{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:XEJOIILMPMF26JKE3D2KXYSU4E","short_pith_number":"pith:XEJOIILM","canonical_record":{"source":{"id":"math/0612778","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.DS","submitted_at":"2006-12-27T10:34:57Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"fa069e5faa50907cbaeafad0e4bd727dd57da6474549a57395fe93e088cf3973","abstract_canon_sha256":"2564c65ab988cfa3841461ea4f7ad644596260a92df04390985b449fde8333f6"},"schema_version":"1.0"},"canonical_sha256":"b912e4216c7b0baf2544d8f4abe254e11c3948e6dc6189c4403f58692086137b","source":{"kind":"arxiv","id":"math/0612778","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0612778","created_at":"2026-05-18T04:36:42Z"},{"alias_kind":"arxiv_version","alias_value":"math/0612778v1","created_at":"2026-05-18T04:36:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0612778","created_at":"2026-05-18T04:36:42Z"},{"alias_kind":"pith_short_12","alias_value":"XEJOIILMPMF2","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"XEJOIILMPMF26JKE","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"XEJOIILM","created_at":"2026-05-18T12:25:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:XEJOIILMPMF26JKE3D2KXYSU4E","target":"record","payload":{"canonical_record":{"source":{"id":"math/0612778","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.DS","submitted_at":"2006-12-27T10:34:57Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"fa069e5faa50907cbaeafad0e4bd727dd57da6474549a57395fe93e088cf3973","abstract_canon_sha256":"2564c65ab988cfa3841461ea4f7ad644596260a92df04390985b449fde8333f6"},"schema_version":"1.0"},"canonical_sha256":"b912e4216c7b0baf2544d8f4abe254e11c3948e6dc6189c4403f58692086137b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:36:42.386791Z","signature_b64":"gPsLjTJ/+cADM6GoMPc9NxbsLsnZcmvcCvbFuwX23RzFuiRLj/WWta6PPFCd1PFfo1CAJ0rspuWj3w2J+4yfAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b912e4216c7b0baf2544d8f4abe254e11c3948e6dc6189c4403f58692086137b","last_reissued_at":"2026-05-18T04:36:42.386373Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:36:42.386373Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0612778","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":"2026-05-18T04:36:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4bSrrVJqy7IlVcbRepbaUsNe2DX491qJgobvqkPsERZx4ucuKFMLEtsiq53edO805kqtxhfvwZK1ciFhqaZkCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T12:38:56.128155Z"},"content_sha256":"4f4c0226ced1d05aa7d5e59ad4ef609a3e1c234de68febf3fe243becb8d6ee14","schema_version":"1.0","event_id":"sha256:4f4c0226ced1d05aa7d5e59ad4ef609a3e1c234de68febf3fe243becb8d6ee14"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:XEJOIILMPMF26JKE3D2KXYSU4E","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Probabilistic Inductive Classes of Graphs","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.DS","authors_text":"Nata\\v{s}a Kej\\v{z}ar, Vladimir Batagelj, Zoran Nikoloski","submitted_at":"2006-12-27T10:34:57Z","abstract_excerpt":"Models of complex networks are generally defined as graph stochastic processes in which edges and vertices are added or deleted over time to simulate the evolution of networks. Here, we define a unifying framework - probabilistic inductive classes of graphs - for formalizing and studying evolution of complex networks. Our definition of probabilistic inductive class of graphs (PICG) extends the standard notion of inductive class of graphs (ICG) by imposing a probability space. A PICG is given by: (1) class B of initial graphs, the basis of PICG, (2) class R of generating rules, each with distin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0612778","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":"2026-05-18T04:36:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"R0j9hG2/laH3JY2SSHg87jD3Pij11pEcUKBAVrx1rtpd2CXHdVxbNX2u1ZRXptAIWxlWscK+XkBaeFseLEWWCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T12:38:56.129076Z"},"content_sha256":"3c55bbcf0676f7f7f5bd19da44b550a5ae2723afee05eb7e476b753cde10451c","schema_version":"1.0","event_id":"sha256:3c55bbcf0676f7f7f5bd19da44b550a5ae2723afee05eb7e476b753cde10451c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XEJOIILMPMF26JKE3D2KXYSU4E/bundle.json","state_url":"https://pith.science/pith/XEJOIILMPMF26JKE3D2KXYSU4E/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XEJOIILMPMF26JKE3D2KXYSU4E/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":"2026-05-31T12:38:56Z","links":{"resolver":"https://pith.science/pith/XEJOIILMPMF26JKE3D2KXYSU4E","bundle":"https://pith.science/pith/XEJOIILMPMF26JKE3D2KXYSU4E/bundle.json","state":"https://pith.science/pith/XEJOIILMPMF26JKE3D2KXYSU4E/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XEJOIILMPMF26JKE3D2KXYSU4E/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:XEJOIILMPMF26JKE3D2KXYSU4E","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":"2564c65ab988cfa3841461ea4f7ad644596260a92df04390985b449fde8333f6","cross_cats_sorted":["math.CO"],"license":"","primary_cat":"math.DS","submitted_at":"2006-12-27T10:34:57Z","title_canon_sha256":"fa069e5faa50907cbaeafad0e4bd727dd57da6474549a57395fe93e088cf3973"},"schema_version":"1.0","source":{"id":"math/0612778","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0612778","created_at":"2026-05-18T04:36:42Z"},{"alias_kind":"arxiv_version","alias_value":"math/0612778v1","created_at":"2026-05-18T04:36:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0612778","created_at":"2026-05-18T04:36:42Z"},{"alias_kind":"pith_short_12","alias_value":"XEJOIILMPMF2","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"XEJOIILMPMF26JKE","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"XEJOIILM","created_at":"2026-05-18T12:25:54Z"}],"graph_snapshots":[{"event_id":"sha256:3c55bbcf0676f7f7f5bd19da44b550a5ae2723afee05eb7e476b753cde10451c","target":"graph","created_at":"2026-05-18T04:36:42Z","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":"Models of complex networks are generally defined as graph stochastic processes in which edges and vertices are added or deleted over time to simulate the evolution of networks. Here, we define a unifying framework - probabilistic inductive classes of graphs - for formalizing and studying evolution of complex networks. Our definition of probabilistic inductive class of graphs (PICG) extends the standard notion of inductive class of graphs (ICG) by imposing a probability space. A PICG is given by: (1) class B of initial graphs, the basis of PICG, (2) class R of generating rules, each with distin","authors_text":"Nata\\v{s}a Kej\\v{z}ar, Vladimir Batagelj, Zoran Nikoloski","cross_cats":["math.CO"],"headline":"","license":"","primary_cat":"math.DS","submitted_at":"2006-12-27T10:34:57Z","title":"Probabilistic Inductive Classes of Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0612778","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":"sha256:4f4c0226ced1d05aa7d5e59ad4ef609a3e1c234de68febf3fe243becb8d6ee14","target":"record","created_at":"2026-05-18T04:36:42Z","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":"2564c65ab988cfa3841461ea4f7ad644596260a92df04390985b449fde8333f6","cross_cats_sorted":["math.CO"],"license":"","primary_cat":"math.DS","submitted_at":"2006-12-27T10:34:57Z","title_canon_sha256":"fa069e5faa50907cbaeafad0e4bd727dd57da6474549a57395fe93e088cf3973"},"schema_version":"1.0","source":{"id":"math/0612778","kind":"arxiv","version":1}},"canonical_sha256":"b912e4216c7b0baf2544d8f4abe254e11c3948e6dc6189c4403f58692086137b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b912e4216c7b0baf2544d8f4abe254e11c3948e6dc6189c4403f58692086137b","first_computed_at":"2026-05-18T04:36:42.386373Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:36:42.386373Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gPsLjTJ/+cADM6GoMPc9NxbsLsnZcmvcCvbFuwX23RzFuiRLj/WWta6PPFCd1PFfo1CAJ0rspuWj3w2J+4yfAw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:36:42.386791Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0612778","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4f4c0226ced1d05aa7d5e59ad4ef609a3e1c234de68febf3fe243becb8d6ee14","sha256:3c55bbcf0676f7f7f5bd19da44b550a5ae2723afee05eb7e476b753cde10451c"],"state_sha256":"d1426f8aee44d8bc51d2a4b6d5e1f6f455a900dd3f3f923c9bdd07cdcdfcaf09"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JRHcf9ScAMfEQTMq9dzvAUuTIvNOJXMhxu5tVe4C7G9Ny6P/TFMGFuDRqt5kkcxmdSg8a9nTD8Lv8YfXOQJPCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T12:38:56.133569Z","bundle_sha256":"67c984fdafd93db532a2239d3b083166e8a273ed0b590338d5d947ec1e11a2de"}}