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Understanding friendship formation with explainable machine learning

Using an Explainable Boosting Machine to analyze signed social relationships among 3,395 students, the study reveals that local network structure (triadic influence) overwhelmingly drives friendship formation, while individual traits only explain a tiny subset of weaker, less embedded ties, thereby supporting a layered view of social tie formation where structural mechanisms dominate globally.

Original authors: María Pereda

Published 2026-03-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: María Pereda

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to figure out why two people in a high school become best friends, or why they end up hating each other. Is it because they are just naturally similar people (maybe they both love math and are very kind)? Or is it because of the "social web" around them (maybe they have three mutual friends who push them together)?

This paper is like a detective story that uses a special kind of "smart microscope" (Machine Learning) to answer that question. Here is the breakdown in simple terms:

1. The Detective Tool: The "Explainable" AI

Usually, when computers predict things, they act like a "black box." You put data in, and a result comes out, but you have no idea why the computer made that choice.

The researchers used a special tool called an Explainable Boosting Machine (EBM). Think of this not as a black box, but as a transparent glass box. You can see exactly how the computer is thinking. It can tell you: "I predicted they are friends because 90% of the reason is their mutual friends, and only 10% is because they both like pizza."

2. The Two Main Suspects

The study looked at two main suspects for why friendships form:

Suspect A: The Individual Traits. (Who you are: Are you kind? Are you smart? What's your gender?)
Suspect B: The Social Web (Triadic Influence). (Who you know: Do you share friends? If Alice likes Bob, and Bob likes Charlie, does that make Alice like Charlie?)

3. The Big Verdict: The Social Web Wins (Huge)

The computer analyzed over 3,000 students and their relationships. The result was clear: The Social Web is the boss.

The Analogy: Imagine trying to predict if two people will get along by looking at their resumes (Suspect A). It's okay, but not great. Now, imagine looking at a map of who hangs out with whom (Suspect B). If they are already in the same circle of friends, the prediction becomes almost perfect.
The Stat: The "Social Web" factor (Triadic Influence) explained 91% of the friendships and feuds. The personal traits (like being kind or smart) only explained a tiny sliver.

4. The Mystery of the "Tiny 0.24%"

If the Social Web is so powerful, why did the researchers bother looking at personal traits? Because they found a tiny, weird group of relationships (only 0.24% of all links) where the Social Web didn't seem to matter. In these rare cases, the computer said, "Hey, these two are friends/foes just because of who they are, not because of their friends."

Who are these people?

They are mostly first-year students.
They are often weaker relationships (not best friends, maybe even enemies).
They are less "balanced."

The Analogy: Think of a new student walking into a cafeteria. They haven't met anyone yet. They might sit next to someone just because they both like the same weird music (Individual Trait). But once they start hanging out with the same group of people, the "Social Web" takes over, and their relationship becomes stable and predictable based on the group.

5. The "Empty Room" Surprise

A previous study suggested that if two people have no mutual friends (an "empty room" with no social web), then their relationship must be driven entirely by their personalities.

The researchers found this was WRONG.
Even when two people have zero mutual friends, the computer still said, "The reason they are friends is the absence of a social web."

The Analogy: It's like walking into a quiet room. The silence isn't caused by a specific sound; the silence is the signal. The computer realized that "having no friends in common" is actually a very strong signal in itself, even if it's a zero value. It's not the personality that drives it; it's the lack of structure.

6. The Final Picture: A Layered Cake

The paper concludes that friendship formation isn't a simple "Nature vs. Nurture" fight. It's a layered process:

The Foundation (Global): For almost everyone, the "Social Web" (triadic influence) is the main engine. If you share friends, you are likely to be friends. This creates stable, positive, and balanced relationships.
The Top Layer (Local): In specific, early, or weak situations (like new students or shaky relationships), personal traits can peek through. But these are the exceptions, not the rule.

The Takeaway

This study shows that we are less "individuals" and more "nodes in a network" than we think. While our personalities matter a little bit, the structure of our social circle is the giant puppet master pulling the strings of who we like and who we dislike.

The researchers also showed us that AI doesn't have to be a magic trick. By using "Explainable" AI, we can finally understand why the computer thinks what it thinks, turning a black box into a clear window into human behavior.

1. Problem Statement

The formation of social ties is a fundamental question in network science, specifically regarding the relative influence of individual traits (e.g., personality, cognitive style, gender) versus local network structure (e.g., triadic closure, indirect paths).

The Conflict: Previous research suggested that structural metrics like "triadic influence" (indirect paths of length two) could predict relationship signs (friendship vs. enmity) better than individual traits. However, a lingering hypothesis suggested that in the absence of structural paths (isolated links), individual traits might drive the initial formation of ties.
The Gap: Traditional "black-box" machine learning models (like deep neural networks) can predict link polarity accurately but fail to explain why or which features drive specific predictions. There is a need for a model that offers both high predictive accuracy and interpretability to disentangle these competing mechanisms.

