{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:6M4MOBSVIEJTTQXYHKNC2IKBFE","short_pith_number":"pith:6M4MOBSV","canonical_record":{"source":{"id":"1906.11083","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-06-25T16:54:04Z","cross_cats_sorted":[],"title_canon_sha256":"8b008a0e2945bd75b0221b40f4d46fa73645ceb89071da6c56d6d16239a6c81d","abstract_canon_sha256":"20599cab8edf521d023e9f6b12e5bea85ab458b75252a1abd32a49ffcab688d0"},"schema_version":"1.0"},"canonical_sha256":"f338c70655411339c2f83a9a2d2141290ff70c65d6a4d1e158ed9fe0e02ce72b","source":{"kind":"arxiv","id":"1906.11083","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.11083","created_at":"2026-05-17T23:42:09Z"},{"alias_kind":"arxiv_version","alias_value":"1906.11083v1","created_at":"2026-05-17T23:42:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.11083","created_at":"2026-05-17T23:42:09Z"},{"alias_kind":"pith_short_12","alias_value":"6M4MOBSVIEJT","created_at":"2026-05-18T12:33:10Z"},{"alias_kind":"pith_short_16","alias_value":"6M4MOBSVIEJTTQXY","created_at":"2026-05-18T12:33:10Z"},{"alias_kind":"pith_short_8","alias_value":"6M4MOBSV","created_at":"2026-05-18T12:33:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:6M4MOBSVIEJTTQXYHKNC2IKBFE","target":"record","payload":{"canonical_record":{"source":{"id":"1906.11083","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-06-25T16:54:04Z","cross_cats_sorted":[],"title_canon_sha256":"8b008a0e2945bd75b0221b40f4d46fa73645ceb89071da6c56d6d16239a6c81d","abstract_canon_sha256":"20599cab8edf521d023e9f6b12e5bea85ab458b75252a1abd32a49ffcab688d0"},"schema_version":"1.0"},"canonical_sha256":"f338c70655411339c2f83a9a2d2141290ff70c65d6a4d1e158ed9fe0e02ce72b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:42:09.899495Z","signature_b64":"abKtjx1gRrEGZz4MtA8QrxmcrxvLsYx1RuFeHOeOhzDNCVc1vhCwstJetxXnCxe3nF5YMFH9X973YrkMD0khDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f338c70655411339c2f83a9a2d2141290ff70c65d6a4d1e158ed9fe0e02ce72b","last_reissued_at":"2026-05-17T23:42:09.898834Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:42:09.898834Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1906.11083","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":"2026-05-17T23:42:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2SNMxAsW+c34FaLDxYxM2cLDBfoxU3k3aaWzyzoKG0DBY3R6gmc7y/xutetcRAsZifYxmenAouhSlye8YcziAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T21:24:26.934407Z"},"content_sha256":"b18948089d6008ede9b3027b72a3ccdd03f223ec8ab801f21058d34ad2eac45a","schema_version":"1.0","event_id":"sha256:b18948089d6008ede9b3027b72a3ccdd03f223ec8ab801f21058d34ad2eac45a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:6M4MOBSVIEJTTQXYHKNC2IKBFE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Using Markov chains to determine expected propagation time for probabilistic zero forcing","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Emelie Curl, Issac Odegard, Jesse Geneson, Kevin Liu, Leslie Hogben, Michael S. Ross, Yu Chan","submitted_at":"2019-06-25T16:54:04Z","abstract_excerpt":"Zero forcing is a coloring game played on a graph where each vertex is initially colored blue or white and the goal is to color all the vertices blue by repeated use of a (deterministic) color change rule starting with as few blue vertices as possible. Probabilistic zero forcing yields a discrete dynamical system governed by a Markov chain. Since in a connected graph any one vertex can eventually color the entire graph blue using probabilistic zero forcing, the expected time to do this studied. Given a Markov transition matrix for a probabilistic zero forcing process, we establish an exact for"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.11083","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":"2026-05-17T23:42:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nXZUd3XiZj2t9AhgfGBEW8l1tFBBOF8m0Abb6lKvJs6jlIp9yOj8p7FTYgTjZZhPXfcOE9o6qtQfEk3hagbnCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T21:24:26.934981Z"},"content_sha256":"ec1008e83a13bee622bbc54d937afad0f5f20658f5246a3ead8158e86a9a51bb","schema_version":"1.0","event_id":"sha256:ec1008e83a13bee622bbc54d937afad0f5f20658f5246a3ead8158e86a9a51bb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6M4MOBSVIEJTTQXYHKNC2IKBFE/bundle.json","state_url":"https://pith.