{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:4ASM66XAGTDBSRVSZ7WGBUOGOK","short_pith_number":"pith:4ASM66XA","canonical_record":{"source":{"id":"1511.07734","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-11-24T14:43:39Z","cross_cats_sorted":[],"title_canon_sha256":"cfcb93de4ff4c6cc8ce6ebcba913a54108adfd4e2a084950da391b945f3d63c3","abstract_canon_sha256":"74eae5ca196967fe37b1442d315f5050bdebfa4b1e8608302157973263931b23"},"schema_version":"1.0"},"canonical_sha256":"e024cf7ae034c61946b2cfec60d1c672b0db5edfee4b884847345232f1424c9e","source":{"kind":"arxiv","id":"1511.07734","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.07734","created_at":"2026-05-18T00:56:46Z"},{"alias_kind":"arxiv_version","alias_value":"1511.07734v3","created_at":"2026-05-18T00:56:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.07734","created_at":"2026-05-18T00:56:46Z"},{"alias_kind":"pith_short_12","alias_value":"4ASM66XAGTDB","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_16","alias_value":"4ASM66XAGTDBSRVS","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_8","alias_value":"4ASM66XA","created_at":"2026-05-18T12:29:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:4ASM66XAGTDBSRVSZ7WGBUOGOK","target":"record","payload":{"canonical_record":{"source":{"id":"1511.07734","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-11-24T14:43:39Z","cross_cats_sorted":[],"title_canon_sha256":"cfcb93de4ff4c6cc8ce6ebcba913a54108adfd4e2a084950da391b945f3d63c3","abstract_canon_sha256":"74eae5ca196967fe37b1442d315f5050bdebfa4b1e8608302157973263931b23"},"schema_version":"1.0"},"canonical_sha256":"e024cf7ae034c61946b2cfec60d1c672b0db5edfee4b884847345232f1424c9e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:56:46.603103Z","signature_b64":"9jk4W4e08ILKqVBN3hmLTn06cvTIX8i9Mbbry1toSQMLqshZybNgEQYHvlTLSeNoQYp3YIwauykZJP2BzvAsCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e024cf7ae034c61946b2cfec60d1c672b0db5edfee4b884847345232f1424c9e","last_reissued_at":"2026-05-18T00:56:46.602643Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:56:46.602643Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1511.07734","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:56:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JQX9AWhfxjoZmNiOOQdRn1lSyF+AO5jYgsuW/0+F2otdQsQ2JRXq6XNLkKLDQlGJAnbvTyubS3neAN8Ajdy0Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T22:35:19.153914Z"},"content_sha256":"9bdb3bd007ba42954618e1dd62475d26edb29358bf558b6f3cc945c02bdd2825","schema_version":"1.0","event_id":"sha256:9bdb3bd007ba42954618e1dd62475d26edb29358bf558b6f3cc945c02bdd2825"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:4ASM66XAGTDBSRVSZ7WGBUOGOK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A generating function approach to branching random walks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Daniela Bertacchi, Fabio Zucca","submitted_at":"2015-11-24T14:43:39Z","abstract_excerpt":"It is well known that the behaviour of a branching process is completely described by the generating function of the offspring law and its fixed points. Branching random walks are a natural generalization of branching processes: a branching process can be seen as a one-dimensional branching random walk. We define a multidimensional generating function associated to a given branching random walk. The present paper investigates the similarities and the differences of the generating functions, their fixed points and the implications on the underlying stochastic process, between the one-dimensiona"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.07734","kind":"arxiv","version":3},"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":"2026-05-18T00:56:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yAzlhkKO7elDuiLmQS0b2m8fjMdC99U+3zavf4aDHyaCtgbQN1lpM3SVTUOvvByWnDyiGm7KsNJeiXpLMeNvBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T22:35:19.154566Z"},"content_sha256":"2087de18656f587aaa13632ab92dd93de353c730445d3df8902ca3ddb6c72336","schema_version":"1.0","event_id":"sha256:2087de18656f587aaa13632ab92dd93de353c730445d3df8902ca3ddb6c72336"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4ASM66XAGTDBSRVSZ7WGBUOGOK/bundle.json","state_url":"https://pith.science/pith/4ASM66XAGTDBSRVSZ7WGBUOGOK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4ASM66XAGTDBSRVSZ7WGBUOGOK/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":"2026-05-28T22:35:19Z","links":{"resolver":"https://pith.science/pith/4ASM66XAGTDBSRVSZ7WGBUOGOK","bundle":"https://pith.science/pith/4ASM66XAGTDBSRVSZ7WGBUOGOK/bundle.json","state":"https://pith.science/pith/4ASM66XAGTDBSRVSZ7WGBUOGOK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4ASM66XAGTDBSRVSZ7WGBUOGOK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:4ASM66XAGTDBSRVSZ7WGBUOGOK","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":"74eae5ca196967fe37b1442d315f5050bdebfa4b1e8608302157973263931b23","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-11-24T14:43:39Z","title_canon_sha256":"cfcb93de4ff4c6cc8ce6ebcba913a54108adfd4e2a084950da391b945f3d63c3"},"schema_version":"1.0","source":{"id":"1511.07734","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1511.07734","created_at":"2026-05-18T00:56:46Z"},{"alias_kind":"arxiv_version","alias_value":"1511.07734v3","created_at":"2026-05-18T00:56:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1511.07734","created_at":"2026-05-18T00:56:46Z"},{"alias_kind":"pith_short_12","alias_value":"4ASM66XAGTDB","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_16","alias_value":"4ASM66XAGTDBSRVS","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_8","alias_value":"4ASM66XA","created_at":"2026-05-18T12:29:05Z"}],"graph_snapshots":[{"event_id":"sha256:2087de18656f587aaa13632ab92dd93de353c730445d3df8902ca3ddb6c72336","target":"graph","created_at":"2026-05-18T00:56:46Z","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":"It is well known that the behaviour of a branching process is completely described by the generating function of the offspring law and its fixed points. Branching random walks are a natural generalization of branching processes: a branching process can be seen as a one-dimensional branching random walk. We define a multidimensional generating function associated to a given branching random walk. The present paper investigates the similarities and the differences of the generating functions, their fixed points and the implications on the underlying stochastic process, between the one-dimensiona","authors_text":"Daniela Bertacchi, Fabio Zucca","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-11-24T14:43:39Z","title":"A generating function approach to branching random walks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.07734","kind":"arxiv","version":3},"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":"sha256:9bdb3bd007ba42954618e1dd62475d26edb29358bf558b6f3cc945c02bdd2825","target":"record","created_at":"2026-05-18T00:56:46Z","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":"74eae5ca196967fe37b1442d315f5050bdebfa4b1e8608302157973263931b23","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-11-24T14:43:39Z","title_canon_sha256":"cfcb93de4ff4c6cc8ce6ebcba913a54108adfd4e2a084950da391b945f3d63c3"},"schema_version":"1.0","source":{"id":"1511.07734","kind":"arxiv","version":3}},"canonical_sha256":"e024cf7ae034c61946b2cfec60d1c672b0db5edfee4b884847345232f1424c9e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e024cf7ae034c61946b2cfec60d1c672b0db5edfee4b884847345232f1424c9e","first_computed_at":"2026-05-18T00:56:46.602643Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:56:46.602643Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9jk4W4e08ILKqVBN3hmLTn06cvTIX8i9Mbbry1toSQMLqshZybNgEQYHvlTLSeNoQYp3YIwauykZJP2BzvAsCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:56:46.603103Z","signed_message":"canonical_sha256_bytes"},"source_id":"1511.07734","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9bdb3bd007ba42954618e1dd62475d26edb29358bf558b6f3cc945c02bdd2825","sha256:2087de18656f587aaa13632ab92dd93de353c730445d3df8902ca3ddb6c72336"],"state_sha256":"2e422693cb0868e3879971b6beefd251b5886b1fdb17adca9c7f568af27cf05b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"R6vwKFwKcCIryJlfhGRlrnxXMQhmhKpx8Szs8x2bbJ2sk97Q29+OsEpsvPncxr8yL70H8C1uFYmwMoUP0J4FBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T22:35:19.161089Z","bundle_sha256":"d40b626f8f0038e6f62bfdd6cdfb0a110ec46fa36896f60a5bfc78fb859743a1"}}