{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:VQ6DQVS6P744425FLHHHKPXMLP","short_pith_number":"pith:VQ6DQVS6","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"},"canonical_sha256":"ac3c38565e7ff9ce6ba559ce753eec5bedbe0488f15580578df223135e8923c5","source":{"kind":"arxiv","id":"1712.04341","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1712.04341","created_at":"2026-05-18T00:17:32Z"},{"alias_kind":"arxiv_version","alias_value":"1712.04341v2","created_at":"2026-05-18T00:17:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.04341","created_at":"2026-05-18T00:17:32Z"},{"alias_kind":"pith_short_12","alias_value":"VQ6DQVS6P744","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_16","alias_value":"VQ6DQVS6P744425F","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_8","alias_value":"VQ6DQVS6","created_at":"2026-05-18T12:31:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:VQ6DQVS6P744425FLHHHKPXMLP","target":"record","payload":{"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"},"canonical_sha256":"ac3c38565e7ff9ce6ba559ce753eec5bedbe0488f15580578df223135e8923c5","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"},"source_kind":"arxiv","source_id":"1712.04341","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:17:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7xskmoDfC/hBTPySAcs9Z1mnjlhglqPdn8sJZVHcl7+t2UvIsvccw0nhBEvkSqiYen+KTQAIEUEbX8fLcsZPDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T19:52:47.429370Z"},"content_sha256":"ba51f55ccfb514e8ec8bbcef07ef5897d7f95e52e67b4bd1eed3cb75de1841b4","schema_version":"1.0","event_id":"sha256:ba51f55ccfb514e8ec8bbcef07ef5897d7f95e52e67b4bd1eed3cb75de1841b4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:VQ6DQVS6P744425FLHHHKPXMLP","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:17:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7U3fdGMr8KWLIoFg9CLi8W2PZrgwW0B0zDz8R7vDJeEptbFdo3uDubYcrrpWPLm0IhynBiCw5R5+sfYjVzSBAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T19:52:47.429967Z"},"content_sha256":"ece2bf56e311f182c8ea90fd803caad942601aa90bf7353fb72f0c07ee597868","schema_version":"1.0","event_id":"sha256:ece2bf56e311f182c8ea90fd803caad942601aa90bf7353fb72f0c07ee597868"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VQ6DQVS6P744425FLHHHKPXMLP/bundle.json","state_url":"https://pith.science/pith/VQ6DQVS6P744425FLHHHKPXMLP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VQ6DQVS6P744425FLHHHKPXMLP/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-06T19:52:47Z","links":{"resolver":"https://pith.science/pith/VQ6DQVS6P744425FLHHHKPXMLP","bundle":"https://pith.science/pith/VQ6DQVS6P744425FLHHHKPXMLP/bundle.json","state":"https://pith.science/pith/VQ6DQVS6P744425FLHHHKPXMLP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VQ6DQVS6P744425FLHHHKPXMLP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:VQ6DQVS6P744425FLHHHKPXMLP","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"8b2452a32490f5fb2b89a297fa54d3b37ec1655b3cd6edbde500e3369fe20396","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-12-10T04:02:09Z","title_canon_sha256":"45fe6a3a01992f1f2a8d9dffb45d8809e3422fd65371a49b59c9c622f8de3ed9"},"schema_version":"1.0","source":{"id":"1712.04341","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1712.04341","created_at":"2026-05-18T00:17:32Z"},{"alias_kind":"arxiv_version","alias_value":"1712.04341v2","created_at":"2026-05-18T00:17:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.04341","created_at":"2026-05-18T00:17:32Z"},{"alias_kind":"pith_short_12","alias_value":"VQ6DQVS6P744","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_16","alias_value":"VQ6DQVS6P744425F","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_8","alias_value":"VQ6DQVS6","created_at":"2026-05-18T12:31:49Z"}],"graph_snapshots":[{"event_id":"sha256:ece2bf56e311f182c8ea90fd803caad942601aa90bf7353fb72f0c07ee597868","target":"graph","created_at":"2026-05-18T00:17:32Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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.","authors_text":"Feng Wei","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-12-10T04:02:09Z","title":"Investigate Invertibility of Sparse Symmetric Matrix"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.04341","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:ba51f55ccfb514e8ec8bbcef07ef5897d7f95e52e67b4bd1eed3cb75de1841b4","target":"record","created_at":"2026-05-18T00:17:32Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"8b2452a32490f5fb2b89a297fa54d3b37ec1655b3cd6edbde500e3369fe20396","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-12-10T04:02:09Z","title_canon_sha256":"45fe6a3a01992f1f2a8d9dffb45d8809e3422fd65371a49b59c9c622f8de3ed9"},"schema_version":"1.0","source":{"id":"1712.04341","kind":"arxiv","version":2}},"canonical_sha256":"ac3c38565e7ff9ce6ba559ce753eec5bedbe0488f15580578df223135e8923c5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ac3c38565e7ff9ce6ba559ce753eec5bedbe0488f15580578df223135e8923c5","first_computed_at":"2026-05-18T00:17:32.992572Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:17:32.992572Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"51gaELd04hl06Wx9mSLJSuePTT3zwrRmlJ9DJijSxGG9gwssKhEefJCrvxoUEuYhTaIcDBAomA5KBJZXUe8zDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:17:32.993252Z","signed_message":"canonical_sha256_bytes"},"source_id":"1712.04341","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ba51f55ccfb514e8ec8bbcef07ef5897d7f95e52e67b4bd1eed3cb75de1841b4","sha256:ece2bf56e311f182c8ea90fd803caad942601aa90bf7353fb72f0c07ee597868"],"state_sha256":"cf75b21579f219c43df5d5e98fbc9b713012873090a4eb67a58e964698bebd8e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ey214lppkzbta884MJ/IpEmcBKTD1ierTAlqe49iFBhwjRUxSsYilzX+8ntlXxqqmrFe1V6qJNd/jQma+W25BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T19:52:47.433258Z","bundle_sha256":"de226a25ab639c2dfda0f3aa55fd8eb1aeb889dd5671e247e31ccca672bdf013"}}