Learning Social Influence
with Unobserved Mechanisms

The Santa Fe Institute
October 21st, 2026

Phil Chodrow
Department of Computer Science
Middlebury College

Hi everyone! I’m Phil Chodrow

Prof. of CS at Middlebury College, a small liberal arts college in Vermont. I teach courses in discrete math, machine learning, data science, and network science.

Junior folks: ask me about #LiberalArtsLife if you’re curious.

I like aikido, tea, my cats, being outside, hypergraphs, and math models of social systems.

From 0-5, how’s this talk?

Autokinetic effect: a still point of light in a dark room will appear to move if you stare at it long enough.

“So, how far did the light move?”

Sherif (1935)

“So, how far did the light move?”

Sherif (1935)

Social Influence

Any change in an individual’s thoughts, feelings, or behaviors caused by other people, who may be actually present or whose presence is imagined, expected, or only implied.

- American Psychological Association (APA) Dictionary of Psychology, 2026

Models of social influence address:

• Which other people?
• What kind of change?

Towards Formalization: Which Other People?

Towards Formalization: What Kind of Change?

Discrepancy

0 inches 10 inches Opinion x: how much
has the dot moved?

\(x_t\): opinion of the target \(x_s\): opinion of a source at discrepancy \(d\)

Kernel

The kernel \(k(d)\) sets how far the target moves

\(\delta = k(d) + \text{noise}\)

Outcome

The target’s opinion moves by \(\delta\)

Slide h/t Will Thompson

• 2026-10-11: kernel panel is now a mostly attractive kernel with a shallow repulsive dip (x(0.45-x), damped past the zero crossing at d = 0.45, dip about a quarter of the peak; x axis “similar” to “different”) with the #bipolarization shading: top half “I move toward you”, bottom half “I move away from you”. Scaled so k(d) at d = 0.3 equals the World’s mean move (0.21), so p(delta) shares the kernel’s y axis. The toy numbers below about the sigmoid gain describe the old curve; the outcome panel still uses that run (same mean move at d, so the jump is consistent). [WILL: one dyad, built up in six clicks. Start from the opinion line [0, 1]; (1) the target’s opinion x_t; (2) a source x_s at discrepancy d. (3) The kernel k(d) says how far the target moves at that discrepancy. (4) The update: delta = k(d) + noise, a move toward the source plus zero-mean noise. (5) Outcome: the target’s opinion moves by delta, one step in a trajectory. This is the paper’s hero, piece by piece; retention and selection get added to it later]

Facts, with sources:

• Notation: this slide absorbs both lambda and d into k, so the slide’s k(d) is the paper’s lambda k(|d|) d, the expected move. d = x_s - x_t. Linear (DeGroot) is a straight line through the origin, k(d) = lambda d. If asked: “in the paper k is a gain, and we split it into a scale lambda and a shape”. Every later slide uses the same k(d): #explanations and the hero kernel panels (#model-attention, #model-hero) draw the move against the discrepancy, not the gain [scripts/talk/scenes/_hero.py kernel_panel].
• Toy numbers: x_t = 0.40, x_s = 0.70, d = 0.30, paper gain k(|d|) = 0.92, lambda = 0.75, so k(d) = 0.21 is the mean move; sigma = 0.035. The paper gain here is a sigmoid with half-point 0.6 and steepness 0.12, wider than the hero’s World (0.35, 0.07), so d sits on the rising flank of k [scripts/talk/scenes/_scene.py, World; panels/model_kernel.py].
• The density p(delta) is scaled so its centre sits level with k(d) on the kernel [panels/model_kernel.py, kernel()].
• The paper writes the update as delta_e ~ q(. | mu, eta) with mu = lambda k(|d_e|) d_e [Overleaf sections/method_v2.tex:52]; “noise” on the slide is q.
• The noise around the mean move is one of the paper’s noise families; the talk holds it fixed (Laplace in the paper’s selected models) [paper Methods].

A first model: random neighbor choice and linear influence

DeGroot or linear kernel (DeGroot 1974): \(k(d) = \lambda d\).

Across a broad range of model choices, linear influence with \(\lambda > 0\) tends to produce consensus.

