Title: Fast Computation and Optimization for Opinion-Based Quantities of Friedkin-Johnsen Model
Venue: NeurIPS 2025
Accepted: yes
Keywords: forest matrix

fast algorithm

FJ model

opinion dynamics，optimization problems
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 and present an efficient algorithm for computing these 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 sampling. 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.


Peer reviews:
comments: SUMMARY:
This paper proposes an efficient algorithm for computing the quantities in FJ model and applies the quantities estimation to the Opinion Minimization Problem and the Polarization and Disagreement Minimization Problem.

STRENGTHS AND WEAKNESSES:
Strengths:

1.	The motivation of partial rooted forests and the quantities estimation is clear.

2.	The paper provides a solid analysis of the performance guarantee.

Weaknesses:

1. The sampling based method is an extension of existing method. The existing method is sufficient to scale large graphs, which may limit the scope of readers.

2. The quantities estimation has been investigated previously and the method seems an application of the absorbing random walk, which limits the technical novelty.

QUESTIONS:
Questions

1. In the experiments, for OpMin problem, the paper compares with FAST; for PDMin, the paper compares with FASTGREEDY. In the related work, the authors refer to many other recent baselines, which should be compared.

2. The experiments report Relative Error in the tables. In the greedy algorithms, the estimation error might lead to choosing different nodes or edges. Since the problems are usually not submodular, small estimation error could lead to completely different final results. It is better to present the final performance for OpMin and PDMin.

3. Some important references are missing, for example,

Sijing Tu and Stefan Neumann. A Viral Marketing-Based Model For Opinion Dynamics in Online Social Networks. WWW 2022.

Random-Walk Computation of Similarities between Nodes of a Graph with Application to Collaborative Recommendation, TKDE, 2007.

4. Small typos, e.g., “Reletive Error”->” Relative”

LIMITATIONS:
Please see the weakness of the questions.
TITLE: Official Review
CLARITY: 3
IMPACT: 2
ORIGINALITY: 2
DATE: 2025-06-14

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies algorithms for efficiently computing the “final expressed opinions” in the standard Friedkin-Johnson (FJ) model of opinion dynamics. For a social network with Laplacian L, these opinions are equal to z = (L+I)^-1 s, where s is a vector of n “innate” opinions. The authors are specifically interested in computing some function of the opinions, like their inner product with a vector (with the all ones vector would give the average), their second moment z^Tz, and various other quantities related to “polarization and disagreement” of the final opinions. To do so, they adapt an approach based on random spanning forests from prior work. In particular, a natural approach to estimating a *single entry* is to write (L+I)^-1 as a power series in involving the random walk matrix D^-1A, exactly as is done when computing PageRank or personalized PageRank. The conclusion is that the i,j entry of  (L+I)^-1 is exactly equal to the chance that a random walk with absorption probability 1/1+d_k at node k that starts at i lands exactly on j. Thus, we can estimate entries in the ith row of (L+I)^-1 by running a bunch of random walks with this absorption probability, and seeing the percentage of times we land on various j. Recent work proposes and alternative approach based on the “Matrix Forest Theorem” which basically says that the probability an absorbing walk goes from j to i is also proportional to the probability i and j land in the same connected component if we construct a random spanning forest of G. \n\nThe authors observation is that, if we want to estimate a quantity like c^Tz for some vector v, we can subsample a set of indices, S, and return a Monte Carlo estimate. Accordingly, instead of building an *entire* random spanning forest for G, we should only build the part necessary to see what components nodes of S land in. This can be done easily by “early stopping” standard iterative algorithms for constructing random forests. The analysis of the result is thus straightforward (see Section A.3, A.4, A.5, A.6). Everything follows from prior work + Hoeffding’s inequality. \n\nThe authors show their methods can be used as subroutines to solve various optimization problems involving functions of the expressed opinions and compare the accuracy and runtime of their methods to prior work. There wasn’t sufficient discussion of the methods compared to — for example, it wasn’t clear to me what algorithm “LazyWalk” implements — however, it seems like the authors at least beat this baseline. At least in the main body also didn’t see to be a direct comparison to the “FAST” method (based on constructing a full random spanning forest), expect for on the downstream optimization problems. It would be helpful if the authors could comment on this, or provide results for FAST e.g. in Table 2.\n\nSTRENGTHS AND WEAKNESSES:\n+ Interesting problem that has been of recent interest. \n+ Reasonable algorithm that might very well be the \"right\" algorithm for the problem of estimating functions of the expressed opinions z. \n\n- I don't see any technical lift of previous work -- unless I misunderstood, the main result amounts to an observation that you can early stop an algorithm proposed in prior work if you only care about estimating entries of z for a subset of nodes. This is a nice observation, but I think more would be required to elevate it to publication level. \n- I found the writing informal and unclear at points. For example, the first theorem stated in the paper (Theorem 4.2)  is imprecise -- it is not clear in what way the partial random forest method can be directly compared to the efficiency of independent walk sampling, and the proof of the theorem doesn't make any such comparison.\n\nQUESTIONS:\nMix of questions and small comments below. Don't feel the need to respond to everything: \n\n- when discussing the prior work in Section 4, what is “l” when you say prior methods require O(ln) time?\n- is “absorbing random walk” (e.g., as in Lemma 4.1) ever formally defined? At the beginning of Section 4.1 you define what it means for a walk to “absorb” at q, but not what it means to “be absorbing”.\n- The transition from Section 4 to 5 could use work. At the end of 4, it’s not clear why Algorithm PFS is useful, or why we should care about it. In Theorem 4.2, what does it mean to be “more efficient than performing absorbing random walks for eahc node S”? At this point, the algorithm and the proposed alternative output different things — a partially rooted forest vs a bunch of walks. More efficient for what task or by what measure?\n- Theorem 5.2 doesn’t seem scale invariant to me as stated. It implicitly depends on the assumption that s in [0,1], which was stated at some point, but not emphasized. I would remind the reader directly in the statement of Theorem 5.2. \n- It would be helpful to clarify how Theorem 5.2 compares with what would be obtained with independent random walk samples. To me this would be a more meaningful comparison than what is currently in Theorem 4.2.\n- It feels like to estimate something like z^Tz we should be able to randomly seleclt 1/epsilon^2 entries, and also get relative error to each entry with roughly 1/epsilon^2 walks. Average length is bar{d} so is the main imrpovement going from r to bar{d}?\n- Could you add more details on the “average expressed minimm opinion problem”? In the *undirected setting* (I believe the settin of this work), I think it is well-known that the mean of z is equal to a weighted mean of s (entries weighted by degree). It seems like is should be straightforward to solve the problem — set the opinions to 0 for the k nodes with highest product of initial opinion and degree. The cited paper [44] handles the directed case, which is maybe more interesting? If the problem is only interesting in your setting when c is not the all ones vector, I think this is worth pointing out. It might also be helpful to clarify why vectors c are natural besides all ones. \n- Some more discussion of what the baseline methods are in the experiments section would be helpful. E.g, what is te Lazy walk method?\n- Does Lemma 5.1 require S to random? This isnt’ stated. Also, is it fine if Algorithm PFS is run on the elements of S in any order? I.e., the order doesn’t need to be random?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 1, "DATE": "2025-07-02"}

