Learning Social Influence with Unobserved Attention

1 Probabilistic Generative Agent-Based Model (PGABM)

• Each event centers on a focal target and its available information environment, which may be a stimulus or a network neighborhood.The model distinguishes the potential sources available at an event from the source actually attended to.Source: §1 · pp. 3–4

1.1 Selection of information source

• Averaging over all neighbors can hide the attended source’s influence, even making movement toward that source look like movement away from the average.The model instead treats the attended source as a latent, uniformly selected source; more general attention rules are left for future work.Source: §1.1 · p. 4

1.2 Retention

• A latent response state distinguishes social influence from retention, with a shared influence rate π giving the probability of a social response.Retention means no socially driven opinion change, apart from changes independent of social influence.Source: §1.2 · pp. 4–5

1.3 Response variability

• Opinion updates include intrinsic variability modeled with Gaussian or Laplace noise, and allow baseline exogenous drift when the target does not respond socially.The authors compare the noise distributions because residuals are empirically heavy-tailed.Source: §1.3 · p. 5

1.4 Opinion updates

• The model compares four parametric kernels—DeGroot, bounded confidence, sigmoidal bounded confidence, and SAR—with a nonparametric RFF kernel.The kernel sets response shape, while λ sets its scale and sign; only SAR is repulsive.Source: §1.4 · pp. 5–6, Figure 2 · p. 7

2.1 Likelihoods

• The observed-data likelihood factors over update events and marginalizes latent response and source assignments; EM estimates parameters by alternating posterior inference and maximization.The initial opinion state is treated as given because the model describes updates, not initial values.Source: §2.1 · p. 8

2.2 Model selection

• Candidate models are compared by five-fold cross-validation using held-out marginal log likelihood per transition, with folds blocked by individual.The nested model ladder adds mechanisms stepwise, and RFF bandwidth is selected by the one-standard-error rule.Source: §2.2 · p. 8
• Omitting either latent mechanism roughly halves the inferred kernel while leaving its shape close to the synthetic truth.Figure 3 also compares held-out MLL per transition for linear and nonparametric kernels, with and without each mechanism.Source: Figure 3 · p. 9

2.3 Size and curvature of a kernel

• Kernel strength measures overall influence over observed discrepancies, while curvature measures variation in the kernel’s gain.Both metrics are computed over the observed support, ending at the 99th percentile of discrepancies.Source: §2.3 · p. 9

3.1 Model misspecification induces bias in synthetic data

• Omitting retention left a roughly half-strength kernel nearly indistinguishable from a linear fit; omitting source selection reduced strength 59% and curvature 55%.The selection result came from replacing the chosen source’s discrepancy with the neighborhood average; under Laplace noise, retention shrinkage is not a simple factor of the influence rate π.Source: §3.1 · p. 10

3.2 Modeling latent mechanisms improves out-of-sample prediction across empirical datasets

• Laplace noise was preferred in 14/15 datasets; where noise choice changed, RFF size fell in 11/13 datasets and curvature rose in 8/13.Heavy-tailed noise turned a linear verdict nonlinear on six datasets, but latent mechanisms reversed that verdict on ADAMS-low, ADAMS-medium, and MARKETS.Source: §3.2 · p. 12
• Retention improved held-out prediction in 13/15 datasets, while source selection beat neighborhood averaging in 6/8 networked datasets, with approximate ties on two.A better fit is not necessarily evidence of the psychological mechanism; the model predicts updates from neighbors’ opinions but cannot make causal claims about influence.Source: §3.2 · pp. 10, 12, 14
• Retention raised kernel size in 9/13 datasets where selected, while source selection exposed bounded-confidence-like behavior in HOUSE and SENATE hidden by neighborhood averaging.Retention’s effect on shape depends on where retained updates occur; selection changes shape where the neighborhood mean poorly summarizes neighbors.Source: §3.2 · pp. 14–15

3.3 Latent mechanisms change the shape of interaction kernels

• Across 14 datasets with a kernel beating the null, nonlinear preference rose from 5 to 9 after mechanisms and noise-model selection; verdicts changed in eight.Six shifted toward nonlinear preference and two the other way; mechanisms do not resolve empirical disagreement about nonlinearity.Source: §3.3 · p. 16