Many Kernels, Many Mechanisms

Linear

Compromise –> consensus

convergence

Empirical justification

• Vande Kerckhove et al. 2016
• Takács et al. 2016
• Clemm von Hohenberg et al. 2017
• Friedkin & Bullo 2017
• Friedkin et al. 2021

Repulsion

Compromise with similar opinions, repulsed by distant opinions

polarization

• Mazen & Leventhal 1972
• van Knippenberg et al. 1990
• Krizan & Baron 2007
• Liu & Srivastava 2015

Bounded confidence

Compromise with similar opinions, ignore distant opinions

clusters

• Moussaïd et al. 2013

Or do it nonparametrically!

Rahimi + Recht (2007)

[WILL: the first models said influence is linear: DeGroot, a flat kernel, every update a fixed fraction of the gap. Run that and everyone meets. Our world does not look like that; it is dominated by dissensus and polarization. So what kernel makes this world? Each kernel shape is a hypothesis about why we disagree, and each has its backers. Walk the columns left to right: the shape, who reports it, and the world it makes. Linear pulls everyone together; repulsion pushes far opinions away and leaves two poles; bounded confidence stops listening past a radius and leaves clusters. Same kind of experiment, every possible answer. Last click: no one agrees on the shape]

Three clicks: repulsion column, bounded confidence column, then the punch line. (2026-10-11: order swapped from BC then repulsion.)

Facts, with sources:

• One curve per column, drawn as influence k(d) (each column on its own y scale; bounded confidence peaks at 0.045, just inside its radius), from the paper’s gain functions: DeGroot k = 1, the diagonal; hard bounded confidence, radius 0.15 (full gain inside, zero beyond: Deffuant/Hegselmann-Krause); SAR with crossover 0.4 [presentations/ccs_2026/panels/explanations.py; kernel_inference/paper_figures/kernel_gains.py].
• Bottom row is simulated, not data: 60 agents, each step every agent hears one agent drawn at random and moves 0.3 k(d), clipped to [0, 1], 80 steps, seed 1. Linear ends in consensus, bounded confidence in clusters, repulsion in two poles at the edges [same].
• Citations are Kozitsin (2021) Table 1, “Studies testing influence assumptions against real data”, by its three columns: linear positive, moderated positive, positive + negative. All approved (+) except Krizan & Baron 2007, which rejected the positive + negative hypothesis. Kozitsin’s footnotes, dropped from the slide: Mazen & Leventhal 1972 ignores trends in opinion shifts; van Knippenberg et al. 1990 does not separate in-group and out-group influence; Liu & Srivastava 2015 is nonlinear, with an increasing rate [~/Zotero/storage/2Q4MVCCW, Table 1].
• Column headings are ours: Kozitsin’s “moderated positive” is influence that fades with distance, which is the bounded confidence shape; “positive + negative” is assimilation near, repulsion far.

Literature (verified 2026-10-05 against the papers where reachable; otherwise the manuscript):

• Takács, Flache & Mäs 2016, PLoS ONE 11(6):e0157948: “positive and rather linear” (Study 1, distance coefficient 0.34); “We did not find support for the negative influence hypothesis.” One manuscript sentence (background.tex:10) calls Takács bounded confidence; that contradicts the paper.
• Vande Kerckhove et al. 2016, PLOS ONE 11(6):e0157685: constant influenceability alpha per player; bounded confidence only mentioned as an alternative.
• Kozitsin 2021, J. Math. Sociol. 46(1): positive shifts “largely through a linear form or an inverted U-shaped form”, hence in two columns. Its negative shifts “may be partially explained … without referring to the notion of repulsive influence”.
• Deffuant et al. 2000; Hegselmann & Krause 2002: step kernels, theory only. DeGroot 1974: constant gain, theory. Noise destroys BC clusters: Pineda, Toral & Hernández-García 2009 [not re-verified].
• Moussaïd et al. 2013, PLOS ONE 8(11):e78433: “Social influence is negligible when opinions are initially too distant.” The far zone is fading influence, not repulsion.
• Sherif, Taub & Hovland 1958: assimilation near the anchor, contrast far from it. Latitudes of acceptance are Sherif & Hovland 1961, not 1958.
• Adams et al. 2022, PLOS ONE 17(10):e0275473: the manuscript (introduction.tex:4) says it observed repulsive dynamics, reported as an anomaly. The paper itself tests the Martins model. [WILL: check before saying this out loud]
• If asked about backfire in the field: Bail et al. 2018 (PNAS); failures to find it: Wood & Porter 2019, Guess & Coppock 2020 [not re-verified]. Mäs & Flache 2013, PLoS ONE 8(11):e74516: “bi-polarization can be explained without negative influence”.
• Becker et al. 2017 left off: no kernel-shape claim found.

