Title: Discovering Opinion Intervals from Conflicts in Signed Graphs
Venue: NeurIPS 2025
Accepted: yes
Keywords: Signed graphs

intervals graphs

social networks

correlation clustering

opinions
Abstract: Online social media provide a platform for people to discuss current events and exchange opinions with their peers. While interactions are predominantly positive, in recent years, there has been a lot of research to understand the conflicts in social networks and how they are based on different views and opinions.  In this paper, we ask whether the conflicts in a network reveal a small and interpretable set of prevalent opinion ranges that explain the users' interactions.  More precisely, we consider signed graphs, where the edge signs indicate positive and negative interactions of node pairs, and our goal is to infer opinion intervals that are consistent with the edge signs.  We introduce an optimization problem that models this question, and we give strong hardness results and a polynomial-time approximation scheme by utilizing connections to interval graphs and the Correlation Clustering problem.  We further provide scalable heuristics and show that in experiments they yield more expressive solutions than Correlation Clustering baselines. We also present a case study on a novel real-world dataset from the German parliament, showing that our algorithms can recover the political leaning of German parties based on co-voting behavior.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors introduce the Opinion Interval Representation problem: given a signed graph, assign each node to an interval on the real line in a way that maximizes the number of positive edges between nodes whose intervals overlap and the number of negative edges between nodes whose intervals are disjoint. They show the problem is NP hard and relate it to Interval Editing and Correlation Clustering. The authors provide a PTAS for the case when only $k$ distinct intervals are allowed and two greedy and simulated annealing heuristics. In experiments on real-world signed graph datasets, the authors find that interval representations found by their heuristics are more expressive than Correlation Clustering solutions. In synthetic data, they find that their heuristics can recover true interval structure. Finally, in a dataset of German parliamentary votes, the authors show one-dimensional ideologies of politicians can be recovered using overlapping intervals.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. The new algorithmic problem is explored thoroughly, with hardness, approximation, and heuristics.\n2. The experiments do a good job of showcasing the problem and the heuristics\n\nWeaknesses:\n1. The experimental comparison to Correlation Clustering is presented in a misleading way in certain places\n\nQuality: Seems high throughout the paper.  Good grounding in existing work and a well-executed paper overall.\n\nClarity:\nThe paper is generally well-written, with only a few unclear points (see suggestions below.)\n\nSignificance: I think this paper would be of interest to the the community interested in signed networks, community detection, and ideology estimation. \n\nOriginality: This paper explores a novel problem (to my knowledge), but the techniques are standard. Another piece of related work in the line of recovering ideological positions of legislators from co-voting data is the (DW-)NOMINATE algorithm of Poole and Rosenthal. That method has the advantage of providing continuous estimates for each politician rather than an interval representation.\n\nQUESTIONS:\nI don't have any major questions, but I do have the following suggestions/comments:\n\n1. I don't understand Figure 3... It introduces many symbols not mentioned in the text (S, M, T, $H_S$, ...) and it's not at all clear how this relates to a signed graph. The text/caption needs to provide enough detail for the reader to gain something from the figure (if there isn't enough space, it should be in the appendix).\n2. I think it's important to make clear that GAIA and VENUS are not finding better Correlation Clustering solutions in Table 1: the 8-chain structure is intrinsically more forgiving than clustering, since edges between nodes assigned to an interval and those on either side are not penalized. (i.e., the authors are assigning 2-3 cluster labels to each node and counting agreement if any labels match between nodes connected by a positive edge). In particular, \"our novel methods outperform state-of-the-art heuristics for CORRELATION CLUSTERING. On 8 real-world datasets, the results are at least 20% better and over 38% on average\" is misleading--they're being compared on two different tasks. \n\nMinor:\n\"in recent years we are increasingly witnessing conflicts based on different views and opinions.\" is this really a recent phenomenon? Seems like people have been disagreeing since Plato and Aristotle (and earlier...). I don't think the motivation needs to rely on an unbacked claim that conflict is increasing (I would add sources or cut the claim).\n\nLIMITATIONS:\nThe authors are clear about the extent of their results, and I don't see any potential for negative societal impact.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe manuscript's main contribution is the opinion interval representation, a framework for analyzing signed graphs. The proposed method assigns intervals on the real line to nodes, maximizing overlap for positively connected pairs and disjointness for negatively connected pairs (see Section 2). This method differs from correlation clustering, which partitions nodes into disjoint clusters. Another contribution of the paper is establishing NP-hardness for general instances (see Section 2.1). It also provides a polynomial-time approximation scheme (PTAS) for a restricted complete-graph scenario with a fixed number of intervals (see Section 2.2). Finally, the manuscript presents two scalable heuristics, GAIA and VENUS (see Section 3), for inference in the interval representation. Experiment results are encouraging and outperform state-of-the-art correlation clustering approaches on both synthetic and real-world datasets.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**:\n\n* Novelty and a principled approach to a solution. The proposed framework for opinion interval representation allows for higher expressivity when modelling real-world scenarios (as shown in Figure 1 in the German parliament case study). Moreover, the theoretical results in Section 2 establish clear boundaries for the application of this framework. For instance, Theorem 2.3 provides grounding for an approximate solution, given that Section 2.1 shows that the problem is NP-hard.\n* Experiment results are thorough and convincing. The manuscript compares the method with four SOTA baselines, including a novel dataset based on voting data from the German Parliament. Table 1, which compares the proposed heuristics with the baseline, achieves an improvement of 20% to 90%.\n\n**Weaknesses**:\n\n* The polynomial-time approximation in Theorem 2.3 could limit the method's application due to graph completeness and fixed interval numbers. It is unclear from the manuscript the impact and size of practical limitations on real-world cases.