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Bernoulli random variables with $\\mathbb{P} \\left(\\xi_{ij}=1 \\right) =p \\ge n^{-c}$, $\\xi_{ij}, i\\ge j$ are i.i.d. random variables with mean 0, variance 1 and finite forth moment $M_4$, and $c$ is constant depending on $M_4$. More precisely, $$ s_{\\rm min} (A) > \\varepsilon \\sqrt{\\frac{p}{n}}. $$ with high probability."},"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":"1712.04341","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-12-10T04:02:09Z","cross_cats_sorted":[],"title_canon_sha256":"45fe6a3a01992f1f2a8d9dffb45d8809e3422fd65371a49b59c9c622f8de3ed9","abstract_canon_sha256":"8b2452a32490f5fb2b89a297fa54d3b37ec1655b3cd6edbde500e3369fe20396"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:17:32.993252Z","signature_b64":"51gaELd04hl06Wx9mSLJSuePTT3zwrRmlJ9DJijSxGG9gwssKhEefJCrvxoUEuYhTaIcDBAomA5KBJZXUe8zDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ac3c38565e7ff9ce6ba559ce753eec5bedbe0488f15580578df223135e8923c5","last_reissued_at":"2026-05-18T00:17:32.992572Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:17:32.992572Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Investigate Invertibility of Sparse Symmetric Matrix","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Feng Wei","submitted_at":"2017-12-10T04:02:09Z","abstract_excerpt":"In this paper, we investigate the invertibility of sparse symmetric matrices. We show that for an $n\\times n$ sparse symmetric random matrix $A$ with $A_{ij} = \\delta_{ij} \\xi_{ij}$ is invertible with high probability. Here, $\\delta_{ij}$s, $i\\ge j$ are i.i.d. Bernoulli random variables with $\\mathbb{P} \\left(\\xi_{ij}=1 \\right) =p \\ge n^{-c}$, $\\xi_{ij}, i\\ge j$ are i.i.d. random variables with mean 0, variance 1 and finite forth moment $M_4$, and $c$ is constant depending on $M_4$. More precisely, $$ s_{\\rm min} (A) > \\varepsilon \\sqrt{\\frac{p}{n}}. $$ with high probability."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.04341","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"1712.04341","created_at":"2026-05-18T00:17:32.992697+00:00"},{"alias_kind":"arxiv_version","alias_value":"1712.04341v2","created_at":"2026-05-18T00:17:32.992697+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.04341","created_at":"2026-05-18T00:17:32.992697+00:00"},{"alias_kind":"pith_short_12","alias_value":"VQ6DQVS6P744","created_at":"2026-05-18T12:31:49.984773+00:00"},{"alias_kind":"pith_short_16","alias_value":"VQ6DQVS6P744425F","created_at":"2026-05-18T12:31:49.984773+00:00"},{"alias_kind":"pith_short_8","alias_value":"VQ6DQVS6","created_at":"2026-05-18T12:31:49.984773+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/VQ6DQVS6P744425FLHHHKPXMLP","json":"https://pith.science/pith/VQ6DQVS6P744425FLHHHKPXMLP.json","graph_json":"https://pith.science/api/pith-number/VQ6DQVS6P744425FLHHHKPXMLP/graph.json","events_json":"https://pith.science/api/pith-number/VQ6DQVS6P744425FLHHHKPXMLP/events.json","paper":"https://pith.science/paper/VQ6DQVS6"},"agent_actions":{"view_html":"https://pith.science/pith/VQ6DQVS6P744425FLHHHKPXMLP","download_json":"https://pith.science/pith/VQ6DQVS6P744425FLHHHKPXMLP.json","view_paper":"https://pith.science/paper/VQ6DQVS6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1712.04341&json=true","fetch_graph":"https://pith.science/api/pith-number/VQ6DQVS6P744425FLHHHKPXMLP/graph.json","fetch_events":"https://pith.science/api/pith-number/VQ6DQVS6P744425FLHHHKPXMLP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VQ6DQVS6P744425FLHHHKPXMLP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VQ6DQVS6P744425FLHHHKPXMLP/action/storage_attestation","attest_author":"https://pith.science/pith/VQ6DQVS6P744425FLHHHKPXMLP/action/author_attestation","sign_citation":"https://pith.science/pith/VQ6DQVS6P744425FLHHHKPXMLP/action/citation_signature","submit_replication":"https://pith.science/pith/VQ6DQVS6P744425FLHHHKPXMLP/action/replication_record"}},"created_at":"2026-05-18T00:17:32.992697+00:00","updated_at":"2026-05-18T00:17:32.992697+00:00"}