Folded in from the dissensus slide (#why-dissensus, now in backup), which showed a DeGroot consensus run against a bounded-confidence dissensus run and a tangle of candidate kernels with a question mark:

• Its trajectory panels are simulated, not data: 60 agents, each step every agent hears one agent drawn at random and moves 0.3 k(d), clipped to [0, 1], 80 steps, seed 1. DeGroot, k = 1, ends in consensus. Sigmoidal bounded confidence (sharpness 200, squared radius 0.05, so radius 0.22) ends in two groups near 0.31 and 0.79, away from the edges: dissensus without extremes [panels/why_dissensus.py]. The linear column’s dynamics panel on this slide is the same consensus run.
• DeGroot in the paper’s notation: k(|d|) = 1, so the mean move is lambda d.
• Its tangle of candidate kernels was illustrative: sigmoidal BC at four radii, SAR at four crossovers, an inverted U and a concave decline; not a list of fitted kernels [panels/why_dissensus.py, CANDIDATES].
• Picture options if the slide gets real data later: Voteview’s party means over time, or a protest photo shot from a distance with no identifiable faces.

Slide h/t Will Thompson

How do we learn the kernel from data?

Easiest case: for each \(i\) there is only one neighbor \(j\).

\[ \begin{aligned} x_i(t+1) - x_i(t) &= k(|d|) + \text{noise}\\ d &= x_j(t) - x_i(t) \end{aligned} \]

Adams, White, and Araujo (2022). “Person-to-person opinion dynamics: An empirical study using an online game.” PLoS ONE.

A more complex case: multiple neighbors

\[ \begin{aligned} \delta_i &\approx \frac{1}{|\partial i|} \sum_{j \in \partial i} k(d_{ij}) \;?\\ \delta_i &\approx k\left( \frac{1}{|\partial i|} \sum_{j \in \partial i} d_{ij} \right) \;? \end{aligned} \]

Vande Kerckhove + al. (2016). “Modelling Influence and Opinion Evolution in Online Collective Behaviour.” PLoS ONE.

Some common implicit assumptions

At each timestep each agent
always
interacts with
all
of their neighbors.

\[ \begin{aligned} \delta_i &\approx \frac{1}{|\partial i|} \sum_{j \in \partial i} k(d_{ij}) \;? \end{aligned} \]

Core idea for today: selective attention

At each timestep each agent
always sometimes
interacts with
all some
of their neighbors.

This is complex because we typically don’t observe these attentional mechanisms: we only observe the opinion changes that result.

The Team

	Will Thompson UVM		Moyi Tian CU Boulder		Linh Huynh Texas Tech
	Alex Wiedemann Hamline University				Vicki Modisette Eastern Kentucky
	Phil Chodrow Middlebury College				Heather Brooks Harvey Mudd

Work initiated at the AMS Mathematics Research Community on Complex Social Systems (2023)!

Three Innovations

Adams 2022, n = 3,760 updates

Standard: average over all neighbors

Novel: listen to one neighbor, chosen at random

Standard: you always update

Novel: sometimes you don’t

Standard: Gaussian noise (least-squares methods)

Problem: real updates often appear heavy tailed

Novel: select among noise models like Laplace or Gaussian

After Kozitsin (2021)

I can move toward a neighbor but away from the neighborhood average

Moussaïd et al. (2013): far apart, people keep their opinion

→

Chacoma & Zanette (2015): keeping happens at every distance

Ignore retention and the kernel looks like bounded confidence

[WILL, 2026-10-11 night: #attention-confounds merged back into this slide. Say the old lead aloud at the start: “Many studies have pointed out ways attention can be confounded with kernel shape. But it is rarely included in models, because it cannot be directly measured.” Click 1: the selection box pops up at the bottom: “I can move toward a neighbor but away from the neighborhood average” (Kozitsin cartoon; say aloud that averaging therefore fakes repulsion). Click 2: it goes, selection solution. Click 3: retention traditional. Click 4: the retention box (Moussaid says far apart people keep; Chacoma & Zanette show keeping at every distance; ignore retention and you see bounded confidence). Click 5: it goes, retention solution and the gate. Noise and fit beats unchanged. The click numbers below predate this and are otherwise still right.]