3.4 Exploring learned kernels

• Every fitted kernel was attractive over observed discrepancies, but shared discrepancy-only kernels cannot rule out repulsion specific to identities or topics.The authors note that conclusions about interaction kernels may depend on which network edges are treated as carrying social influence.Source: §3.4 · p. 17

4 Discussion

• Omitting either latent mechanism reduces inferred kernel strength and obscures evidence of nonlinearity on synthetic data; modeling attention improves held-out prediction on most empirical datasets.Latent mechanisms contribute more to prediction than the choice of parametric kernel family.Source: §4 · p. 18
• Retention enlarges kernel strength on 9 of 13 datasets, while selection changes curvature according to how far single neighbors lie from the neighborhood average.With heavy-tailed Laplace noise, kernels tend to have smaller overall strength and larger curvature.Source: §4 · p. 18
• Held-out predictive fit supports models with retention and selection but does not identify those latent variables as psychological mechanisms.Retention can absorb other near-zero updates, and exact zeros are removed before fitting, so estimated π describes nonzero updates only.Source: §4 · p. 18
• Future work could explore non-translation-invariant kernels, other noise models, richer source selection, multiple simultaneous neighbors, and discrepancy-dependent retention.The considered kernels depend on focal and source opinions only through their difference.Source: §4 · p. 18
• Controlled experiments could test latent retention and source selection, which the empirical datasets yield at different rates and with different kernel shapes.The authors suggest pairing such studies with framework extensions to connect model-based and empirical understanding.Source: §4 · pp. 18–19

Data Availability

• The project repository provides all code needed to reproduce results, and manuscript AI use was limited to coding assistance and grammatical edits.The authors say they reviewed and edited the code and take responsibility for the results.Source: Data Availability · p. 19

A Generative model

• In source-identified exposure studies, opinion change δe follows a measured single exposure, and plotting discrepancy against δe recovers gain kϕ(|de|).Exposures include another participant’s opinion, a signal, or a persuasive article.Source: A Generative model · p. 24

A.1 Parametric kernels

• The four parametric kernels map opinion distance |de| to a gain kϕ(|de|), with parameters ϕ chosen from the kernel family’s parameter space.Source: §A.1 · p. 24

A.1.1 DeGroot: linear assimilation

• The DeGroot kernel is constant, with influence strength carried by coupling λ; fixing the kernel constant at one avoids confounding it with λ.Under relatively broad settings, this model tends toward consensus.Source: §A.1.1 · p. 24

A.1.2 Smoothed attractive-repulsive kernel

• The smoothed attractive–repulsive kernel switches from assimilation to repulsion as opinion distance crosses scale ϕ.It is proposed as a smoothed alternative to thresholded or clipped backfire constructions.Source: §A.1.2 · p. 24

A.1.3 Bounded confidence kernel

• The bounded-confidence kernel uses a hard threshold: a target is influenced only when opinion distance is below confidence threshold ϕ.Unlike DeGroot dynamics, bounded-confidence models can yield consensus, fragmentation, or polarization.Source: §A.1.3 · p. 25

A.1.4 Sigmoidal bounded-confidence

• The sigmoidal bounded-confidence kernel smooths the hard cutoff: ϕ0 sets squared confidence radius and ϕ1 controls sharpness.Large positive ϕ1 approaches an HK-like threshold; small ϕ1 gives near-constant gain, and the fit allows either parameter’s sign.Source: §A.1.4 · p. 25

A.2 Learning the RFF kernel

• RFF weights are fit by weighted ridge regression; under Gaussian noise the step is exact, while under Laplace noise it is a surrogate that can lower likelihood.Each EM run returns its best iterate; the method uses 256 random features and a bandwidth grid with a one-standard-error selection rule.Source: §A.2 · p. 25

B.1 Likelihood

• A transition likelihood includes latent response state and source; its retention and influence distributions use candidate noise families whose parameters are learned by expectation maximization.Source: §B.1 · pp. 25–26

B.2 Generalized expectation maximization

• GEM estimates latent-state probabilities in its E-step and updates parameters in its M-step, which need only increase the objective; the best marginal-likelihood iterate is retained.The procedure alternates until relative marginal-likelihood improvement falls below a tolerance.Source: §B.2 · pp. 26–27

B.2.1 Influence Probability ˆπ

• The influence probability estimate is the expected frequency of influence events under the current latent-state distribution.Source: §B.2.1 · pp. 27–28

