{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:CIY45MMTS44BWXRFLYEEWA47RO","short_pith_number":"pith:CIY45MMT","canonical_record":{"source":{"id":"cs/0609116","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"cs.DS","submitted_at":"2006-09-20T14:17:34Z","cross_cats_sorted":["cond-mat.stat-mech","cs.NI"],"title_canon_sha256":"6fd0ec7a14f9de780cfe563a5de08019626d8c5c6e9903799604003b0b703a21","abstract_canon_sha256":"51d8b1c0564c39f36ef6201cc877e06c0e5234bd7900e8716ee623986657353c"},"schema_version":"1.0"},"canonical_sha256":"1231ceb19397381b5e255e084b039f8bb4223ec844681c39180989d3b076b643","source":{"kind":"arxiv","id":"cs/0609116","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"cs/0609116","created_at":"2026-07-04T14:56:27Z"},{"alias_kind":"arxiv_version","alias_value":"cs/0609116v1","created_at":"2026-07-04T14:56:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.cs/0609116","created_at":"2026-07-04T14:56:27Z"},{"alias_kind":"pith_short_12","alias_value":"CIY45MMTS44B","created_at":"2026-07-04T14:56:27Z"},{"alias_kind":"pith_short_16","alias_value":"CIY45MMTS44BWXRF","created_at":"2026-07-04T14:56:27Z"},{"alias_kind":"pith_short_8","alias_value":"CIY45MMT","created_at":"2026-07-04T14:56:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:CIY45MMTS44BWXRFLYEEWA47RO","target":"record","payload":{"canonical_record":{"source":{"id":"cs/0609116","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"cs.DS","submitted_at":"2006-09-20T14:17:34Z","cross_cats_sorted":["cond-mat.stat-mech","cs.NI"],"title_canon_sha256":"6fd0ec7a14f9de780cfe563a5de08019626d8c5c6e9903799604003b0b703a21","abstract_canon_sha256":"51d8b1c0564c39f36ef6201cc877e06c0e5234bd7900e8716ee623986657353c"},"schema_version":"1.0"},"canonical_sha256":"1231ceb19397381b5e255e084b039f8bb4223ec844681c39180989d3b076b643","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:56:27.647444Z","signature_b64":"wT8u58TZUmJMAiPF7dauGssY4vHNjAzSv85M4KjZopXOcM8rPOvmPD2u52zOQlx1fDfQxcvxGWorVMRq0YtRAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1231ceb19397381b5e255e084b039f8bb4223ec844681c39180989d3b076b643","last_reissued_at":"2026-07-04T14:56:27.646930Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:56:27.646930Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"cs/0609116","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T14:56:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NZmJDWdFoNgqopWdwh6fCvwgFF8mnHFaNjE27PomVXyiUDeHZzBskAaQpC3VqaM/wQ/2x6Wb0L27LZebQ/0XDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T03:42:17.989762Z"},"content_sha256":"61b05c10a566bfd501c2cae87fc59ba5ede63ee5fcf3c8e9bca4ed6299658265","schema_version":"1.0","event_id":"sha256:61b05c10a566bfd501c2cae87fc59ba5ede63ee5fcf3c8e9bca4ed6299658265"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:CIY45MMTS44BWXRFLYEEWA47RO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Theory and Practice of Triangle Problems in Very Large (Sparse (Power-Law)) Graphs","license":"","headline":"","cross_cats":["cond-mat.stat-mech","cs.NI"],"primary_cat":"cs.DS","authors_text":"Matthieu Latapy (LIAFA - CNRS, Universite Paris 7)","submitted_at":"2006-09-20T14:17:34Z","abstract_excerpt":"Finding, counting and/or listing triangles (three vertices with three edges) in large graphs are natural fundamental problems, which received recently much attention because of their importance in complex network analysis. We provide here a detailed state of the art on these problems, in a unified way. We note that, until now, authors paid surprisingly little attention to space complexity, despite its both fundamental and practical interest. We give the space complexities of known algorithms and discuss their implications. Then we propose improvements of a known algorithm, as well as a new alg"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"cs/0609116","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/cs/0609116/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T14:56:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Shx+sG2staswISLXACrvpz1cYtKUx+2csMVLNjaQQ2CITJQABiJMrMsv6M19M0cqLTslGnNkTGpGuWU+TihGAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T03:42:17.990250Z"},"content_sha256":"50bdceacfb466ae9c0bfea4a97f3306f3a48dbf69f6fdd60b02c945da37c4270","schema_version":"1.0","event_id":"sha256:50bdceacfb466ae9c0bfea4a97f3306f3a48dbf69f6fdd60b02c945da37c4270"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CIY45MMTS44BWXRFLYEEWA47RO/bundle.json","state_url":"https://pith.science/pith/CIY45MMTS44BWXRFLYEEWA47RO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CIY45MMTS44BWXRFLYEEWA47RO/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":"2026-08-08T03:42:17Z","links":{"resolver":"https://pith.science/pith/CIY45MMTS44BWXRFLYEEWA47RO","bundle":"https://pith.science/pith/CIY45MMTS44BWXRFLYEEWA47RO/bundle.json","state":"https://pith.science/pith/CIY45MMTS44BWXRFLYEEWA47RO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CIY45MMTS44BWXRFLYEEWA47RO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:CIY45MMTS44BWXRFLYEEWA47RO","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":"51d8b1c0564c39f36ef6201cc877e06c0e5234bd7900e8716ee623986657353c","cross_cats_sorted":["cond-mat.stat-mech","cs.NI"],"license":"","primary_cat":"cs.DS","submitted_at":"2006-09-20T14:17:34Z","title_canon_sha256":"6fd0ec7a14f9de780cfe563a5de08019626d8c5c6e9903799604003b0b703a21"},"schema_version":"1.0","source":{"id":"cs/0609116","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"cs/0609116","created_at":"2026-07-04T14:56:27Z"},{"alias_kind":"arxiv_version","alias_value":"cs/0609116v1","created_at":"2026-07-04T14:56:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.cs/0609116","created_at":"2026-07-04T14:56:27Z"},{"alias_kind":"pith_short_12","alias_value":"CIY45MMTS44B","created_at":"2026-07-04T14:56:27Z"},{"alias_kind":"pith_short_16","alias_value":"CIY45MMTS44BWXRF","created_at":"2026-07-04T14:56:27Z"},{"alias_kind":"pith_short_8","alias_value":"CIY45MMT","created_at":"2026-07-04T14:56:27Z"}],"graph_snapshots":[{"event_id":"sha256:50bdceacfb466ae9c0bfea4a97f3306f3a48dbf69f6fdd60b02c945da37c4270","target":"graph","created_at":"2026-07-04T14:56:27Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/cs/0609116/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Finding, counting and/or listing triangles (three vertices with three edges) in large graphs are natural fundamental problems, which received recently much attention because of their importance in complex network analysis. We provide here a detailed state of the art on these problems, in a unified way. We note that, until now, authors paid surprisingly little attention to space complexity, despite its both fundamental and practical interest. We give the space complexities of known algorithms and discuss their implications. Then we propose improvements of a known algorithm, as well as a new alg","authors_text":"Matthieu Latapy (LIAFA - CNRS, Universite Paris 7)","cross_cats":["cond-mat.stat-mech","cs.NI"],"headline":"","license":"","primary_cat":"cs.DS","submitted_at":"2006-09-20T14:17:34Z","title":"Theory and Practice of Triangle Problems in Very Large (Sparse (Power-Law)) Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"cs/0609116","kind":"arxiv","version":1},"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":"sha256:61b05c10a566bfd501c2cae87fc59ba5ede63ee5fcf3c8e9bca4ed6299658265","target":"record","created_at":"2026-07-04T14:56:27Z","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":"51d8b1c0564c39f36ef6201cc877e06c0e5234bd7900e8716ee623986657353c","cross_cats_sorted":["cond-mat.stat-mech","cs.NI"],"license":"","primary_cat":"cs.DS","submitted_at":"2006-09-20T14:17:34Z","title_canon_sha256":"6fd0ec7a14f9de780cfe563a5de08019626d8c5c6e9903799604003b0b703a21"},"schema_version":"1.0","source":{"id":"cs/0609116","kind":"arxiv","version":1}},"canonical_sha256":"1231ceb19397381b5e255e084b039f8bb4223ec844681c39180989d3b076b643","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1231ceb19397381b5e255e084b039f8bb4223ec844681c39180989d3b076b643","first_computed_at":"2026-07-04T14:56:27.646930Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T14:56:27.646930Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wT8u58TZUmJMAiPF7dauGssY4vHNjAzSv85M4KjZopXOcM8rPOvmPD2u52zOQlx1fDfQxcvxGWorVMRq0YtRAg==","signature_status":"signed_v1","signed_at":"2026-07-04T14:56:27.647444Z","signed_message":"canonical_sha256_bytes"},"source_id":"cs/0609116","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:61b05c10a566bfd501c2cae87fc59ba5ede63ee5fcf3c8e9bca4ed6299658265","sha256:50bdceacfb466ae9c0bfea4a97f3306f3a48dbf69f6fdd60b02c945da37c4270"],"state_sha256":"e7705f44c7ecc153484b4d7222d448b622eb162ba8b96d9d4fe7028bd5206359"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"m7ncxpBH9pHt2LTgNjNR2QY2Ud7J3Kae/WwMT+0frdH1VkikFZb5wjrhpj3CrfDqt3vJDnzgWFwNfIy3+3x2Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T03:42:17.993625Z","bundle_sha256":"59e7e7c2a5aab3c642e2de5a8f925e16788e6293d98719bc2c8861b371378099"}}