[WILL: one slide for both latent mechanisms (was #model-selection-only + #model-retention, both now in backup). Opens on the averaging model with “Traditional method” under selection. Click 1: the averaging problem. Click 2: the scene switches to selection, with the solution. Clicks 3-5: the same three beats under retention, and the gate appears with the retention solution. Clicks 6-8: the same beats under the kernel column for noise. Click 7 pops the Adams residual histogram over the figure; click 8 hides it and draws a Gaussian (grey) and a Laplace (teal) dotted over p(delta). Click 9: the text goes, the figure shrinks and the data come in on the right, the network and the opinion time series. Click 10: unobserved and observed boxes. Click 11: the EM arrow back from the data to kernel shape, retention and selection. The goal card that used to be click 12 is now its own slide, #goal, right after this one. 2026-10-11: the selection and retention “Problem” lines (old clicks 1 and 4) moved to #attention-confounds, the slide before, so every click number above after 0 is now one or two lower; reveal compacts the gaps. 2026-10-11 late: column text trimmed to a few words each (was full sentences, e.g. “latent neighbor selection. Every time step the agent chooses one neighbor uniformly at random to listen to”); say the long versions aloud. Removed text: “Problem: an agent can move toward a neighbor but away from the average, creating spurious repulsion (Kozitsin 2021; Mäs & Flache 2013)” and “Problem: behavior is better modeled as a probabilistic mixture of retention and influence (Chacoma & Zanette 2015)”]

Facts, with sources:

• Moussaid et al. 2013, PLOS ONE 8(11):e78433: “Social influence is negligible when opinions are initially too distant.” Regions keep / compromise / adopt over opinion distance and confidence. Figure number not re-verified [planning/history_figures.md item 8]. CC BY 4.0.
• Chacoma & Zanette 2015, PLOS ONE 10(10):e0140406, Fig. 2A: keep / adopt / compromise over opinion difference and confidence difference, overlapping ellipses. “opinion changes are not influenced by the initial difference with the reference opinion”. CC BY 4.0.
• Kozitsin 2021 (J. Math. Sociol.): negative shifts “may be partially explained … without referring to the notion of repulsive influence”. Mäs & Flache 2013, PLoS ONE 8(11):e74516: “bi-polarization can be explained without negative influence”. [WILL: confirm which Kozitsin paper and figure number the cartoon is]
• Caveat if pressed: in our synthetic 2x2, omitting retention mostly shrinks the kernel (by about pi) and only sometimes fakes bounded confidence; omitting selection smooths the kernel, and the “fake repulsion” there is a repulsive truth whose zero crossing moves, not repulsion from nothing [#hypotheses notes; #mech-retention notes].
• Built from backup slides #retention-why (Moussaid + Chacoma), #selection-why (Kozitsin cartoon) and the flash illustrations on #omission-flow.
• Figures are the existing hero stages, cropped to the selection, retention and kernel columns (the outcome panel is cut by .attn-fig in styles/components.scss): scene_hero_selection steps 1-2, scene_hero_retention step 2 [scripts/talk/scenes/scene_hero_selection.py, scene_hero_retention.py].
• Averaging can look like repulsion: Kozitsin 2021 (J. Math. Sociol.): negative shifts “may be partially explained … without referring to the notion of repulsive influence”; Mäs & Flache 2013, PLoS ONE 8(11):e74516: “bi-polarization can be explained without negative influence”.
• Clicks 9-11 are scene_hero_attn_fit [scripts/talk/scenes/scene_hero_attn_fit.py], drawn at the hero’s coordinates so stage 1 lies exactly on the noise stage; the shrink is the .attn-fig crop window opening up (styles/components.scss). The network is the toy graph the time series were simulated on (scene_data.EDGES), focal i in black as its trace.
• Noise popup is the #noise slide’s histogram with both fits [assets/panels/noise/fits.svg]: Adams, n = 3,760 updates, excess kurtosis about 10. The Laplace there is the ML fit, the Gaussian the moment fit.
• On the scene the two dotted curves share the influence bump’s mean and variance (Laplace scale sigma/sqrt 2), so the only difference is shape: a sharper peak and fatter tails [scripts/talk/scenes/scene_hero_noise.py].
• Chacoma & Zanette 2015, PLOS ONE 10(10):e0140406: keep, adopt and compromise overlap in opinion distance, so each response is a draw from a mixture of modes; the probability of keeping rises with relative confidence, not distance.
• Toy numbers as on the old slides: uniform choice of j_e is the model’s prior; pi = 0.6, retention arm N(mu_0 = 0, 0.015) drawn wider than a real spike [scripts/talk/scenes/_scene.py, World].
• Exact-zero updates are removed before fitting, not modelled; the retention arm sees only near-zero moves [paper Table 1 caption and App. D].

