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. THE PHENOMENON

Polarized Diffusion: Flow Networks

The Asymmetry: Directional Friction by Domain

Asymmetry means skills face different “friction” when diffusing up vs. down the wage ladder. The direction of this friction reverses between cognitive and physical skills.

	Cognitive	Physical	Interpretation
Upward rate	20.2%	11.1%	Cognitive climbs easily
Downward rate	15.6%	19.0%	Physical falls easily
Gap (Up − Down)	+4. 7 p.p.	−8.0 p. p.	Opposite signs
Implication	Facilitates ascent	Blocks ascent	Polarizing force

Physical skills face 1.7× more friction going up than going down

Average shift in wage quintiles (destination - origin). Core cognitive skills drift upward

Most physical skills drift downward or lateral (static strength, inspection, coordination)

Raw Gravity Patterns

Adoption rate by education gap, wage gap, and structural distance deciles

II. THE MODEL

Gravity Logic for Skill Diffusion

Classic Gravity Model:

The expected flow from origin \(i\) to destination \(j\) follows:

\[T_{ij} = k \cdot \frac{M_i M_j}{D_{ij}^{\gamma}}\]

where \(M_i, M_j\) = origin/destination mass, \(D_{ij}\) = distance, \(\gamma\) = distance sensitivity.

Log-linear form:

\[\log \mathbb{E}[T_{ij}] = \beta_0 + \alpha_i + \beta_j - \gamma \log D_{ij}\]

Application to skill diffusion:

Some occupations are stronger senders (innovation-intensive)
Others are more capable receivers (high absorptive capacity)
Diffusion declines with cognitive, educational, or wage/status distance

Triadic Gravity: Incorporating Skills

Skill diffusion is inherently triadic: source occupation \(i\), target occupation \(j\), and skill \(k\).

Binary outcome:

\[T_{ijk} = \begin{cases} 1 & \text{if skill } k \text{ is newly adopted in } j \text{ after being established in } i \\ 0 & \text{otherwise} \end{cases}\]

Triadic gravity with skill-specific mass \(S_k\):

\[\lambda_{ijk} \propto \frac{M_i \cdot M_j \cdot S_k}{D_{ij}^{\gamma}}\]

Log-linear form:

\[\log \lambda_{ijk} = \beta_0 + \alpha_i + \beta_j + \delta_k - \gamma \log D_{ij}\]

Asymmetric Frictions: The ATC Mechanism

Standard gravity assumes symmetric distance costs. In occupational diffusion, this is unrealistic.

Directional decomposition of wage/status gap \(G_{ij}\):

\[\Delta^{\uparrow}_{ij} = \max(G_{ij}, 0) \quad \text{(upward: target higher than source)}\]

\[\Delta^{\downarrow}_{ij} = \max(-G_{ij}, 0) \quad \text{(downward: target lower than source)}\]

Core hypothesis — Asymmetric Trajectory Channeling (ATC):

Socio-cognitive skills face low upward friction; physical skills face high upward friction

This decomposition enables distinct coefficients for upward vs. downward flows — the central feature of ATC.

Modeling Skill Diffusibility

Instead of absorbing skill heterogeneity via unrestricted fixed effects (\(\delta_k\)), we model skill “mass” structurally:

Two theoretically grounded attributes:

Domain (\(D_k\)): Physical vs. cognitive/socio-cognitive
Nestedness (\(N_k\)): Low vs. high dependency in skill hierarchy

Skill mass parameterization:

\[\text{SkillMass}_k = \eta_0 + \eta_D D_k + \eta_N N_k + \eta_{DN} D_k N_k\]

This enforces meaningful structure in the skill space, rather than treating each skill as categorical noise.

Triadic Gravity Specification (ClogLog)

With a binary outcome in discrete time, we use a complementary log-log link:

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

Interactions (e.g., \(D_k \cdot \Delta^{\uparrow}_{ij}\), \(N_k \cdot \Delta^{\downarrow}_{ij}\)) reveal whether:

Certain skill types face penalties in upward diffusion
Physical/nested skills are preferentially funneled downward

Fixed effects:

\(\alpha_i\): Source sending capacity
\(\beta_j\): Target receiving capacity
No full \(\delta_k\): Skill heterogeneity explained via domain × nestedness

Identification Strategy

Challenge: O*NET rolling panel creates “false zeros”

Solution: Interval-Censored Design

Adoption = latent event within window
Opportunity: source specializes, target does not
Outcome: target crosses threshold

Estimation:

Two-way fixed effects (source, target)
Node-level bootstrap (B=1000) for valid inference

Diagnostics:

Placebo leads (future gaps have no effect)
Within-cluster permutation
Worker-flow distances replication
Out-of-sample prediction (2015-19 to 2020-24)

Data:

O*NET 2015-2024 (161 descriptors)
BLS wages and education
About 5M dyadic opportunities

III. MODEL RESULTS

ClogLog discrete-time hazard. Two-way FE. 90% CI from node bootstrap

High-nestedness physical skills face more than 30 p.p. penalty at 2-quartile upward gap

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

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.

IV. IMPLICATIONS

What This Tells Us

The asymmetry is structural, not incidental:

Physical skills don’t just happen to stay low — they face systematic friction
Cognitive skills don’t just happen to climb — the path is smoother
Nestedness amplifies these differences

This reframes polarization:

Not just what skills exist where, but how skill requirements flow — and why certain flows are blocked

Open questions:

Does reducing occupational distance (via training, credentialing) actually change diffusion?
Which specific skills are “near the boundary” and most amenable to intervention?
How do firm-level decisions interact with these macro patterns?

Conclusion

Main finding:

Skill diffusion follows an asymmetric gravity rule that actively filters physical requirements out of high-status occupations

Three key mechanisms:

Directional friction: Physical upward friction much greater than cognitive
Distance decay: Steeper for physical than cognitive skills
Nestedness amplification: Enabling skills show strongest asymmetries

Contribution:

Provides the missing bridge from static architecture to dynamic reproduction
Specifies testable propagation rules with concrete policy levers
Demonstrates that stratification is reproduced through selective diffusion

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.

B5: 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 \[\text{cloglog}\big(\Pr(T_{ijk}=1)\big) = \beta_0 + \alpha_i + \beta_j + \eta_0 + \eta_D D_k + \eta_N N_k + \eta_{DN} D_k N_k + \beta^{\uparrow} \Delta^{\uparrow}_{ij} + \beta^{\downarrow} \Delta^{\downarrow}_{ij}\]