science/pith/6M4MOBSVIEJTTQXYHKNC2IKBFE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6M4MOBSVIEJTTQXYHKNC2IKBFE/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":"2026-06-23T21:24:26Z","links":{"resolver":"https://pith.science/pith/6M4MOBSVIEJTTQXYHKNC2IKBFE","bundle":"https://pith.science/pith/6M4MOBSVIEJTTQXYHKNC2IKBFE/bundle.json","state":"https://pith.science/pith/6M4MOBSVIEJTTQXYHKNC2IKBFE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6M4MOBSVIEJTTQXYHKNC2IKBFE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:6M4MOBSVIEJTTQXYHKNC2IKBFE","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":"20599cab8edf521d023e9f6b12e5bea85ab458b75252a1abd32a49ffcab688d0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-06-25T16:54:04Z","title_canon_sha256":"8b008a0e2945bd75b0221b40f4d46fa73645ceb89071da6c56d6d16239a6c81d"},"schema_version":"1.0","source":{"id":"1906.11083","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.11083","created_at":"2026-05-17T23:42:09Z"},{"alias_kind":"arxiv_version","alias_value":"1906.11083v1","created_at":"2026-05-17T23:42:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.11083","created_at":"2026-05-17T23:42:09Z"},{"alias_kind":"pith_short_12","alias_value":"6M4MOBSVIEJT","created_at":"2026-05-18T12:33:10Z"},{"alias_kind":"pith_short_16","alias_value":"6M4MOBSVIEJTTQXY","created_at":"2026-05-18T12:33:10Z"},{"alias_kind":"pith_short_8","alias_value":"6M4MOBSV","created_at":"2026-05-18T12:33:10Z"}],"graph_snapshots":[{"event_id":"sha256:ec1008e83a13bee622bbc54d937afad0f5f20658f5246a3ead8158e86a9a51bb","target":"graph","created_at":"2026-05-17T23:42:09Z","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":"Zero forcing is a coloring game played on a graph where each vertex is initially colored blue or white and the goal is to color all the vertices blue by repeated use of a (deterministic) color change rule starting with as few blue vertices as possible. Probabilistic zero forcing yields a discrete dynamical system governed by a Markov chain. Since in a connected graph any one vertex can eventually color the entire graph blue using probabilistic zero forcing, the expected time to do this studied. Given a Markov transition matrix for a probabilistic zero forcing process, we establish an exact for","authors_text":"Emelie Curl, Issac Odegard, Jesse Geneson, Kevin Liu, Leslie Hogben, Michael S. Ross, Yu Chan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-06-25T16:54:04Z","title":"Using Markov chains to determine expected propagation time for probabilistic zero forcing"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.11083","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:b18948089d6008ede9b3027b72a3ccdd03f223ec8ab801f21058d34ad2eac45a","target":"record","created_at":"2026-05-17T23:42:09Z","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":"20599cab8edf521d023e9f6b12e5bea85ab458b75252a1abd32a49ffcab688d0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-06-25T16:54:04Z","title_canon_sha256":"8b008a0e2945bd75b0221b40f4d46fa73645ceb89071da6c56d6d16239a6c81d"},"schema_version":"1.0","source":{"id":"1906.11083","kind":"arxiv","version":1}},"canonical_sha256":"f338c70655411339c2f83a9a2d2141290ff70c65d6a4d1e158ed9fe0e02ce72b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f338c70655411339c2f83a9a2d2141290ff70c65d6a4d1e158ed9fe0e02ce72b","first_computed_at":"2026-05-17T23:42:09.898834Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:42:09.898834Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"abKtjx1gRrEGZz4MtA8QrxmcrxvLsYx1RuFeHOeOhzDNCVc1vhCwstJetxXnCxe3nF5YMFH9X973YrkMD0khDQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:42:09.899495Z","signed_message":"canonical_sha256_bytes"},"source_id":"1906.11083","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b18948089d6008ede9b3027b72a3ccdd03f223ec8ab801f21058d34ad2eac45a","sha256:ec1008e83a13bee622bbc54d937afad0f5f20658f5246a3ead8158e86a9a51bb"],"state_sha256":"e99939d150ac692350d2627c94e2f8c47e8d68f59236cf11e0090b3f00afe335"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RwA1evGttSZYvzghFl6EzAKGINtC49i3coGL57NnQhTUMU4iV2fe4NQddjoyEd8A+4QLJcv15qMEy2KWXtFFDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T21:24:26.937686Z","bundle_sha256":"5d2e35bb1eb19b661c2c1cc075a953eddaad048ae9a28e9f03e1430de33b8e73"}}