Slide h/t Will Thompson

Learning the model

Our stochastic model defines a likelihood function over observed opinion trajectories, given an observation of a fixed network of possible interactions.

We learn the kernel parameters by maximizing this likelihood via the expectation-maximization (EM) algorithm (Dempster et al., 1977).

We select between competing models (kernels, mechanisms, etc.) by evaluating their predictive likelihoods on held-out validation data.

Omitting retention and neighbor selection alters learned kernels

But what about real people?

Controlled, single source

Controlled, network

Social media

Prediction markets

Congress

ADAMS
lab, pairs · also TAKACS, BECKER

VK-counting
lab, groups of six · also VK-gauging

CMV
Reddit threads · also SPINOS, VKontakte

MARKETS
Manifold

SENATE
roll calls, cosponsorship · also HOUSE

controlled lab experiments

the social wilds

[WILL: five very different worlds; each click brings in one whole column (screenshot and network; the opinion-trace row was dropped 2026-10-11, the svgs are still in assets/figures/datasets/*_x.svg). The punch at the end: to us they are all the same object, a network and opinion traces, and an event is one agent’s x before and after while we see everyone else’s x]

Reading the panels: the black agent is one focal agent; its edges are black, the agents it sees grey, everyone else faint (greyscale since 2026-10-11; FOCAL in scripts/draw_specimens.py). Same grammar in every column.

Facts, with sources (exporters in presentations/ccs_2026/scripts/specimen_*.py, panels drawn by draw_specimens.py):

• ADAMS: Adams et al. 2022 (PLOS ONE), the Moussaid 2013 estimation game. The participant places a circle, sees a partner’s (red) circle, places again. Panel plots |x|, the distance to the partner, before and after: nearly every line slopes down. One update per pair. 40 random pairs of 3,760. ADAMS splits into low/medium/high by the red circle’s reliability (20/50/80%). [WILL: confirm what the traffic light in the screenshot shows]
• VK-counting: Vande Kerckhove et al. 2016 (PLOS ONE). Six players count dots over three rounds and see the others’ previous estimates. x = count / 500. One median group (gameset 61, image 4); true count dashed. Fully connected group of 6.
• CMV: r/ChangeMyView. x = classifier stance of a commenter’s latest stance-bearing comment. The near 0/1 jumps are the classifier saturating, not people flipping that hard. Network: largest reply component of the thread (126 of 281 commenters); traces for the 15 most active. Screenshot thread is not the plotted one; plotted thread is the median of the 9 threads rebuildable offline.
• MARKETS: Manifold. x = a trader’s implied probability at their latest bet; network is who replied to whom. Median market by event count (101 events, 12 traders, 6 trade). Screenshot is a different market, cropped to the price chart.
• SENATE: sim 118633. Network: cosponsorship pairs (>= 3 shared bills, Congresses 108-119), Serrano disparity backbone at alpha = 0.10; 232 senators, 1,600 edges, largest component drawn. x = Nokken-Poole first-dimension score per Congress (not DW-NOMINATE: Nokken-Poole lets a member move between Congresses). Focal: median |mean score| among the 13 senators who served all 12 Congresses. The House panel (118639) is built the same way. Screenshot: Voteview’s 119th Senate DW-NOMINATE plot (credit voteview.com).
• Screenshots: from Will’s Keynote and Voteview; CMV usernames blurred; the Manifold title is cropped off because it names a private person.

