{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:3NLOIHUZHODFV2Q2CQMIVT5PFZ","short_pith_number":"pith:3NLOIHUZ","canonical_record":{"source":{"id":"math/0603412","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.PR","submitted_at":"2006-03-16T22:24:28Z","cross_cats_sorted":[],"title_canon_sha256":"762ff8b993aa84ae68c8712a27b14c46f2a03bcb4dd3f466872d62ebf9419568","abstract_canon_sha256":"a58c3f76ecec40923abb23b9d5289952f70a1bbc2503d7aa4a15435fc5e6705c"},"schema_version":"1.0"},"canonical_sha256":"db56e41e993b865aea1a14188acfaf2e42c43991295a3dd2bc8e7cdc2fc82dc1","source":{"kind":"arxiv","id":"math/0603412","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0603412","created_at":"2026-05-18T04:31:00Z"},{"alias_kind":"arxiv_version","alias_value":"math/0603412v2","created_at":"2026-05-18T04:31:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0603412","created_at":"2026-05-18T04:31:00Z"},{"alias_kind":"pith_short_12","alias_value":"3NLOIHUZHODF","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"3NLOIHUZHODFV2Q2","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"3NLOIHUZ","created_at":"2026-05-18T12:25:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:3NLOIHUZHODFV2Q2CQMIVT5PFZ","target":"record","payload":{"canonical_record":{"source":{"id":"math/0603412","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.PR","submitted_at":"2006-03-16T22:24:28Z","cross_cats_sorted":[],"title_canon_sha256":"762ff8b993aa84ae68c8712a27b14c46f2a03bcb4dd3f466872d62ebf9419568","abstract_canon_sha256":"a58c3f76ecec40923abb23b9d5289952f70a1bbc2503d7aa4a15435fc5e6705c"},"schema_version":"1.0"},"canonical_sha256":"db56e41e993b865aea1a14188acfaf2e42c43991295a3dd2bc8e7cdc2fc82dc1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:31:00.296010Z","signature_b64":"XUOZZX/6jmhPYu5XCZNultYOpO9RD2Am36MphGbVKU0icIixrh5WG9V7YXMCXHnKd+4zm/0BF5XSJBLVaFiwBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"db56e41e993b865aea1a14188acfaf2e42c43991295a3dd2bc8e7cdc2fc82dc1","last_reissued_at":"2026-05-18T04:31:00.295080Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:31:00.295080Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0603412","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:31:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"E6msHEi1+KvGMUt/V72WjKHK2b9CjcUhLljonKIVRllov+nVbaGZ7/+/ZKLnRlm0TR4h/o7GuATlc4J8RLUKAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T21:45:35.395265Z"},"content_sha256":"b5d463110ce22d9d4b24fd6e739fdaf97ae631fb58ecf9d06bb0454d4b6095d7","schema_version":"1.0","event_id":"sha256:b5d463110ce22d9d4b24fd6e739fdaf97ae631fb58ecf9d06bb0454d4b6095d7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:3NLOIHUZHODFV2Q2CQMIVT5PFZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Weak survival for branching random walks on graphs","license":"","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Daniela Bertacchi, Fabio Zucca","submitted_at":"2006-03-16T22:24:28Z","abstract_excerpt":"We study weak and strong survival for branching random walks on multigraphs. We prove that, for a large class of multigraphs, weak survival is related to a geometrical parameter of the multigraph and that the existence of a pure weak phase is equivalent to nonamenability. Finally we study weak and strong critical behaviors of the branching random walk."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0603412","kind":"arxiv","version":2},"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":"2026-05-18T04:31:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hrstgg7BfNWouiw1EAIIcAh9WqG8SFRflBZJ1NUKZ5JrQKGaolqcRfwGIBkb6yOyGGiGxKCXTdhRh97Vwdd/Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T21:45:35.395632Z"},"content_sha256":"0ea6b20384cb560417c61d2edf978ea0fc98aabb6a891ee476afeab5dd234dda","schema_version":"1.0","event_id":"sha256:0ea6b20384cb560417c61d2edf978ea0fc98aabb6a891ee476afeab5dd234dda"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3NLOIHUZHODFV2Q2CQMIVT5PFZ/bundle.json","state_url":"https://pith.science