\n* The hyperparameter selection for GAIA and VENUS could be challenging for some applications. The manuscript does not offer a data-driven approach for selecting the number of intervals or the overlap configuration.\n\nQUESTIONS:\n* How should practitioners select the amount and overlap configuration for the intervals in GAIA and VENUS? Let us assume no domain expertise with the new datasets (so no reasonable guess can be made for these values). Are there any principled or data-driven approaches for this task?\n* Could the authors discuss potential extensions of this method to multi-dimensional interval representations? Although outside the scope of this work, a discussion on the challenges of higher-dimensional interactions, including their real-world applicability, would enhance the significance of this framework.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a generalization of correlation clustering, which they call the “Opinion Interval Representation” problem. The idea is that we are given a set of nodes and a set of signed edges. In correlation clustering, the goal is to assign nodes to clusters in such a way that we minimize the total number of negative edges within a cluster plus the number of positive edges between two different clusters. This paper consider instead assigning each node to an interval on the real line. If two nodes have overlapping intervals, we recieve a penality if they have a negative edge between them. If nodes have non-overlapping intervals, they receive a penalty if connected by a positive edge. The idea is to choose intervals to minimize the number of penalties. \n\nThe justifcation for this problem is that we might have non-transitive edge forming behavior in social networks — the authors give the compelling example of politicians — a centrist candidate might often vote the same way as both a liberal candidate and a conservative candidate, but those candidates rarely vote in the same way. The authors also provide some nice experimental justification — they apply the clustering approach to a new data set based on German Parliment votes, and recover a natural picture of the political “spread” of different parties. They also show that, on all examples datasets they test, the problem leads to significanty better “explanations” than correlation clustering -- i.e., it is possible to produce an interval representations that gets many more edges correct than a regular clustering. This is despite the fact that these experiments actually solve a restricted version of the problem, where we have a fixed number of intervals k (they choose k = 8 for some reason) and assign every node to one of those intervals. \n\nThe theoretical work of the paper is mainly focused on algorithms, and read more like an algorithms conference paper than a machine learning paper. The authors first prove the problem as stated is NP-hard, which is not too surprising. They then describe a fixed-parameter tractible algorithm for their restricted version of the problem involving a fixed set of k intervals — the runtime is polynomial but exponential in k. This method is based on adapting an FPT algorithm for correlation clustering when the number of clusters is fixed (in general, correlation clustering allows any number of clusters — the idea is to find the least disagreements amoung all possible partitions). Basically, but brute forcing over all possible “overlaps” of the fixed set of intervals, they can reduce to a generalization of correlation clustering where disagreement between some clusters is not penalized.\n\nThe authors also describe some heuristic algorithms for the problem, which they test in an experiments section. The experiments section is complete in that it asks clear questions and tells a clear story: the authors justify they their new problem is interesting (it’s more expressive than correlation clusteirng for real dataset), that their heuristics find reasonable solutions in a reasonable amount of time, and that sovling the problem can product interesting visualizations/explanations (via their German parliment example). \n\nSmall comments:\n- It doesn’t seem necessary to write k = O(1) on Page 2 since you are being explicit about the dependence on k in the stated runtime.\n- In the introduction it is mentioned that the paper approach leads to results that are “20% better than correlation clustering”. It would be helpful to describe here roughy what metric is being considered — how methods for two different problems compared in a head-tohead way?\n- A few times in the intro is was mentioned that something would be discussed later or “below” but I would have preferred a reference to exactly where. E.g.: Below, we will also argue that Interval Editing problems are closely related. Some of these points I also didn’t think were worth delaying — it is pretty obvious why the problem is more expressive than correlation clustering — could be explained in 2 sentences instead of defering until Section 2. \n- The name \"OPINION INTERVAL REPRESENTATION” makes me think of the problem of finding an *exact* representation if one exists, but Problem 2.1 is more general — maybe a better name would be something like “best interval approximation” or something? \n- The text claims that OPINION INTERVAL REPRESENTATION “is not FPT when parameterized by the number of required edge deletions”, but the problem doesn’t even involve edge deletions at all. I think it would be better (if it is correct) to say when parameterized by OPT where OPT is the minimum cost solution of OPINION INTERVAL REPRESENTATION? Or at least to explain things more.\n- Why in the experiments did you restrict to exactly 8 intervals?\n- Should there be a last line of Theorem 2.2 that drives the point home, e.g. “Thus, the Opinion Interval Representation problem is NP-hard”.\n\nSTRENGTHS AND WEAKNESSES:\nWhile I am not an expert on the topic, this feels like a very complete paper to me. The paper is well-written and easy to follow. The authors introduce a natural and easy to understand clustering objetive. Given its connection to correlation clustering, and how well interval graphs have been studied, the problem is not too far afield from prior work. At the same time, it appears significantly more expressive than correlation clustering, while still resulting in an easy to interpret visualization for structure that could possibly have lead to the observed signed graph (k 1-d intervals instead of k clusters). At least this is true of the restricted version of the problem. While none of the theoretical results in the paper are ground breaking, the authors take meaningful first steps towards understanding algorithms for their objective, and applying it to a case study. Perhaps the only “negative” I can see for this paper is that it’s not an amazing fit for a machine learning conference, although correlation clustering seems to have been studied quite a bit in the ML community.\n\nQUESTIONS:\nI listed maybe one small question with my summary, and a few other small suggestions, but honestly I don't have any major suggestions. I thought the paper was well written and an interesting read.\n\nLIMITATIONS:\nna", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-03"}