From the roster slide (#data-roster, back in the main deck right after this one, with the full table of panels, node and event counts): the “also” names under each column are the sibling panels in the same setting.

• Known single source: ADAMS, TAKACS, BECKER. A few sources: VK-gauging, VK-counting. Many sources: SENATE, HOUSE, MARKETS, SPINOS, VKontakte, CMV [paper Results, dataset paragraph].
• ADAMS has three subsets by the red circle’s reliability (low, medium, high = 20/50/80%), a property of the source, not the participant’s confidence [plan 01_plan.md §1.1; CLAUDE.md panel table].
• TAKACS has two studies; Study 2 has the within-subject disliking treatment [paper Results, dataset paragraph].
• The ladder has 15 rows, not 16 [plan 01_plan.md §6 Q3, §7].
• Open wording: “11 studies, 15 panels” vs “12 datasets, 15 panels” [plan 01_plan.md §6 Q5]. [WILL: the title says fifteen panels; confirm the count you want to say out loud]

Slide h/t Will Thompson

Impact of Attentional Mechanisms

Example: Senate political opinions

latent selection

latent retention

heavy tail

baseline

Adding attention changes our conclusion about the shape of the kernel!

[WILL: the circles on the left are the nesting: each rung contains the one before it, so a rung can only fit as well or better. WILL: introduce the ladder on one panel. Start from the classical model, then switch on one capability at a time: heavy-tailed noise, latent retention, latent selection. At every rung we fit both a linear kernel and a flexible one (RFF) and score them on held-out data. On the Senate the linear kernel wins until selection is switched on, and then the flexible kernel wins: the verdict on shape depends on which mechanisms you allow]

Facts, with sources:

• Source: figure_data/nonlinearity_by_rung_rff.json, panel senate, rungs 2 to 5 (baseline = rung “2 + best kernel”: Gaussian noise, one arm, mean-field neighbours). Drawn by panels/ladder_intro.py.
• y is held-out MLL per update minus the rung-0 null (0.876). Solid: best RFF model with at most that rung’s capabilities. Dashed: best DeGroot in the same feasible set (degroot_mll). Filled marker: the higher raw held-out score; no test.
• SENATE above null, linear vs RFF: baseline 0.020 vs 0.001; heavy tail 0.170 vs 0.159; retention 0.170 vs 0.164 (linear unchanged, retention adds nothing for DeGroot on SENATE); selection 0.174 vs 0.192.
• Gaps are small at some rungs (0.006 nats at retention). The ladder figure and the nonlinearity scatter make the cross-panel claims.
• Alternatives on the same file: HOUSE flips at every rung (L, NL, L, NL) but its gaps are under 0.005 nats until selection, so the two lines overlap; PLOS counting is L, L, L, NL like SENATE.

Slide h/t Will Thompson

Latent mechanisms make models more predictive

One bar per dataset: its held-out gain over a random-walk null, segmented into mechanisms by order of addition.

Influence kernel Presence of any social influence (rather than random drift)

Heavy tail Selected on 13/15 datasets; on average 28% more predictive power per update.

Latent retention Selected on 13/15 datasets; on average 21% more.

Latent selection Selected on 6/8 networked datasets; on average 8% more.

On average, heavy tails, retention and selection tend to matter more than the actual kernel shape!

Adding attention makes

the model more predictive (on heldout data!) on 14/15 datasets

the inferred kernel less linear on 6/15 (upper left)

the inferred kernel more linear on 2/15 (lower right)

[WILL, second half (clicks 3-11): the text goes and we walk the ladder one dataset at a time. SENATE: its bar lit up to selection, and the card shows the fit without selection, a linear kernel on the neighbour average. Click: the selection segment, and the card shows the hump the neighbour average hid. Click: its point on the scatter, x the kernel alone, y with the mechanisms; it sits upper left, linear without, nonlinear with. BECKER: retention was shrinking the influence arm; once unmixed it is steeper and curves up, upper left too. ADAMS-high: the stayers made a fake falloff, so with retention it is linear, lower right. Then every dataset, and the verdict]