2. Methodology

Dataset

Source: Publicly available data from a previous study involving 3,395 students across 13 secondary schools in Spain.
Data Type: Self-reported signed social relationships (directed links).
Labels: Relationships were categorized into four levels (−2: "very bad" to +2: "very good"). For binary classification, the study focused on Friendship (+2) vs. Enmity (−1 or −2).
Features:
- Individual Traits: Gender, Cognitive Reflection (measured via the 3-question CRT), and Prosociality (scaled economic choice preferences).
- Structural Metric: Triadic Influence ( $I_{ij}$ ), defined as the sum of directed paths of length two between node $i$ and node $j$ ( $I_{ij} = \sum_k w_{ik}w_{kj}$ ).

Model Selection and Training

Comparative Analysis: The authors compared four classifiers: TensorFlow Neural Network, PyTorch Neural Network, XGBoost, and the Explainable Boosting Machine (EBM).
Selection Criteria: All models achieved similar balanced accuracy (~0.824). The EBM was selected for the final analysis because it offered the lowest variance (most stable) and, crucially, provides transparent, additive feature contributions.
EBM Architecture:
- A type of Generalized Additive Model (GAM) extended with pairwise interactions.
- Formula: $g(E[Y]) = \beta_0 + \sum f_j(x_j) + \sum f_{ij}(x_i, x_j)$ .
- Uses gradient boosting with shallow trees in a round-robin fashion to ensure robustness against collinearity while maintaining interpretability via partial dependence plots.
Validation: Stratified nested cross-validation (5 outer, 5 inner folds) with Random OverSampling (ROS) applied only to training folds to handle class imbalance without data leakage.

3. Key Contributions

Application of EBM to Social Networks: Demonstrates the utility of Explainable Boosting Machines in social network analysis to move beyond prediction accuracy to mechanism discovery.
Quantification of Structural Dominance: Provides rigorous evidence that triadic influence is the overwhelming driver of link polarity, accounting for the vast majority of predictive power.
Characterization of the "Individual-Dominant" Subset: Identifies and deeply analyzes the tiny fraction of links (0.24%) where individual traits, rather than structure, drive the prediction.
Refutation of the "Isolated Link" Hypothesis: Challenges the idea that links without structural paths are driven by personal traits. The study shows these links are predicted by the absence of structural reinforcement (a zero value for triadic influence), not by latent personal affinities.

4. Key Results

A. Global Feature Importance

Triadic Influence Dominance: Triadic influence is the single most important predictor, contributing significantly more to the log-odds of a link than any individual trait.
Distribution of Dominance:
- 91.47% of links are dominated by triadic influence (>50% contribution).
- 0.24% of links are dominated by individual traits.
- 1.27% are dominated by interaction terms.
- 7.02% show no clear dominance.

B. Dependence Plots (Mechanism Analysis)

Triadic Influence: Exhibits a strong, smooth, monotonic relationship. Positive values strongly increase the probability of friendship; negative values increase the probability of enmity.
Individual Traits (CRT & Prosociality): Effects are weak and non-monotonic.
- Cognitive Reflection: Minimal impact; slightly negative for the source node, negligible for the target.
- Prosociality: Shows a modest positive trend for the source node (higher prosociality $\rightarrow$ higher chance of friendship) but negligible effect for the target.

C. Characterization of the Individual-Dominant Subset (0.24%)

Links where individual traits drive the prediction differ structurally and socially from the majority:

Structural Embedding: These links are less embedded. They have fewer indirect paths and are more likely to be "weak" connections (no length-2 paths).
Link Polarity: They are significantly more likely to be negative ties (35.9% negative vs. 23.5% for triadic-dominant links).
Structural Balance: They exhibit lower balanced ratios (mean $\approx$ 0.68) compared to triadic-dominant links (mean $\approx$ 0.75), indicating they are less constrained by the principle of structural balance.
Demographics: These links are disproportionately associated with second-year students, suggesting a temporal dynamic where individual traits play a larger role before structural patterns fully stabilize.
Centrality: No significant difference in node centrality (degree, betweenness) between the two groups.

D. The "No Path" Paradox

The study specifically tested links where $A^2_{ij} = 0$ (no indirect paths).
Finding: Contrary to the hypothesis that personal traits drive these isolated links, the EBM attributed 100% of these predictions to the triadic influence feature (specifically, its value being zero).
Implication: The model interprets the lack of structural reinforcement as a distinct signal, rather than defaulting to individual traits when structure is absent.

5. Significance and Conclusion

Layered View of Tie Formation: The paper proposes a nuanced, layered model of social network formation. Structural mechanisms (triadic closure) dominate globally and govern stable, positive, and balanced relationships. Individual-level effects emerge only in specific, less constrained contexts (weak, negative, or newly formed ties).
Dynamic Interpretation: The findings support a two-stage dynamic: early or weak links may be sensitive to individual traits (e.g., prosociality), but as the network matures, structural reinforcement takes over as the primary organizing principle.
Methodological Impact: The work validates Explainable Machine Learning (EBM) as a critical tool for network science. It proves that high-accuracy models can be used not just for prediction, but to uncover the functional forms and relative weights of the mechanisms driving complex social behaviors, offering a transparent alternative to black-box deep learning.

In summary, the study confirms that while individual traits matter, local network topology is the primary architect of social relationships, with individual characteristics playing a secondary, context-dependent role in the formation of weaker or negative ties.