Structurally Conditioned Diffusion Reproduces Skills-Based Stratification

How Skill Requirements Propagate — And Why They Preserve Hierarchies

Roberto Cantillan & Mauricio Bucca

Department of Sociology | Pontificia Universidad Católica de Chile

The Puzzle

What we know:

• Labor markets are organized around unequal occupations differing in skills, wages, and credentials
• A polarized skill space: socio-cognitive vs. sensory/physical domains align with education and wages
• A nested hierarchy: enabling capabilities sit higher, command wage premia, require longer education

What we don’t know:

How do skill requirements propagate between occupations — and why does this process reproduce hierarchies?

Occupation	1990	2010
Secretary	✓	✓
Nurse	✗	✓

The Architecture We Build Upon

Polarization (Alabdulkareem et al. 2018)

• Skills cluster into two domains
• Socio-cognitive: high education, high wages
• Sensory/physical: lower education, lower wages

The Architecture We Build Upon

Nestedness (Hosseinioun et al. 2025)

• Asymmetric dependencies: some skills “enable” others
• Nested architecture has intensified over time
• Position in hierarchy predicts wages, automation risk

I. A GRAVITY MODEL FOR SKILL DIFFUSION

Skill diffusion as a gravity process

• Skill specific diffusivity (g)

• Masses (m)
  

• Distance (d)

  • Task similarity (symmetric)
  • Status gap (asymmetric)

Our theory: Asimmetric trayectory channeling

• Socio-cognitive requirements are preferentially adopted by occupations of higher status than the source, successfully climbing the status gradient.

• Physical requirements face a structural “upward veto,” being predominantly relegated to adoption paths toward lower-status occupations.

• This asymmetric dynamic would explain the observed polarized skill architecture of the labor market.

Our theory: Asimmetric trayectory channeling

Alternative accounts

• Skills travel between similar occupations: We control for occupational task relatedness (structural distance).
• Diffusion is driven by occupation size, not skills: We account for occupational “mass” using occupation fixed effects (Source and Target).
• Some skills travel more easily than others: We account for intrinsic diffusivity using skill-level fixed effects (\(g_s\)).

II. DATA & METHODS

Data

• O*NET 2015-2024 (161 skill taxonomy)
• BLS wages and education
• About 17M dyadic opportunities

Measures I

• Dependent Variable: Skill Adoption (\(Y_{ijs}\)). Where \(Y=1\) if a target (2024) adopts a skill previously held by a source (2015), and \(Y=0\) if the opportunity for adoption was not taken.

• Main Independent Variables:
  • Wage Gap (Piecewise):
    • \(\text{wage_gap}_{ij}{\uparrow} = \max(0, G_{ij})\)
    • \(\text{wage_gap}_{ij}{\downarrow} = \min(0, G_{ij})\)
    • Where \(G_{ij} = \log(\overline{W}_{target}) - \log(\overline{W}_{source})\)

• Critical control variables:
  • Structural Proximity: Occupational Task Relatedness calculated via cosine similarity
  • Source, Target and Skill fixed effects.

Measures II

Skill type: Skill types might respond to wage gaps differently

The ATC Model

For each skill type, I estimate a linearized version of the gravity model

\[ \begin{aligned} \mathrm{logit}\,P(Y_{ijs}=1) = \, & \beta_{\text{up}}(\text{wage_gap}_{ij})^\uparrow + \beta_{\text{down}}(\text{wage_gap}_{ij})^\downarrow \\ & + \delta \cdot \text{relatedness}_{ij} \\ & + \text{source}_{i} / \text{target}_{j} + \text{skill}_{s} \end{aligned} \]

• \(\beta_{\text{up}}/ \beta_{\text{down}}\) measure the specific resistance (or facilitation) encountered when the target occupation has a higher (\(\uparrow\)) or lower (\(\downarrow\)) wage status than the source.
• \(\delta\) is the effect for the task-relatedness between occupations.
• \(\text{source}_i, \text{target}_j, \text{skill}_s\) are fixed effects measuring occupational emission/absorption mass and intrinsic “diffusivity” of each skill¹.

1. Source and taregt fixed effects are introduced separately to break perfect colinearity

III. FINDINGS

Raw Patterns

Polarized Diffusion: Flow Networks

III. REGRESSION RESULTS

Asymmetric trajectory channeling by skill domain

by domain & Nestedness

Why This Happens: Potential Reasons

This asymmetry reflects structural filtering, not chance:

• Adoption capacity
  High-status occupations have more resources, slack, and organizational capacity to experiment with complex, high-nestedness skills.

• Market shift
  Elite occupations increasingly demand socio-cognitive skills to manage coordination, abstraction, and organizational complexity.

• Status filtering
  High-status occupations actively avoid physical requirements to preserve professional distinction and symbolic boundaries.

Conclusion

Main finding:

Skill diffusion is asymmetric: it pulls socio-cognitive skills upward and pushes physical skills downward.

Key takeaways:

• Directional filtering
  Physical skills face a barrier that prevents them from entering high-wage occupations.