Second-half build (panels/ladder_cases.py, reusing capability_ladder.py, cases.py, arms.py and nl_quadrants.py):

• Clicks: 3 text out, SENATE bar up to selection (baseline + tail + retention) + card without the mechanism; 4 SENATE full bar + card with selection and its tag; 5 SENATE on the scatter; 6 BECKER (bar, card, point); 7 ADAMS-high; 8 all bars back, card out, every point, scatter slides up; 9 verdict line 1; 10 line 2 + upper-left shaded and its bars lit on the ladder (ladder_ul); 11 line 3 + lower-right shaded and its bars lit (ladder_br). The lit rows come from the same quadrant test as the scatter (nl_quadrants.quadrant).
• 2026-10-11: card tags now read in the direction of adding attention, matching the scatter: BECKER and SENATE “more nonlinear” (influence ink), ADAMS-high “more linear” (grey). The taxonomy boxes (“Magnitude of influence shrinks”, “Curvature of the kernel shrinks”, “Spurious bounded confidence”) were removed; say them aloud if wanted. Tags live in panels/ladder_cases.py TAG; cases.py BOX (omission direction) is unchanged for the backup #cases slide.
• Card dashed line = winner at the rung just before the mechanism (BECKER and ADAMS-high rung 3, which already has the heavy tail; SENATE rung 4), as on the old #cases slide (now in backup). 2026-10-12: SENATE now leads (was BECKER), so the slow three-click build is SENATE’s; its _base state lights every segment below selection, so the lit bar and the card’s dashed rung-4 fit agree (ladder_cases.py BASE_STEPS).
• Scatter: held-out MLL per update, RFF minus DeGroot; x at the baseline rung (kernel alone, rung 2), y at the top rung with every selected mechanism, including the heavy tail. Same numbers as #nl-quadrants (backup). TAKACS-S2-control is off the top, drawn as an arrow. Raw held-out comparison, no test.
• 14/15: retention or selection step above selection.TIE_NATS (1e-3) on every panel but SPINOS [capability_ladder_dg_rff.json]. 6/15 upper left: BECKER, TAKACS-S2-disliking, SENATE, VK-counting, TAKACS-S1, HOUSE. 2/15 lower right: ADAMS-high, CMV. Several are near-ties (HOUSE x is -.0004, TAKACS-S1 x is -.0009, VK-counting y is +.004).

[WILL: what each capability buys in held-out likelihood, all 15 panels, one bar each. Open on SENATE alone: the classical model buys 2% over the null, the heavy tail another 16%, retention nothing, selection 2%; the factors multiply to 1.21x. The pictogram in each segment is the selected kernel at that rung: linear until selection, where the curve bends. Then reveal each mechanism on every dataset and read off the right-hand line]

How the right-hand numbers are computed (drawn by panels/capability_ladder.py from figure_data/capability_ladder_dg_rff.json; same 15 rows as the paper figure, takacs_disliking_interior dropped):

• A step’s value is deviance per transition, so its log likelihood ratio is value / 2 and its “% more” is exp(value / 2) - 1: the factor by which that step multiplies the per-update held-out likelihood.
• “Selected on” = step value above selection.TIE_NATS (1e-3) out of the panels where the step exists; raw held-out comparison, no test. Selection exists only on the 8 networked panels.
• “On average” = geometric mean over the panels where it is selected: exp(mean(value) / 2) - 1. Arithmetic means are dragged by CMV (retention alone is x3.2 there): 26% / 29% / 10%. Medians: tail 25%, retention 5%, selection 2%.
• Heavy tail 13/15 (zero on CMV, SPINOS), +28%. Retention 13/15 (zero on SPINOS; SENATE is +0.00004, under the tie), +21%. Selection 6/8 (zero on SPINOS, VKontakte), +8%.
• SENATE segment labels are shares of the total gain, rounded to sum to 100: baseline 10%, tail 78%, retention 0% (3.9e-5 deviance, under the tie; the blue tile is the pictogram’s backing, not a real segment), selection 12%, total 1.21x. As “% more” per update those are +2, +16, +0, +2%.
• Pictograms: the paper’s (fig_capability_ladder kernel_curves + bar_segments), drawn as influence r k(r): dotted diagonal = linear, curve = r times the fold-mean RFF G_grid, each scaled to its own tile.
• Build: SENATE alone, then click 1 jumps straight to every bar complete with all three mechanism lines (the tail-only and retention-only states are no longer shown), click 2 the closing line.
• Closing line: averaged over the 15 rows, the latent mechanisms (tail + retention + selection) carry 65% of each panel’s gain over the null; the kernel-shape step alone (RFF over DeGroot at the baseline rung) carries 0.7%, and influence plus shape together 35%. Mechanisms beat influence plus shape on 11/15 (not on TAKACS-S2-control, TAKACS-S2-disliking, BECKER, SPINOS) [capability_ladder_dg_rff.json, steps clipped at 0, mean of per-panel shares].
• Strictly these are gains over the previous rung, not over the null; the factors multiply to the total over the null.

