{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:YA2IZMXLWUYPZUGXS2RJVJNNPX","short_pith_number":"pith:YA2IZMXL","canonical_record":{"source":{"id":"1202.1146","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-02-06T14:17:57Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"59c3b51dccc7736238d5d5db55393f1018bef8f4ce1d161df544b68beadff841","abstract_canon_sha256":"5415736b2901717f8ac4f8e76bd2debbf5cc36219b1822611b3d18ef2072e9d5"},"schema_version":"1.0"},"canonical_sha256":"c0348cb2ebb530fcd0d796a29aa5ad7dc39847ea314e6def60fa7dff9f54f883","source":{"kind":"arxiv","id":"1202.1146","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.1146","created_at":"2026-05-18T04:02:55Z"},{"alias_kind":"arxiv_version","alias_value":"1202.1146v1","created_at":"2026-05-18T04:02:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.1146","created_at":"2026-05-18T04:02:55Z"},{"alias_kind":"pith_short_12","alias_value":"YA2IZMXLWUYP","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"YA2IZMXLWUYPZUGX","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"YA2IZMXL","created_at":"2026-05-18T12:27:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:YA2IZMXLWUYPZUGXS2RJVJNNPX","target":"record","payload":{"canonical_record":{"source":{"id":"1202.1146","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-02-06T14:17:57Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"59c3b51dccc7736238d5d5db55393f1018bef8f4ce1d161df544b68beadff841","abstract_canon_sha256":"5415736b2901717f8ac4f8e76bd2debbf5cc36219b1822611b3d18ef2072e9d5"},"schema_version":"1.0"},"canonical_sha256":"c0348cb2ebb530fcd0d796a29aa5ad7dc39847ea314e6def60fa7dff9f54f883","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:02:55.016902Z","signature_b64":"TmZDPCtSlkO2osv2bsmLJBTssR2aZBGq0/exP2eHtmSWK/pvWRxuT8UgcfrN89qKY8WJ19vyzWv8SNGoIbiGCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c0348cb2ebb530fcd0d796a29aa5ad7dc39847ea314e6def60fa7dff9f54f883","last_reissued_at":"2026-05-18T04:02:55.016257Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:02:55.016257Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1202.1146","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":"2026-05-18T04:02:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BuDMRCFfw0xN60ly5Z37mHkEc3URNQWOgpuXI/vbcNruaUMiFpBW54z8TQcOsdJ7QXEAMILTLjO0DadHTCHmAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T01:41:23.849932Z"},"content_sha256":"ed9e5f38c9f7a417b7e7e044af3c5a5ba4c3c79f1840d9badc266ca3d11cda6b","schema_version":"1.0","event_id":"sha256:ed9e5f38c9f7a417b7e7e044af3c5a5ba4c3c79f1840d9badc266ca3d11cda6b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:YA2IZMXLWUYPZUGXS2RJVJNNPX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On dynamic monopolies of graphs: the average and strict majority thresholds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Hossein Soltani, Kaveh Khoshkhah, Manouchehr Zaker","submitted_at":"2012-02-06T14:17:57Z","abstract_excerpt":"Let $G$ be a graph and ${\\mathcal{\\tau}}: V(G)\\rightarrow \\Bbb{N}\\cup \\{0\\}$ be an assignment of thresholds to the vertices of $G$. A subset of vertices $D$ is said to be a dynamic monopoly corresponding to $(G, \\tau)$ if the vertices of $G$ can be partitioned into subsets $D_0, D_1,..., D_k$ such that $D_0=D$ and for any $i\\in {0, ..., k-1}$, each vertex $v$ in $D_{i+1}$ has at least $\\tau(v)$ neighbors in $D_0\\cup ... \\cup D_i$. Dynamic monopolies are in fact modeling the irreversible spread of influence in social networks. In this paper we first obtain a lower bound for the smallest size of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.1146","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":"2026-05-18T04:02:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1aK+mMBCQ/jjLWYaU/HlNHz2mwqcqiD3BxBEAvtGF7pzf+TLn4ShTpuV1cumYy65gANIpdOCx2ikIsT7cep2Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T01:41:23.850285Z"},"content_sha256":"ad55570c90118bb2a1a3e27cc31144f041110be21eee4dd65c6c4c09830c585b","schema_version":"1.0","event_id":"sha256:ad55570c90118bb2a1a3e27cc31144f041110be21eee4dd65c6c4c09830c585b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YA2IZMXLWUYPZUGXS2RJVJNNPX/bundle.json","state_url":"https://pith.