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Starting from the empty graph on a fixed vertex set $[n]$, edges are added one by one with probabilities proportional to $(d_u+\\alpha)(d_v+\\alpha)$, where $d_u$ and $d_v$ are the current degrees of $u$ and $v$, and $\\alpha>0$. Let $L_1$ denote the size of the largest component, and set $m_c:=\\frac{\\alpha n}{2(\\alpha+1)}.$ We prove that if $m=m_c(1-\\varepsilon), \\varepsilon=\\varepsilon(n)\\to0, \\varepsilon^3 n\\to\\infty,$ then \\[ L_1=(1+o_p(1))\\frac{2(\\alpha+2)}{\\alpha+1}\\varepsilon^{-2}\\"},"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":"2607.00731","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-07-01T10:17:10Z","cross_cats_sorted":[],"title_canon_sha256":"d90539d126d8c600c260341ce4c0ff85543a1fcbdf3bb887b2bf9296b53d53ae","abstract_canon_sha256":"f71b4e9d2063675628617f64cfd2c815077b55ce2eb297139f46d97828010b7a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-02T01:17:52.853864Z","signature_b64":"2hLcQ3aWAH0ZK+/PsuebEE1tZbX5xEzpEI50z2eC+1k9/lpSM4PAjg9ur9K7bm3KiFOJYyxa9A4OE/yYIi5UAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9d8765518e68ae4ee72e48d4bf94383c452c65bb482063f1ce1c4ebf0abd07c5","last_reissued_at":"2026-07-02T01:17:52.853433Z","signature_status":"signed_v1","first_computed_at":"2026-07-02T01:17:52.853433Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sharp Asymptotics for the Largest Component in the Subcritical Regime of Preferential Attachment Without Vertex Growth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Yiming Chen","submitted_at":"2026-07-01T10:17:10Z","abstract_excerpt":"We study the size of the largest component in Pittel's preferential attachment process without vertex growth. Starting from the empty graph on a fixed vertex set $[n]$, edges are added one by one with probabilities proportional to $(d_u+\\alpha)(d_v+\\alpha)$, where $d_u$ and $d_v$ are the current degrees of $u$ and $v$, and $\\alpha>0$. Let $L_1$ denote the size of the largest component, and set $m_c:=\\frac{\\alpha n}{2(\\alpha+1)}.$ We prove that if $m=m_c(1-\\varepsilon), \\varepsilon=\\varepsilon(n)\\to0, \\varepsilon^3 n\\to\\infty,$ then \\[ L_1=(1+o_p(1))\\frac{2(\\alpha+2)}{\\alpha+1}\\varepsilon^{-2}\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.00731","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.00731/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2607.00731","created_at":"2026-07-02T01:17:52.853508+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.00731v1","created_at":"2026-07-02T01:17:52.853508+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.00731","created_at":"2026-07-02T01:17:52.853508+00:00"},{"alias_kind":"pith_short_12","alias_value":"TWDWKUMONCXE","created_at":"2026-07-02T01:17:52.853508+00:00"},{"alias_kind":"pith_short_16","alias_value":"TWDWKUMONCXE5ZZO","created_at":"2026-07-02T01:17:52.853508+00:00"},{"alias_kind":"pith_short_8","alias_value":"TWDWKUMO","created_at":"2026-07-02T01:17:52.853508+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TWDWKUMONCXE5ZZOJDKL7FBYHR","json":"https://pith.science/pith/TWDWKUMONCXE5ZZOJDKL7FBYHR.json","graph_json":"https://pith.science/api/pith-number/TWDWKUMONCXE5ZZOJDKL7FBYHR/graph.json","events_json":"https://pith.science/api/pith-number/TWDWKUMONCXE5ZZOJDKL7FBYHR/events.json","paper":"https://pith.science/paper/TWDWKUMO"},"agent_actions":{"view_html":"https://pith.science/pith/TWDWKUMONCXE5ZZOJDKL7FBYHR","download_json":"https://pith.science/pith/TWDWKUMONCXE5ZZOJDKL7FBYHR.json","view_paper":"https://pith.science/paper/TWDWKUMO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.00731&json=true","fetch_graph":"https://pith.science/api/pith-number/TWDWKUMONCXE5ZZOJDKL7FBYHR/graph.json","fetch_events":"https://pith.science/api/pith-number/TWDWKUMONCXE5ZZOJDKL7FBYHR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TWDWKUMONCXE5ZZOJDKL7FBYHR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TWDWKUMONCXE5ZZOJDKL7FBYHR/action/storage_attestation","attest_author":"https://pith.science/pith/TWDWKUMONCXE5ZZOJDKL7FBYHR/action/author_attestation","sign_citation":"https://pith.science/pith/TWDWKUMONCXE5ZZOJDKL7FBYHR/action/citation_signature","submit_replication":"https://pith.science/pith/TWDWKUMONCXE5ZZOJDKL7FBYHR/action/replication_record"}},"created_at":"2026-07-02T01:17:52.853508+00:00","updated_at":"2026-07-02T01:17:52.853508+00:00"}