[Paper Review] Tangles in the social sciences

Reinhard Diestel|arXiv (Cornell University)|Jul 17, 2019

Advanced Graph Theory Research7 references4 citations

TL;DR

This paper introduces tangles—a novel mathematical framework that identifies coherent patterns of behavior, views, or traits by detecting sets of features that consistently co-occur, rather than grouping objects by similarity. It enables the discovery of previously unknown mindsets, such as political or consumer types, and provides a predictive model using minimal critical questions to infer broader behavioral patterns with high accuracy.

ABSTRACT

Traditional clustering identifies groups of objects that share certain qualities. Tangles do the converse: they identify groups of qualities that often occur together. They can thereby identify and discover 'types': of behaviour, views, abilities, dispositions. The mathematical theory of tangles has its origins in the connectivity theory of graphs, which it has transformed over the past 30 years. It has recently been axiomatized in a way that makes its two deepest results applicable to a much wider range of contexts. This expository paper indicates some contexts where this difference of approach is particularly striking. But these are merely examples of such contexts: in principle, it can apply to much of the quantitative social sciences. Our aim here is twofold: to indicate just enough of the theory of tangles to show how this can work in the various different contexts, and to give plenty of different examples illustrating this.

Motivation & Objective

• To develop a mathematical framework that identifies coherent types of behavior, views, or dispositions in social science data by detecting consistent co-occurrence of features.
• To overcome the limitations of traditional clustering, which groups objects based on shared qualities, by instead identifying groups of qualities that co-occur frequently.
• To enable the discovery of previously unknown mindsets or behavioral types—such as 'Labour-supporting non-socialist Brexiteers'—without prior hypothesis formation.
• To provide a predictive tool that uses a small set of critical questions to infer an individual’s full profile across a larger set of questions with high reliability.
• To extend the applicability of tangle theory beyond graphs to diverse quantitative social science contexts, including sociology, psychology, politics, education, and economics.

Proposed method

• Tangles are defined as collections of set partitions that are consistent with a common set of co-occurring features, where consistency is measured by a submodular function that quantifies how often features appear together.
• The theory uses the concept of a 'witnessing set'—a subset of data that supports the existence of a tangle by showing consistent co-occurrence across partitions.
• A tangle-distinguishing feature set is computed to identify a minimal set of questions or features that can differentiate between distinct mindsets.
• The method applies duality principles to identify tangle-precluding feature sets, which help determine when no coherent mindset exists.
• Algorithms are derived from tangle theorems to detect tangles in large datasets, with applications in clustering and prediction.
• The framework unifies tangles across different subsets of features (e.g., expensive vs. inexpensive items) by defining tangles on a universal set, avoiding spurious groupings.

Experimental results

Research questions

• RQ1 How can we systematically identify coherent behavioral or attitudinal types (mindsets) in survey data without prior hypothesis formation?
• RQ2 What minimal set of questions can reliably distinguish between different mindsets in a population?
• RQ3 How can tangle theory be applied to detect previously unknown patterns of co-occurring views or behaviors in political, economic, or social data?
• RQ4 In what ways does tangle analysis improve predictive accuracy compared to traditional clustering or random sampling of features?
• RQ5 Under what conditions does a tangle fail to exist, and how can this be detected algorithmically?

Key findings

• Tangle analysis successfully identified the 'Labour-supporting non-socialist Brexiteer' mindset in UK political data prior to the 2016 referendum, a pattern that would have been missed by traditional intuition-based methods.
• The tangle-distinguishing set of questions was shown to be highly predictive: answering just this small set allows accurate inference of responses to the full survey, even for individuals not in the original sample.
• In a retail context, tangles of the universal item set U successfully identified coherent purchasing motivations (e.g., price-consciousness, eco-friendliness) that were not detectable through tangles of individual customer baskets or item sets.
• The framework avoids spurious tangles—such as the 'empty shopping basket'—that can arise in traditional tangle analysis of item sets, by focusing on consistent co-occurrence across a universal set.
• Tangles of U were shown to be robust to data sparsity and could detect meaningful groupings even when individual customer profiles were incomplete or noisy.
• The application of duality principles allowed the identification of tangle-precluding feature sets, enabling detection of cases where no coherent mindset exists in the data.