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Denoting these graphs by $HG_n(R_n ; \\alpha, \\zeta)$, we study asymptotic counts of copies of a fixed tree $\\Ga"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1802.06105","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-02-16T20:11:03Z","cross_cats_sorted":["math.CO","math.GT"],"title_canon_sha256":"9615ec51ede033a162cd40fc50c19182401668ab588932f163abe9a703316eca","abstract_canon_sha256":"2240ae951e0b9665d63e57077bb60b0e9cda4071d0568c6ce3b07cf7efb4797c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:23:04.477502Z","signature_b64":"8T9RCK11Y/ZWqnr53RfYy2y5oEtb80u9EYgX33eY+gj+w7nYisx8aGYjWZd82QcXhRLoD5XEbk7xBYfqHlNNCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5e5e925f841b76abdda86e8e5ccb1d6df20f653a957f3c9b59b5257c90f024a7","last_reissued_at":"2026-05-18T00:23:04.476657Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:23:04.476657Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sub-tree counts on hyperbolic random geometric graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.GT"],"primary_cat":"math.PR","authors_text":"D. 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