/pith/3NLOIHUZHODFV2Q2CQMIVT5PFZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3NLOIHUZHODFV2Q2CQMIVT5PFZ/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":"2026-06-21T21:45:35Z","links":{"resolver":"https://pith.science/pith/3NLOIHUZHODFV2Q2CQMIVT5PFZ","bundle":"https://pith.science/pith/3NLOIHUZHODFV2Q2CQMIVT5PFZ/bundle.json","state":"https://pith.science/pith/3NLOIHUZHODFV2Q2CQMIVT5PFZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3NLOIHUZHODFV2Q2CQMIVT5PFZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:3NLOIHUZHODFV2Q2CQMIVT5PFZ","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":"a58c3f76ecec40923abb23b9d5289952f70a1bbc2503d7aa4a15435fc5e6705c","cross_cats_sorted":[],"license":"","primary_cat":"math.PR","submitted_at":"2006-03-16T22:24:28Z","title_canon_sha256":"762ff8b993aa84ae68c8712a27b14c46f2a03bcb4dd3f466872d62ebf9419568"},"schema_version":"1.0","source":{"id":"math/0603412","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0603412","created_at":"2026-05-18T04:31:00Z"},{"alias_kind":"arxiv_version","alias_value":"math/0603412v2","created_at":"2026-05-18T04:31:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0603412","created_at":"2026-05-18T04:31:00Z"},{"alias_kind":"pith_short_12","alias_value":"3NLOIHUZHODF","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"3NLOIHUZHODFV2Q2","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"3NLOIHUZ","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:0ea6b20384cb560417c61d2edf978ea0fc98aabb6a891ee476afeab5dd234dda","target":"graph","created_at":"2026-05-18T04:31:00Z","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":"We study weak and strong survival for branching random walks on multigraphs. We prove that, for a large class of multigraphs, weak survival is related to a geometrical parameter of the multigraph and that the existence of a pure weak phase is equivalent to nonamenability. Finally we study weak and strong critical behaviors of the branching random walk.","authors_text":"Daniela Bertacchi, Fabio Zucca","cross_cats":[],"headline":"","license":"","primary_cat":"math.PR","submitted_at":"2006-03-16T22:24:28Z","title":"Weak survival for branching random walks on graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0603412","kind":"arxiv","version":2},"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:b5d463110ce22d9d4b24fd6e739fdaf97ae631fb58ecf9d06bb0454d4b6095d7","target":"record","created_at":"2026-05-18T04:31:00Z","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":"a58c3f76ecec40923abb23b9d5289952f70a1bbc2503d7aa4a15435fc5e6705c","cross_cats_sorted":[],"license":"","primary_cat":"math.PR","submitted_at":"2006-03-16T22:24:28Z","title_canon_sha256":"762ff8b993aa84ae68c8712a27b14c46f2a03bcb4dd3f466872d62ebf9419568"},"schema_version":"1.0","source":{"id":"math/0603412","kind":"arxiv","version":2}},"canonical_sha256":"db56e41e993b865aea1a14188acfaf2e42c43991295a3dd2bc8e7cdc2fc82dc1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"db56e41e993b865aea1a14188acfaf2e42c43991295a3dd2bc8e7cdc2fc82dc1","first_computed_at":"2026-05-18T04:31:00.295080Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:31:00.295080Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XUOZZX/6jmhPYu5XCZNultYOpO9RD2Am36MphGbVKU0icIixrh5WG9V7YXMCXHnKd+4zm/0BF5XSJBLVaFiwBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:31:00.296010Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0603412","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b5d463110ce22d9d4b24fd6e739fdaf97ae631fb58ecf9d06bb0454d4b6095d7","sha256:0ea6b20384cb560417c61d2edf978ea0fc98aabb6a891ee476afeab5dd234dda"],"state_sha256":"75969c180983e4727d1e4380ca074ab4555f73403eff8ae546980c0e559bf8ff"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Mmko60HSKnOVPkMA6ybgcRJE/XgziBmQ4Uzz8sv1s7K6AIL21WOvdqCskHglTQrmqMMdtinrsLvAcG5j5n4cAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T21:45:35.397594Z","bundle_sha256":"c8aeb66a09d76b2658997047036934176101b33d125f2afb81c0beadb1db636a"}}