{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:MLHOGDX2RPV6IOH6CD5DYQYKBJ","short_pith_number":"pith:MLHOGDX2","canonical_record":{"source":{"id":"1103.6073","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2011-03-31T01:26:13Z","cross_cats_sorted":["cs.DM","cs.SI"],"title_canon_sha256":"24f2ea518a92faf86d1cf7233152ef51b40c8c4ced2160acb850689890b93b4a","abstract_canon_sha256":"0de3c0279479c7785a022de5ec1c8d711f80668dd71aee413d05bbba5bccf5b6"},"schema_version":"1.0"},"canonical_sha256":"62cee30efa8bebe438fe10fa3c430a0a54e21da4dec75fca71c3fc70512546e3","source":{"kind":"arxiv","id":"1103.6073","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.6073","created_at":"2026-05-18T04:25:25Z"},{"alias_kind":"arxiv_version","alias_value":"1103.6073v1","created_at":"2026-05-18T04:25:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.6073","created_at":"2026-05-18T04:25:25Z"},{"alias_kind":"pith_short_12","alias_value":"MLHOGDX2RPV6","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_16","alias_value":"MLHOGDX2RPV6IOH6","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_8","alias_value":"MLHOGDX2","created_at":"2026-05-18T12:26:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:MLHOGDX2RPV6IOH6CD5DYQYKBJ","target":"record","payload":{"canonical_record":{"source":{"id":"1103.6073","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2011-03-31T01:26:13Z","cross_cats_sorted":["cs.DM","cs.SI"],"title_canon_sha256":"24f2ea518a92faf86d1cf7233152ef51b40c8c4ced2160acb850689890b93b4a","abstract_canon_sha256":"0de3c0279479c7785a022de5ec1c8d711f80668dd71aee413d05bbba5bccf5b6"},"schema_version":"1.0"},"canonical_sha256":"62cee30efa8bebe438fe10fa3c430a0a54e21da4dec75fca71c3fc70512546e3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:25:25.247175Z","signature_b64":"XSEK9jv90SeQwVICRFj0zJXatNiI4TyWFLd2wwK+hq7/Wqz50GLtIyOA565FrXYs8Pmhj32CYg5Cx7WmK98xAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"62cee30efa8bebe438fe10fa3c430a0a54e21da4dec75fca71c3fc70512546e3","last_reissued_at":"2026-05-18T04:25:25.246484Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:25:25.246484Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1103.6073","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":"2026-05-18T04:25:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rwWdDWzIMQvveTcI7/4IKTjve1DEta6UODxsChgh5v2/wtPOpe1ESGDfIwB8yBoDCaDReTu6BVUW6AM+LKc3Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T02:20:53.876906Z"},"content_sha256":"6816ba0a4e69d1dc8faee7c51a93ab89f2f9d03258b87d37be28732f9d540cd4","schema_version":"1.0","event_id":"sha256:6816ba0a4e69d1dc8faee7c51a93ab89f2f9d03258b87d37be28732f9d540cd4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:MLHOGDX2RPV6IOH6CD5DYQYKBJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Colorful Triangle Counting and a MapReduce Implementation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","cs.SI"],"primary_cat":"cs.DS","authors_text":"Charalampos E. Tsourakakis, Rasmus Pagh","submitted_at":"2011-03-31T01:26:13Z","abstract_excerpt":"In this note we introduce a new randomized algorithm for counting triangles in graphs. We show that under mild conditions, the estimate of our algorithm is strongly concentrated around the true number of triangles. Specifically, if $p \\geq \\max{(\\frac{\\Delta \\log{n}}{t}, \\frac{\\log{n}}{\\sqrt{t}})}$, where $n$, $t$, $\\Delta$ denote the number of vertices in $G$, the number of triangles in $G$, the maximum number of triangles an edge of $G$ is contained, then for any constant $\\epsilon>0$ our unbiased estimate $T$ is concentrated around its expectation, i.e., $ \\Prob{|T - \\Mean{T}| \\geq \\epsilon"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.6073","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":""},"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":"2026-05-18T04:25:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QD9yyVRsGd1Bb/L5zCcB5c22f0MYJIKP494j0pMaoJe5Mbaf7CKNRxbxySHFnevmqM7LwRDouE2k96Xy8spADg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T02:20:53.877253Z"},"content_sha256":"95d82cad500bc41d97ce0353690ccba659d0a51702f3a8029d564c120f0f602a","schema_version":"1.0","event_id":"sha256:95d82cad500bc41d97ce0353690ccba659d0a51702f3a8029d564c120f0f602a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MLHOGDX2RPV6IOH6CD5DYQYKBJ/bundle.json","state_url":