science/pith/YA2IZMXLWUYPZUGXS2RJVJNNPX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YA2IZMXLWUYPZUGXS2RJVJNNPX/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":"2026-06-04T01:41:23Z","links":{"resolver":"https://pith.science/pith/YA2IZMXLWUYPZUGXS2RJVJNNPX","bundle":"https://pith.science/pith/YA2IZMXLWUYPZUGXS2RJVJNNPX/bundle.json","state":"https://pith.science/pith/YA2IZMXLWUYPZUGXS2RJVJNNPX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YA2IZMXLWUYPZUGXS2RJVJNNPX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:YA2IZMXLWUYPZUGXS2RJVJNNPX","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":"5415736b2901717f8ac4f8e76bd2debbf5cc36219b1822611b3d18ef2072e9d5","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-02-06T14:17:57Z","title_canon_sha256":"59c3b51dccc7736238d5d5db55393f1018bef8f4ce1d161df544b68beadff841"},"schema_version":"1.0","source":{"id":"1202.1146","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.1146","created_at":"2026-05-18T04:02:55Z"},{"alias_kind":"arxiv_version","alias_value":"1202.1146v1","created_at":"2026-05-18T04:02:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.1146","created_at":"2026-05-18T04:02:55Z"},{"alias_kind":"pith_short_12","alias_value":"YA2IZMXLWUYP","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"YA2IZMXLWUYPZUGX","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"YA2IZMXL","created_at":"2026-05-18T12:27:27Z"}],"graph_snapshots":[{"event_id":"sha256:ad55570c90118bb2a1a3e27cc31144f041110be21eee4dd65c6c4c09830c585b","target":"graph","created_at":"2026-05-18T04:02:55Z","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":"Let $G$ be a graph and ${\\mathcal{\\tau}}: V(G)\\rightarrow \\Bbb{N}\\cup \\{0\\}$ be an assignment of thresholds to the vertices of $G$. A subset of vertices $D$ is said to be a dynamic monopoly corresponding to $(G, \\tau)$ if the vertices of $G$ can be partitioned into subsets $D_0, D_1,..., D_k$ such that $D_0=D$ and for any $i\\in {0, ..., k-1}$, each vertex $v$ in $D_{i+1}$ has at least $\\tau(v)$ neighbors in $D_0\\cup ... \\cup D_i$. Dynamic monopolies are in fact modeling the irreversible spread of influence in social networks. In this paper we first obtain a lower bound for the smallest size of","authors_text":"Hossein Soltani, Kaveh Khoshkhah, Manouchehr Zaker","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-02-06T14:17:57Z","title":"On dynamic monopolies of graphs: the average and strict majority thresholds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.1146","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:ed9e5f38c9f7a417b7e7e044af3c5a5ba4c3c79f1840d9badc266ca3d11cda6b","target":"record","created_at":"2026-05-18T04:02:55Z","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":"5415736b2901717f8ac4f8e76bd2debbf5cc36219b1822611b3d18ef2072e9d5","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-02-06T14:17:57Z","title_canon_sha256":"59c3b51dccc7736238d5d5db55393f1018bef8f4ce1d161df544b68beadff841"},"schema_version":"1.0","source":{"id":"1202.1146","kind":"arxiv","version":1}},"canonical_sha256":"c0348cb2ebb530fcd0d796a29aa5ad7dc39847ea314e6def60fa7dff9f54f883","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c0348cb2ebb530fcd0d796a29aa5ad7dc39847ea314e6def60fa7dff9f54f883","first_computed_at":"2026-05-18T04:02:55.016257Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:02:55.016257Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TmZDPCtSlkO2osv2bsmLJBTssR2aZBGq0/exP2eHtmSWK/pvWRxuT8UgcfrN89qKY8WJ19vyzWv8SNGoIbiGCw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:02:55.016902Z","signed_message":"canonical_sha256_bytes"},"source_id":"1202.1146","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ed9e5f38c9f7a417b7e7e044af3c5a5ba4c3c79f1840d9badc266ca3d11cda6b","sha256:ad55570c90118bb2a1a3e27cc31144f041110be21eee4dd65c6c4c09830c585b"],"state_sha256":"68f7c13ac3ca7750c48dfa895a89dc1d5382413656900d438bd002545e5add83"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xQr+dguubr62QJgC4P6teThz4kMwUpLrEIk5z8SoHSWG9WBpbKlXpqxddcL4DZPnO9cERupCNceACW3ytPs6CQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T01:41:23.852445Z","bundle_sha256":"72987386842a6777f5736eb61820b4624d8cc16a0c43371a36d7e683be69d0e0"}}