B.2.2 Retention arm parameters

• For Gaussian retention noise, location and scale use weighted mean and standard deviation; for Laplace noise, they use weighted median and mean absolute deviation.Source: §B.2.2 · p. 29

B.2.3 Influence arm parameters

• Influence parameters are updated in blocks; λ and η1 are conditional maximizers, while kernel-parameter optimization depends on the kernel.Kernel gains are generally normalized to one at zero discrepancy, identifying λ and ϕ; SBC instead uses λkϕ(0) at zero discrepancy.Source: §B.2.3 · pp. 29–30

B.2.4 Kernel parameter optimizers

• Kernel updates vary by family: bounded confidence uses golden-section search, SAR and SBC use gradient descent, and RFF uses ridge regression or a surrogate.Source: §B.2.4 · pp. 30–31

B.3 Model selection

• Five-fold cross-validation assigns nodes or targets to folds to avoid trajectory leakage, then trains on four folds and scores each held-out fold.The model grid crosses kernel, noise, and retention choices, adding neighbor aggregation or latent neighbor selection for multi-source datasets.Source: §B.3 · p. 31
• Parametric kernels use four discrepancy-based initializations, with four additional steepness values for SBC; RFF sweeps bandwidths and selects the lowest within one standard error of the maximum.For each fold, the highest training-likelihood restart is selected for held-out scoring.Source: §B.3 · p. 31

B.4 EM training

• EM stops at 600 networked-panel or 300 dyadic steps, or after relative marginal-likelihood change stays below 10^-6 for 10 steps; kernel updates follow Section B.2.4.Source: §B.4 · p. 31

• Networked simulations use the Les Misérables character interaction network with N = 77 nodes and T = 500 time steps.Source: C More detail on synthetic data… · pp. 31–32
• In dyadic experiments, each update has a single source.Source: C More detail on synthetic data… · p. 32

C.1 Kernel-family selection

• Table 3 compares held-out MLL gaps across kernels, arms, and noise models for synthetic data generated with a sigmoidal bounded-confidence kernel, latent selection, and latent retention.The study also repeats model selection across true kernels, sample sizes, and influence rates.Source: Table 3 · p. 32, §C.1 · p. 32
• In five of eight settings with DeGroot truth, another kernel is selected, but each beats DeGroot by less than 10−4 nats per transition.Those fits are effectively linear over the data.Source: Table 4 · p. 32

C.2 Nonparametric recovery

• The RFF kernel recovers every planted kernel except bounded confidence, whose sharp cutoff makes it difficult for a smooth basis to approximate.The fit rings near the step; a wavelet basis could capture sharp kernels better.Source: §C.2 · p. 33, Figure 7 · p. 33

• Omitting retention attenuates susceptibility by a factor of π, producing a lower-magnitude kernel whose shape remains close to truth and is harder to distinguish from linear.Source: §C.3 · p. 34
• When source deviation ξ is independent of the aggregate discrepancy, averaging convolves the response with ξ’s distribution and cannot increase total variation.The aggregate discrepancy is the mean available-source opinion minus the target’s opinion.Source: §C.3 · p. 34
• Aggregating over available sources cuts inferred kernel size and total variation by more than half, while modeling source selection returns a nearly identical kernel.Source: §C.3 · p. 34

D.1 Overview

• Regularly reported values, such as congressional or controlled-experiment data, are modeled as synchronous updates in which all nodes update each timestep.Source: §D.1 · pp. 34–35
• Social-media posts are treated as asynchronous updates, with each post representing a timestep; CMV and SPINOS are examples.Source: §D.1 · p. 35
• Numerical estimation gives directly quantifiable opinions but uses unbounded values whose scales and kernel supports are not standardized, and political estimates do not map cleanly to partisan alignment.Source: §D.1 · p. 35
• Ordinal scales are easier to self-report and directly interpretable, but do not naturally fit interaction-kernel models that assume real-valued opinions.Text-based opinions require indirect quantification; stance detection classifies text relative to an anchor.Source: §D.1 · p. 35

D.2 General dataset preparation and filtering

• Exact-zero updates are removed before fitting, so fitted quantities are conditional on nonzero updates; empty candidate sets also reduce held-out scored counts in some panels.With exact-zero share z, the share of all updates that are influence is at most (1 − z)π̂.Source: §D.2 · p. 35, Table 5 · p. 36

