{ "schemaVersion": "1.6", "slug": "multicategory-group-homophily", "title": "Multicategory group homophily: constructive feasibility and sharp limits", "shortTitle": "With three or more categories every category can look homophilous in groups at once; an exact ceiling says how far, and which budget blocks it", "url": "https://evidencepress.org/releases/multicategory-group-homophily/", "oneLine": "With three or more categories the two-category impossibility for simultaneous group homophily disappears; an exact test and a closed-form ceiling with a named binding budget say what is possible, and a school example shows the same 4,600 groups passing for 10/10 classrooms but 0/5 school years.", "abstract": "Group homophily measures compare how often a category's members find themselves in groups with one, two or more others of their own category against a baseline. Veldt, Benson and Kleinberg (2023) proved that with two categories natural forms of simultaneous group homophily are combinatorially impossible, and asked how these limits change with more categories. This candidate answers the question for a single group size. First, with three or more categories and any full-support baseline whose category participation matches the observed participation, arbitrarily small changes that keep every category's (and every individual's) participation fixed make every category's affinity ratios strictly increase with its number of members; connected, simple, regular finite hypergraphs realise this at every rational composition (Theorem 1, Propositions 2–3). This proves an earlier informal remark of Sarker, Northrup and Jadbabaie. Second, an exact finite list of inequalities describes which strict-majority probabilities can coexist at fixed participation (Theorem 2). Third, a closed-form, attained ceiling bounds how far all majority affinity ratios can rise together, as the minimum of a group-count budget and each category's own and complementary participation budgets (Theorem 3, Corollary 1). Consequences: a short certificate for the two-category odd-size impossibility (Corollary 2) and the fact that at matched participation the baseline must allow groups without a strict majority (Corollary 3). In an exploratory reanalysis of 4,600 distinct primary-school proximity triples, between 8 and 10 of 10 pupil classrooms meet strict-majority homophily across four specifications, against 0 or 1 of 5 merged school years; the years fail only at the two-of-three count because merging roughly doubles its chance rate, and a binary pupil/teacher coding gives a ceiling of 0.641. These results concern measurement feasibility; they do not identify preferences or causal mechanisms. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.", "datePublished": "2026-11-03", "dateModified": "2026-11-03", "version": "0.1.0-candidate", "doi": "10.5281/zenodo.23916043", "doiUrl": "https://doi.org/10.5281/zenodo.23916043", "conceptDoi": "10.5281/zenodo.23916042", "pdfUrl": "https://github.com/ipitchford/multicategory-group-homophily/releases/download/v0.1.0-candidate/multicategory-group-homophily-0.1.0-candidate.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23916043", "repoUrl": "https://github.com/ipitchford/multicategory-group-homophily", "releaseUrl": "https://github.com/ipitchford/multicategory-group-homophily/releases/tag/v0.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/multicategory-group-homophily/index.md", "bibtexUrl": "https://evidencepress.org/releases/multicategory-group-homophily/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/multicategory-group-homophily.mp3", "imageUrl": "https://evidencepress.org/assets/og/multicategory-group-homophily.png?v=25ebec385d", "coverArtUrl": "https://evidencepress.org/assets/art/multicategory-group-homophily.svg?v=3168fe02fa", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/multicategory-group-homophily.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/multicategory-group-homophily.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed AI-assisted candidate in mathematical sociology and network science. The theorems have written proofs; the scripts are finite exact and linear-programming consistency checks; the school example is exploratory and descriptive. Internal AI-assisted review in five roles, a separate-model adversarial review and proof audit, a proof audit, a clean replay, one revision round and a re-review. External specialist review, independent reproduction and formal verification have not been established." }, "assurance": [ { "dimension": "availability", "label": "Availability and archiving", "question": "Is the evidence package publicly retrievable from an archive under a persistent identifier?", "state": "passed", "evidenceUrl": "https://zenodo.org/records/23916043", "note": "The versioned research package and paper are publicly archived on GitHub and Zenodo. Provider asset parity was checked against frozen local files by full download; availability does not establish correctness or peer review." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "reproduce.py (five check scripts) was rerun with Python -I -B and -I -B -O from fresh copies before freezing and again from the frozen archive; every regenerated receipt and figure matched the package (the figure PDF differs only in its creation timestamp). GitHub CI repeats the run on Linux. The data-download route was not rerun." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No unaffiliated rerun has been established." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "partial", "note": "Theorems 2–3 are cross-checked by an independent full-composition linear programme, and the school tables by a second parser written without importing the analysis code. This is producer-side diversity, not unaffiliated reimplementation." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "No proof-assistant formalisation." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "AI-assisted internal and adversarial review is not external specialist review." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "No external editorial peer review." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "passed", "note": "Python 3.12 with numpy 2.3.5, scipy 1.17.0 and matplotlib 3.10.8 (tested on macOS and Linux CI); exact rational arithmetic for the constructions; not a locked container. Raw third-party archives are not redistributed; the download route checks their hashes." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted mathematical and social-science research (original research bundle)", "AI-assisted internal review in five roles (Claude Code subagents); adversarial review and proof audit by a separate model (OpenAI Codex); proof audit and re-review", "Claude Code: revision, publication packaging and release" ], "humanRole": "Publication authority only; no human research contribution inferred.", "reviewStatus": "Internal AI-assisted and adversarial review, not external peer review." }, "attributionBoundary": null, "problem": null, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Prepared from the final paper (revision of 3 November 2026 with DOI insertion, 22 pages), its verification scripts and the frozen release archive. The versioned GitHub and Zenodo research archives were checked against the frozen release assets on 3 November 2026.", "grandfatheredAtSchemaVersion": null, "keywords": [ "group homophily", "higher-order homophily", "hypergraphs", "affinity ratio", "impossibility theorem", "network flows", "social networks", "school contact data" ], "keyResults": [ "Theorem 1 (proof): for q ≥ 3 categories and any full-support baseline with matched participation, arbitrarily small participation-preserving (even individual-participation-preserving) perturbations make every category's affinity ratios strictly increase in its count; Proposition 3 realises this in connected, simple, regular finite hypergraphs at every rational composition.", "Theorem 2 (proof): an exact finite list of inequalities decides which strict-majority probabilities can coexist at fixed participation, without enumerating all compositions.", "Theorem 3 and Corollary 1 (proof): the largest common multiplier of all majority affinity ratios is λ_max = min{1/A₀, min_c d_c/T_c⁰, min_c (k − d_c)/(T⁰ − T_c⁰ + S_c⁰)}, attained; simultaneous strict-majority homophily is possible iff λ_max > 1. For q equal categories in a three-person multinomial baseline, λ_max = q²/(3q − 2).", "Corollaries 2 and 3 (proof): with two categories and odd group size, λ_max ≤ 1 for every activity profile (a short certificate for Veldt et al.'s Theorem 5); at matched participation λ_max ≤ 1/A₀, so the baseline must allow groups without a strict majority.", "Section 7 (exploratory, descriptive): in 4,600 school proximity triples, 10/10 classrooms but 0/5 merged school years meet strict-majority homophily under the primary specification (8–10/10 vs 0–1/5 across four); ceilings 3.92 (classrooms), 2.07 (years), 0.641 for pupils versus teachers." ], "reviews": [], "evidencePackage": "Written proofs of Theorems 1–3, Propositions 2–3 and Corollaries 1–3 (22-page paper); exact-rational checks of Theorem 1 (42 cases), an explicit simple regular cyclic-lift construction, exact checks of Corollaries 2–3 and the free-participation remark (999, 466 and 360 random cases), linear-programming cross-checks of Theorems 2–3 (560 and 864 problems), an independent second parser of the school data, and aggregate-only recomputation of the school ratios, deletion counterfactual, fragility counts and pupil/teacher ceiling; all checks fail closed under optimised Python, a corrupted copy is detected, and a clean replay from fresh copies reproduced every receipt.