comments: SUMMARY:
The paper provides a cost-efficient way to compute and optimize opinion-based quantities in the Friedkin–Johnsen (FJ) social-influence model. Central to these tasks is the forest matrix $\Omega$, which has connections to absorbing random walks. The authors present a method for approximating the final opinions $z$ in FJ model by providing a sampling method. Building on Wilson's loop‑erased random‑walk sampler, they introduce Partial Forest Sampling (PFS), a local algorithm that generates only the few forest components needed for a given query. Unbiasedness and Hoeffding-type error bounds guarantee that averaging over all sampled forests yields accurate estimates of node opinions, polarization, and disagreement.

Leveraging the same sampler, the paper presents two optimization algorithms: OPMIN, which selects $k$ internal opinions to minimize overall deviation from truth, and PDMIN, which adds up to $k$ edges to minimize polarization and disagreement index, which is convex in nature. Both achieve sub-linear expected runtimes and come with provable approximation guarantees, and it's a good asymptotic improvement over earlier approaches.

STRENGTHS AND WEAKNESSES:
Strengths:

1) Sublinear time algorithms

2)  Practical scalability across various datasets

Overall the paper is a well‑written contribution that combines sound theory with convincing large‑scale experiments. 

Weakness:

1) Maybe I missed it but the choice of $r$ is not very clear from the text. Can the authors elaborate on it?

QUESTIONS:
Can you provide some insights if this can also be employed in approximately measuring polarization index which is non-convex in nature?

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 4
IMPACT: 3
ORIGINALITY: 4
DATE: 2025-07-02

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes Partial Rooted Forest Sampling (PFS), a fast sublinear-time method for estimating key quantities in the Friedkin-Johnsen opinion dynamics model. By using loop-erased absorbing random walks to construct partial rooted forests, the method enables efficient, unbiased approximation of the forest matrix and related opinion metrics. It also supports fast optimization for tasks like reducing overall opinion or polarization, outperforming prior methods on large-scale real-world networks.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\nThe PFS method is a novel and elegant idea that localizes sampling to relevant graph regions, reducing complexity from linear to sublinear and enabling practical analysis of massive networks. The paper is theoretically solid with clear proofs, and the experiments show dramatic speedups and superior scalability compared to strong baselines. Besides, the paper is exceptionally well-written, with clear explanations, consistent notation, and well-designed visuals that effectively support its claims.\n\n\nWeaknesses:\nThe method’s efficiency relies on a parameter r that depends on the graph structure. Although empirical results show r is small in practice, a deeper theoretical analysis of how graph properties affect r would enhance understanding and applicability.\nThe paper briefly mentions limitations only in the conclusion. Including a dedicated \"Limitations\" section with discussion on issues like variance in Disagreement estimation and nonlinear objectives would improve transparency and completeness.\n\nQUESTIONS:\nYour algorithm’s performance depends critically on the graph-dependent ratio r, which is shown empirically to be small. Could you elaborate on which structural properties of graphs influence r, and how it behaves on different types like expanders or grid graphs?\n\nAlgorithm 1 samples S randomly, while Algorithm 3 uses endpoints of candidate edges, suggesting that smarter sampling could help. Could a more strategic selection of S, such as high-degree or central nodes, reduce the number of samples p or forests l needed for accurate estimation?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-07-09"}