D.3.2 Preparation

• ADAMS analyzes 1,051, 1,097, and 1,114 trials at 20%, 50%, and 80% reliability; Becker scores spoke updates responding to the hub, dropping participants whose per-round standard deviation exceeds 3.This removed 67 of 2,274 participants; respondents were also excluded when no data existed.Source: §D.3.2 · p. 37, §D.4 · p. 37

D.5 TAKACS: Tak´acs et al. (2016) [15]

• TAKACS combines questionnaire and paired-chat studies, including a disliking treatment whose alternation by round parity may confound treatment differences with odd-even effects.Opinions are scaled to [0, 1] and de-quantized.Source: §D.5 · p. 37

D.6 VK-gauging and VK-counting: Vande Kerckhove et al. (2016) [9]

• The VK-gauging and VK-counting panels treat each participant group as a six-person clique, while unobserved responses and exact-zero updates are excluded from scoring.The source does not replicate the original trustworthiness filter for VK-counting.Source: §D.6 · pp. 37–38

D.7 VKontakte: Kozitsin (2021) [10]

• VKontakte scores come from three waves and a frozen friendship network, while Senate and House opinions are legislator-term scores paired with bill-cosponsorship networks.VKontakte scores only ego transitions; the congressional network is sparsified with a disparity filter.Source: §D.7 · p. 38, §D.8 · p. 38

D.9 MARKETS: Manifold prediction markets

• Manifold trades provide implied probabilities as opinion updates, and the analysis retains qualifying markets and users with sufficient bets and scored transitions.Edges in the reply network connect users when one replies to the other's comment.Source: §D.9 · p. 39, §D.9.1 · p. 39

D.10 SPINOS: Sakketou et al. (2022) [50]

• SPINOS uses human-rated stance scores but its discrete ratings challenge the model's continuous update density; CMV opinions instead use classifier logits for affirmation and refutation.SPINOS keeps four topics; CMV masks transitions beginning or ending at exactly 0.5.Source: §D.10 · pp. 39–40, §D.11 · p. 40

E.1 Retention

• Without retention, the inferred kernel can be about seven times smaller on ADAMS-low; on ADAMS-high, the matching influence share falls from 0.95 near the source to 0.56 at |d| = 0.6.On ADAMS-medium and ADAMS-high, rising retention with distance creates a spurious downward trend in the kernel.Source: §E.1 · p. 41

E.2 Selection

• On the shared range, selection increases kernel size on every networked panel except CMV; its shape changes most where neighborhood averages poorly summarize neighbors.Selection changes shape only where the neighborhood mean is a poor summary of neighbors.Source: §E.2 · pp. 41–42
• SENATE and HOUSE have 21% and 25% of neighbor distance on the far side of the target from the neighborhood mean, with single-neighbor distances reaching about twice as far.Their neighbor weights extend beyond mean-field support by 14% and 11%, respectively.Source: §E.2 · p. 41, Table 7 · p. 42
• On MARKETS and both VK games, neighbors mostly lie on the same side as their mean, so selection mostly rescales the kernel; VK-counting also gains curvature.CMV has the most weight past the mean-field support, but selection removes curvature there.Source: §E.2 · pp. 41–42

F Full empirical results

• Figure 10 displays every kernel's full-model fit, Figure 9 shows held-out PIT calibration across ladder rungs, and Figure 8 compares fitted kernels with baselines.Some datasets have missing PIT curves where rungs could not be pooled or computed.Source: F Full empirical results · p. 43, Figure 8 · p. 45, Figure 9 · p. 46
• Table 8 reports each dataset's selected highest-held-out-MLL fit, while Table 9 reports held-out MLL gains at each capability-ladder step.Table 8 also records kernel, noise model, and networked-data arm; Table 9 includes total gain over the random-walk null.Source: Table 8 · p. 43, Table 9 · p. 44

F.1 Kernel and noise grids

• Figures 11 and 12 compare held-out MLL per transition across four noise models, respectively for latent selection and neighbor aggregation on networked datasets.Dyadic experiments have one candidate source, so their two arms are the same model.Source: Figure 11 · p. 48, Figure 12 · p. 49