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Veldt, Benson and Kleinberg (2023). Combinatorial characterizations and impossibilities for higher-order homophily. Science Advances. Proved the two-category impossibilities and asked how they change with more categories; Proposition 1 and the odd-size result recovered in Corollary 2 are theirs.", "url": "https://doi.org/10.1126/sciadv.abq3200", "doi": "10.1126/sciadv.abq3200" }, { "citation": "Sarker, Northrup and Jadbabaie (2024). Higher-order homophily on simplicial complexes. Proceedings of the National Academy of Sciences. Studies the same school data and shows group-level class affinities can largely be inherited from pairwise contacts; their earlier preprint remarked informally that three or more labels help.", "url": "https://doi.org/10.1073/pnas.2315931121", "doi": "10.1073/pnas.2315931121" }, { "citation": "Rizi, Michielan, Stegehuis and Kivelä (2025). Homophily within and across groups. Nature Communications. Multicategory and category-coarsening extensions of a maximum-entropy model; does not characterise all count-specific ratios jointly.", "url": "https://doi.org/10.1038/s41467-025-66415-2", "doi": "10.1038/s41467-025-66415-2" }, { "citation": "Kumano, Nishimura and Kimura (2026). Higher-order homophily analysis for hypergraphs with time-varying node categories. Applied Network Science. Temporal category and category-pair measures.", "url": "https://doi.org/10.1007/s41109-026-00801-7", "doi": "10.1007/s41109-026-00801-7" }, { "citation": "Wang, Zhou, Li, Zhao and Qi (2026). Measuring the higher-order homophily in a general social hypergraph. Chaos. Extends measurement to groups of different sizes.", "url": "https://doi.org/10.1063/5.0345566", "doi": "10.1063/5.0345566" }, { "citation": "Chun, Yoon, Choe, Lee and Shin (2026). From binary to general attributes: attributed hypergraph generation with realistic interplay between structure and attributes. Knowledge and Information Systems. Generative models with categorical affinity profiles.", "url": "https://doi.org/10.1007/s10115-026-02850-x", "doi": "10.1007/s10115-026-02850-x" }, { "citation": "McPherson, Smith-Lovin and Cook (2001). Birds of a Feather: Homophily in Social Networks. Annual Review of Sociology. The standard account of homophily.", "url": "https://doi.org/10.1146/annurev.soc.27.1.415", "doi": "10.1146/annurev.soc.27.1.415" }, { "citation": "Feld (1981). The Focused Organization of Social Ties. American Journal of Sociology. Shared settings create opportunities for contact, so homophily reflects opportunity as well as preference.", "url": "https://doi.org/10.1086/227352", "doi": "10.1086/227352" }, { "citation": "Gale (1957). A theorem on flows in networks. Pacific Journal of Mathematics. The supply-demand feasibility theorem behind Theorems 2 and 3.", "url": "https://doi.org/10.2140/pjm.1957.7.1073", "doi": "10.2140/pjm.1957.7.1073" }, { "citation": "Ford and Fulkerson (1956). Maximal Flow Through a Network. Canadian Journal of Mathematics. Maximum-flow/minimum-cut duality used in the proofs.", "url": "https://doi.org/10.4153/CJM-1956-045-5", "doi": "10.4153/CJM-1956-045-5" }, { "citation": "Wang and Wang (2016). Joint Mixability. Mathematics of Operations Research. Constant-sum couplings related to the composition geometry.", "url": "https://doi.org/10.1287/moor.2015.0755", "doi": "10.1287/moor.2015.0755" }, { "citation": "Stehlé et al. (2011). High-Resolution Measurements of Face-to-Face Contact Patterns in a Primary School. PLoS ONE. The original sensor study of the school data used in Section 7.", "url": "https://doi.org/10.1371/journal.pone.0023176", "doi": "10.1371/journal.pone.0023176" }, { "citation": "Gemmetto, Barrat and Cattuto (2014). Mitigation of infectious disease at school: targeted class closure vs school closure. BMC Infectious Diseases. Documents the classroom and school-year structure of the same data.", "url": "https://doi.org/10.1186/s12879-014-0695-9", "doi": "10.1186/s12879-014-0695-9" }, { "citation": "Benson, Abebe, Schaub, Jadbabaie and Kleinberg (2018). Simplicial closure and higher-order link prediction. Proceedings of the National Academy of Sciences. Source of the temporal group (clique) release.", "url": "https://doi.org/10.1073/pnas.1800683115", "doi": "10.1073/pnas.1800683115" }, { "citation": "Chodrow, Veldt and Benson (2021). Generative hypergraph clustering: From blockmodels to modularity. Science Advances. Source of the labelled hypergraph release analysed in Section 7.", "url": "https://doi.org/10.1126/sciadv.abh1303", "doi": "10.1126/sciadv.abh1303" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:multicategory-group-homophily", "attemptIds": [ "ep-attempt:multicategory-group-homophily-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "communication" ], "lineageId": null, "accelerationPrimitives": [ "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "bound", "description": "Before interpreting a failed group-homophily test, compute from category participation and the chosen baseline whether all categories could have passed a strict-majority test at all (λ_max > 1), and which budget blocks it.", "scope": "One group size; categories forming a partition; relaxed composition model in which all integer compositions are available. 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