"https://pith.science/pith/MLHOGDX2RPV6IOH6CD5DYQYKBJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MLHOGDX2RPV6IOH6CD5DYQYKBJ/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":"2026-06-20T02:20:53Z","links":{"resolver":"https://pith.science/pith/MLHOGDX2RPV6IOH6CD5DYQYKBJ","bundle":"https://pith.science/pith/MLHOGDX2RPV6IOH6CD5DYQYKBJ/bundle.json","state":"https://pith.science/pith/MLHOGDX2RPV6IOH6CD5DYQYKBJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MLHOGDX2RPV6IOH6CD5DYQYKBJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:MLHOGDX2RPV6IOH6CD5DYQYKBJ","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":"0de3c0279479c7785a022de5ec1c8d711f80668dd71aee413d05bbba5bccf5b6","cross_cats_sorted":["cs.DM","cs.SI"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2011-03-31T01:26:13Z","title_canon_sha256":"24f2ea518a92faf86d1cf7233152ef51b40c8c4ced2160acb850689890b93b4a"},"schema_version":"1.0","source":{"id":"1103.6073","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.6073","created_at":"2026-05-18T04:25:25Z"},{"alias_kind":"arxiv_version","alias_value":"1103.6073v1","created_at":"2026-05-18T04:25:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.6073","created_at":"2026-05-18T04:25:25Z"},{"alias_kind":"pith_short_12","alias_value":"MLHOGDX2RPV6","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_16","alias_value":"MLHOGDX2RPV6IOH6","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_8","alias_value":"MLHOGDX2","created_at":"2026-05-18T12:26:34Z"}],"graph_snapshots":[{"event_id":"sha256:95d82cad500bc41d97ce0353690ccba659d0a51702f3a8029d564c120f0f602a","target":"graph","created_at":"2026-05-18T04:25:25Z","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 note we introduce a new randomized algorithm for counting triangles in graphs. We show that under mild conditions, the estimate of our algorithm is strongly concentrated around the true number of triangles. Specifically, if $p \\geq \\max{(\\frac{\\Delta \\log{n}}{t}, \\frac{\\log{n}}{\\sqrt{t}})}$, where $n$, $t$, $\\Delta$ denote the number of vertices in $G$, the number of triangles in $G$, the maximum number of triangles an edge of $G$ is contained, then for any constant $\\epsilon>0$ our unbiased estimate $T$ is concentrated around its expectation, i.e., $ \\Prob{|T - \\Mean{T}| \\geq \\epsilon","authors_text":"Charalampos E. Tsourakakis, Rasmus Pagh","cross_cats":["cs.DM","cs.SI"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2011-03-31T01:26:13Z","title":"Colorful Triangle Counting and a MapReduce Implementation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.6073","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":"sha256:6816ba0a4e69d1dc8faee7c51a93ab89f2f9d03258b87d37be28732f9d540cd4","target":"record","created_at":"2026-05-18T04:25:25Z","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":"0de3c0279479c7785a022de5ec1c8d711f80668dd71aee413d05bbba5bccf5b6","cross_cats_sorted":["cs.DM","cs.SI"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2011-03-31T01:26:13Z","title_canon_sha256":"24f2ea518a92faf86d1cf7233152ef51b40c8c4ced2160acb850689890b93b4a"},"schema_version":"1.0","source":{"id":"1103.6073","kind":"arxiv","version":1}},"canonical_sha256":"62cee30efa8bebe438fe10fa3c430a0a54e21da4dec75fca71c3fc70512546e3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"62cee30efa8bebe438fe10fa3c430a0a54e21da4dec75fca71c3fc70512546e3","first_computed_at":"2026-05-18T04:25:25.246484Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:25:25.246484Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XSEK9jv90SeQwVICRFj0zJXatNiI4TyWFLd2wwK+hq7/Wqz50GLtIyOA565FrXYs8Pmhj32CYg5Cx7WmK98xAg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:25:25.247175Z","signed_message":"canonical_sha256_bytes"},"source_id":"1103.6073","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6816ba0a4e69d1dc8faee7c51a93ab89f2f9d03258b87d37be28732f9d540cd4","sha256:95d82cad500bc41d97ce0353690ccba659d0a51702f3a8029d564c120f0f602a"],"state_sha256":"eb028593d204a5ec1d836d96a2b0eba5615f0adbcc46a6d098c6b62f572a5e4c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3t5YfcjKsDJoMbqN14ViEL3RY+NndeHcCDu7UVRmvLzBYcHlBdLm9SJJp1OHLxH6znEl66y+nqsUd5ETrVFZBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T02:20:53.879263Z","bundle_sha256":"b836fff53afc674891e31198970d139677c0d2c60a2b57e7066b1d36242a72e9"}}