{"paper":{"title":"The interplay between network transitivity and community structure","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"Clustering coefficient limits in geometric block models undergo phase transitions between weak and strong community regimes and prove non-monotonic in strength.","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Chengay S Wangchuk, Irin Rahman, Mingao Yuan, Minglian Lin","submitted_at":"2026-04-21T04:19:17Z","abstract_excerpt":"Recent empirical observations suggest that network transitivity is highly correlated with community structure in many real-world networks. In this paper, we theoretically investigate this relationship by deriving the limits of the global and average clustering coefficients for the geometric block model (GBM). Both limits exhibit a phase transition; specifically, the functional forms of the limit functions differ between the weak and strong community structure strength regimes. For a GBM with balanced communities, the limits of the global and average clustering coefficients are identical, where"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Both limits exhibit a phase transition; specifically, the functional forms of the limit functions differ between the weak and strong community structure strength regimes. For a GBM with balanced communities, the limits of the global and average clustering coefficients are identical, whereas these limits differ for unbalanced communities. In general, the clustering coefficients do not exhibit a monotonic relationship with community structure strength.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The geometric block model with its edge probability rules based on community membership and geometry is the correct generative process, and the asymptotic analysis applies under the stated parameter regimes for weak versus strong community structure.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Limits of clustering coefficients in the geometric block model show phase transitions between weak and strong community regimes, with non-monotonic behavior and differences for balanced versus unbalanced communities.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Clustering coefficient limits in geometric block models undergo phase transitions between weak and strong community regimes and prove non-monotonic in strength.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"b0c61221931e3a57455b23aa1fb2cd9f04365a270a938c498c9c716a814003dc"},"source":{"id":"2604.19067","kind":"arxiv","version":2},"verdict":{"id":"78c1bb4e-8091-4203-8eb3-7d5279fc0400","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T01:39:59.773080Z","strongest_claim":"Both limits exhibit a phase transition; specifically, the functional forms of the limit functions differ between the weak and strong community structure strength regimes. For a GBM with balanced communities, the limits of the global and average clustering coefficients are identical, whereas these limits differ for unbalanced communities. In general, the clustering coefficients do not exhibit a monotonic relationship with community structure strength.","one_line_summary":"Limits of clustering coefficients in the geometric block model show phase transitions between weak and strong community regimes, with non-monotonic behavior and differences for balanced versus unbalanced communities.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The geometric block model with its edge probability rules based on community membership and geometry is the correct generative process, and the asymptotic analysis applies under the stated parameter regimes for weak versus strong community structure.","pith_extraction_headline":"Clustering coefficient limits in geometric block models undergo phase transitions between weak and strong community regimes and prove non-monotonic in strength."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.19067/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-20T03:23:23.760376Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"406f745ef5f851b35fc3d85c8d43bff0e897833eaa42feabf1731431e4bdbbc2"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}