[Paper Review] Higher-order Homophily is Combinatorially Impossible.

Nate Veldt, Austin R. Benson+1 more|arXiv (Cornell University)|Mar 22, 2021

Consumer Market Behavior and Pricing24 references4 citations

TL;DR

This paper introduces a hypergraph-based framework to measure homophily in multiway group interactions, revealing that strict higher-order homophily—such as all-gender or all-party groups—is combinatorially impossible due to structural constraints. It shows that observed group-level homophily requires relaxed, minority-participation mechanisms, explaining patterns in congressional co-sponsorship and gender distribution in group photos.

ABSTRACT

Homophily is the seemingly ubiquitous tendency for people to connect with similar others, which is fundamental to how society organizes. Even though many social interactions occur in groups, homophily has traditionally been measured from collections of pairwise interactions involving just two individuals. Here, we develop a framework using hypergraphs to quantify homophily from multiway, group interactions. This framework reveals that many homophilous group preferences are impossible; for instance, men and women cannot simultaneously exhibit preferences for groups where their gender is the majority. This is not a human behavior but rather a combinatorial impossibility of hypergraphs. At the same time, our framework reveals relaxed notions of group homophily that appear in numerous contexts. For example, in order for US members of congress to exhibit high preferences for co-sponsoring bills with their own political party, there must also exist a substantial number of individuals from each party that are willing to co-sponsor bills even when their party is in the minority. Our framework also reveals how gender distribution in group pictures varies with group size, a fact that is overlooked when applying graph-based measures.

Motivation & Objective

• To address the limitation of pairwise homophily measures in capturing real-world group interactions.
• To investigate whether strict homophily in multiway interactions (e.g., same-gender or same-party groups) is combinatorially feasible.
• To develop a formal framework using hypergraphs to quantify group-level homophily and identify structural impossibilities.
• To explain observed social patterns—such as party co-sponsorship in Congress and gender distribution in group photos—through structural constraints rather than behavior alone.

Proposed method

• Model social groups as hypergraphs, where each hyperedge connects multiple individuals, enabling representation of multiway interactions.
• Define higher-order homophily as a preference for groups in which one's own attribute (e.g., gender or party) is the majority.
• Use combinatorial analysis to prove that simultaneous majority homophily across multiple attributes (e.g., men and women both preferring same-gender groups) is structurally impossible.
• Introduce relaxed homophily measures that account for minority participation, showing how such participation enables observed group-level patterns.
• Apply the framework to real datasets, including U.S. congressional co-sponsorship and group photo data, to validate structural predictions.
• Derive constraints on group composition based on hypergraph structure, revealing how gender and party distributions vary with group size.

Experimental results

Research questions

• RQ1 Can strict homophily in multiway group interactions—where individuals prefer groups in which their attribute is the majority—be simultaneously satisfied by multiple attributes (e.g., gender and party)?
• RQ2 What structural constraints in hypergraphs make certain homophilous group preferences combinatorially impossible?
• RQ3 How can observed patterns of group co-sponsorship in Congress be explained if strict homophily is impossible?
• RQ4 How does the distribution of gender in group photos change with group size, and why is this overlooked in graph-based models?
• RQ5 What relaxed forms of homophily are necessary and sufficient to explain real-world group formation patterns?

Key findings

• Simultaneous majority homophily for both men and women in group interactions is combinatorially impossible, as no group can have both genders as majority.
• For U.S. members of Congress to exhibit high homophily in co-sponsoring bills by party, a substantial number of individuals from each party must be willing to co-sponsor even when their party is in the minority.
• The framework explains that gender distribution in group photos becomes more balanced as group size increases, a pattern obscured by pairwise graph models.
• Strict higher-order homophily cannot be realized in hypergraphs when multiple attributes are involved, due to inherent structural constraints.
• Relaxed homophily—where individuals participate in groups even when their attribute is not in the majority—is a necessary condition for observed group-level homophily.
• The model reveals that apparent homophily in group interactions often results from minority participation rather than exclusive preference for same-attribute groups.