximate the linear-system solutions that define all opinion quantities.

If this is right

  • The Opinion Minimization Problem admits a sublinear-time algorithm that returns a near-optimal set of agents to influence.
  • The Polarization and Disagreement Minimization Problem likewise admits a sublinear-time sampling algorithm.
  • Both algorithms remain accurate on networks with millions of nodes where linear-time baselines become impractical.
  • The methods outperform existing state-of-the-art techniques in wall-clock time while matching or exceeding their solution quality.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The sampling framework could be reused for other linear opinion-dynamics models whose solutions are also expressible via forest structures.
  • Real-time intervention design in live social platforms becomes feasible once the per-query cost drops below linear.
  • Extending the same estimators to time-varying or multiplex networks would require only re-deriving the appropriate forest distribution.
  • Synthetic benchmarks with planted optimal solutions would give a direct empirical check on the failure probability claimed for the sampling procedure.

Load-bearing premise

Random sampling of partial rooted forests yields accurate approximations to the opinion quantities and optimization objectives with high probability.

What would settle it

On a network whose exact opinion vector and optimal intervention sets can be computed directly, compare the sampled approximations against the exact values and observe that the error exceeds the claimed high-probability bound even after drawing polynomially many samples.

Figures

Figures reproduced from arXiv: 2606.26601 by Haoxin Sun, Xiaotian Zhou, Yubo Sun, Zhongzhi Zhang.

Figure 1
Figure 1. Figure 1: Comparison of mean relative error for zi and running time under varying parameters for PF-QE and LazyWalk. Internal opinions are generated using the uniform distribution. From [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
read the original abstract

In this paper, we address the problem of fast computation and optimization of opinion-based quantities in the Friedkin-Johnsen (FJ) model. We first introduce the concept of partial rooted forests, based on which we present an efficient algorithm for computing relevant quantities using this method. Furthermore, we study two optimization problems in the FJ model: the Opinion Minimization Problem and the Polarization and Disagreement Minimization Problem. For both problems, we propose fast algorithms based on partial rooted forest samplings. Our methods reduce the time complexity from linear to sublinear. Extensive experiments on real-world networks demonstrate that our algorithms are both accurate and efficient, outperforming state-of-the-art methods and scaling effectively to large-scale networks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper introduces partial rooted forests as a basis for computing opinion quantities in the Friedkin-Johnsen model and proposes sampling-based algorithms for two optimization problems (Opinion Minimization and Polarization/Disagreement Minimization). It claims these methods reduce computation from linear to sublinear time complexity, with experiments on real networks showing accuracy and scalability over prior methods.

Significance. A rigorously justified sublinear-time sampling method with explicit high-probability error bounds for FJ quantities would be a meaningful contribution to scalable opinion dynamics on large networks. The experimental claims of accuracy are promising but cannot be assessed without the missing analysis.

major comments (2)
  1. [Abstract] Abstract: the claim that sampling partial rooted forests reduces time complexity from linear to sublinear is unsupported; no sample-complexity bound, variance analysis, or concentration inequality is supplied to show that o(n) samples suffice for fixed error with high probability, independent of spectral gap or degree.
  2. [Optimization algorithms] The optimization sections: the reduction for both the Opinion Minimization Problem and the Polarization and Disagreement Minimization Problem rests on the same unanalyzed sampling procedure; without explicit error propagation from the Monte-Carlo estimator to the optimization objective, the sublinear claim for the end-to-end algorithms cannot be verified.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. The points raised correctly identify that the current manuscript presents the partial rooted forest sampling algorithms and empirical results but does not supply the formal sample-complexity and error-propagation analysis needed to rigorously support the sublinear-time claims. We will revise the paper to add this analysis.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that sampling partial rooted forests reduces time complexity from linear to sublinear is unsupported; no sample-complexity bound, variance analysis, or concentration inequality is supplied to show that o(n) samples suffice for fixed error with high probability, independent of spectral gap or degree.

    Authors: We agree. The manuscript introduces the sampling procedure and reports empirical performance but omits the theoretical analysis. In the revision we will add a dedicated section deriving sample-complexity bounds via concentration inequalities (Hoeffding or Bernstein) that establish the number of samples required for additive error ε with probability 1-δ, showing that this number can be made independent of n under the standard FJ-model assumptions, thereby justifying the sublinear claim. revision: yes

  2. Referee: [Optimization algorithms] The optimization sections: the reduction for both the Opinion Minimization Problem and the Polarization and Disagreement Minimization Problem rests on the same unanalyzed sampling procedure; without explicit error propagation from the Monte-Carlo estimator to the optimization objective, the sublinear claim for the end-to-end algorithms cannot be verified.

    Authors: We concur that error propagation must be analyzed explicitly. The revised manuscript will include a subsection that propagates the Monte-Carlo estimation error through the objective functions of both optimization problems, yielding high-probability guarantees on the quality of the returned solutions while preserving the sublinear running time of the overall algorithms. revision: yes

Circularity Check

0 steps flagged

No circularity; new sampling algorithms presented without self-referential reduction

full rationale

The paper defines partial rooted forests as a new concept and builds sampling algorithms for FJ quantities and optimizations directly from this definition, claiming sublinear time via Monte-Carlo sampling. No equations equate a derived quantity to its own fitted input by construction, no load-bearing self-citations justify uniqueness or ansatzes, and no renaming of known results occurs. The central claims rest on the (unproven in the provided text) sample complexity of the new method rather than any circular reduction, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities are described.

pith-pipeline@v0.9.1-grok · 5651 in / 942 out tokens · 46946 ms · 2026-06-26T02:26:12.089807+00:00 · methodology

discussion (0)

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