Dropped stage claims, from fig_capability_ladder.stages.json (full six-stage build in backup, #backup-ladder-build): stage 1, “How to read a bar: the share of the held-out log likelihood ratio over the random-walk null, left to right as capabilities switch on; the total is the per-transition likelihood multiplier.” Stage 2, “The classical model, influence plus kernel shape, alone: its share runs from 0.6% on VKontakte to 98% on SPINOS.”

Facts, with sources:

• Step 1, SENATE row: shares 0.10 / 0.78 / 0.00 / 0.11 of its gain, total 1.21x [figure_data/capability_ladder_dg_rff.json; plan 01_plan.md §3.2].
• Baseline share runs from 0.6% (VKontakte) to 98% (SPINOS) [fig_capability_ladder.stages.json step 2; capability_ladder_dg_rff.json].
• Laplace over Gaussian is selected on 13 of 15 panels (not on CMV and SPINOS per the ladder JSON) and is the largest segment on 9 [fig_capability_ladder.stages.json step 3; capability_ladder_dg_rff.json]. The paper text says “all but two of 15” [paper Results, heavy-tailed paragraph]; note the paper’s own margin comment says Laplace is slightly selected on CMV.
• Retention adds on 13 of 15 panels (zero on SENATE and SPINOS) [capability_ladder_dg_rff.json; fig_capability_ladder.stages.json step 4]. DISCREPANCY: the paper prose says 14 of 15 [paper Results, “Latent retention is a common feature”]; write 13/15 until the prose is fixed.
• Retention share: CMV 63% of its gain, VKontakte 42% [fig_capability_ladder.stages.json step 4].
• Selection beats aggregation on 6 of 8 networked panels, an approximate tie on the other two (VKontakte, SPINOS) [paper Results, source selection paragraph; capability_ladder_dg_rff.json]. Shares: CMV 22%, SENATE 12%, HOUSE 9% [paper Results; plan 01_plan.md §3.2].
• Totals: CMV 6.26x, VKontakte 2.90x, ADAMS-high 2.03x, SPINOS 1.08x [fig_capability_ladder.stages.json step 5].
• ADAMS: estimated response probability 78% (ADAMS-high, red circle 80% reliable) vs 44% (ADAMS-low, 20% reliable) [paper Results, retention paragraph]. The paper calls it “retention rate”; per the plan it is pi-hat, the probability of a social response [plan 01_plan.md §6 Q4].
• SPINOS: no retention, no selection, lambda not distinguishable from zero [paper Results, source selection paragraph].
• Pictograms show the fitted kernel at each step; vertical scales are normalised per dataset [paper Fig. explained_deviance caption].

Slide h/t Will Thompson

Some Learnings

Potential exposure \(\neq\) interaction

In experiments and online settings, relationship networks (proximity, following, etc.) define possible rather than actual interactions.

Model latent mechanisms

We can model what we don’t observe directly: we just need to tie it statistically to things that we do observe.

We direct our attention!

Even simple representations of attention can improve prediction and alter our conclusions.

Thanks everyone!

arxiv:2610.12616

	Will Thompson UVM		Moyi Tian CU Boulder
	Linh Huynh Texas Tech		Alex Wiedemann Hamline University
	Vicki Modisette Eastern Kentucky		Heather Brooks Harvey Mudd