• Nestedness matters
  Skills that anchor and enable other skills exhibit the strongest asymmetries.

• Systemic Reproduction
  This selective channeling acts as the dynamic engine that maintains and reinforces the labor market’s polarized skill architecture.

Thank You

Roberto Cantillan

Department of Sociology, PUC Chile

rcantillan@uc.cl

Paper and Replication: github.com/rcantillan/skill_diffusion

Backup Slides

B1: Formal Definitions

Revealed Comparative Advantage:

\[\mathrm{RCA}(j,s) = \frac{\mathrm{onet}(j,s)/\sum_{s'}\mathrm{onet}(j,s')}{\sum_{j'}\mathrm{onet}(j',s)/\sum_{j',s''}\mathrm{onet}(j',s'')}\]

Event definitions for skill s:

• Specialists at baseline: RCA greater than 1 at t0
• Users at follow-up: RCA greater than 1 at t1
• New adopters: Users minus Specialists

Directional gaps:

\[\Delta^{\uparrow}_{ij} = \max(0, \text{status}_j - \text{status}_i)\]

\[\Delta^{\downarrow}_{ij} = \max(0, \text{status}_i - \text{status}_j)\]

B2: Nestedness Definition

Contribution to nestedness (cs):

\[c_s = \frac{\mathrm{NODF}_{\text{obs}} - \mathbb{E}[\mathrm{NODF}^{(s)}_{\text{rand}}]}{\mathrm{sd}[\mathrm{NODF}^{(s)}_{\text{rand}}]}\]

Interpretation:

• cs high: Skill acts as scaffolding (analytics, management, communication)
• cs low: Terminal or narrow capability (specific technique)

Reach:

• Size of forward-reachable set in dependency network
• High reach means skill enables many downstream capabilities

In our models:

• Terciles: Low, Mid, High nestedness
• Interact with directional gaps
• High nestedness amplifies ATC asymmetry

B3: Full Coefficient Table

Term	Coefficient	SE
Upward (base)	+0.08	0.04
Downward (base)	+0.13	0.11
Domain: Physical	+0.72	0.30
Upward × Physical	−0.42	0.07
Downward × Physical	+0.52	0.18
Structural distance	−1.90	0.20
Distance × Physical	−1.33	0.52
Nestedness: Mid	+0.00	0.04
Nestedness: High	−0.13	0.05
Upward × Nest-Mid	+0.12	0. 03
Upward × Nest-High	+0. 07	0.04

ClogLog link. Two-way FE (source, target). Clustered SE.

B4: Complete Gravity Model Derivation

Step 1: Classic Gravity \[T_{ij} = k \cdot \frac{M_i M_j}{D_{ij}^{\gamma}} \quad \Rightarrow \quad \log \mathbb{E}[T_{ij}] = \beta_0 + \alpha_i + \beta_j - \gamma \log D_{ij}\]

Step 2: Triadic Extension (adding skill \(k\)) \[\lambda_{ijk} \propto \frac{M_i \cdot M_j \cdot S_k}{D_{ij}^{\gamma}} \quad \Rightarrow \quad \log \lambda_{ijk} = \beta_0 + \alpha_i + \beta_j + \delta_k - \gamma \log D_{ij}\]

Step 3: Asymmetric Frictions \[\Delta^{\uparrow}_{ij} = \max(G_{ij}, 0), \quad \Delta^{\downarrow}_{ij} = \max(-G_{ij}, 0)\]

Step 4: Skill Mass via Attributes \[\text{SkillMass}_k = \eta_0 + \eta_D D_k + \eta_N N_k + \eta_{DN} D_k N_k\]

Step 5: Full Specification \[\text{cloglog}\big(\Pr(T_{ijk}=1)\big) = \beta_0 + \alpha_i + \beta_j + \eta_0 + \eta_D D_k + \eta_N N_k + \eta_{DN} D_k N_k + \beta^{\uparrow} \Delta^{\uparrow}_{ij} + \beta^{\downarrow} \Delta^{\downarrow}_{ij}\]

B5: Key Magnitudes

Coefficients (wage gap, cloglog scale):

Term	Cognitive	Physical	Interaction
Upward	+0.08.	−0.34	−0.42***
Downward	+0.13 (ns)	+0.65	+0.52**
Distance	−1.90***	−3.23	−1.33*

Interpretation:

• Upward gap slightly boosts cognitive adoption, penalizes physical
• Downward gap has no effect on cognitive, strongly boosts physical
• Distance decay is 70% steeper for physical skills

B6: Robustness Summary

Test	Specification	Result
Thresholds	RCA 0.9, 1.0, 1.1	Pattern unchanged
Distances	Shortest path, resistance, cosine	Same relative slopes
Periods	2015-18, 2019-21, 2022-24	ATC persists
Bootstrap	Node-level B=1000 vs clustered SE	Ratio around 1.0
Representation	Skill-skill vs occ-occ network	Ordinal match

The asymmetry is not an artifact of thresholds, distance measures, or time periods.