{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:3FPVXT2PMG6CSAMEER3AES4GEK","short_pith_number":"pith:3FPVXT2P","canonical_record":{"source":{"id":"2607.10997","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"econ.TH","submitted_at":"2026-07-13T01:49:15Z","cross_cats_sorted":[],"title_canon_sha256":"cf86090ce3ee95d91d434456b7f9ae84d3367f05f1cebe4f12f9725567e2208d","abstract_canon_sha256":"c9ec46dcf2bc568072fdcd529161c42b13238a30e04fb6a9437b33decf45333e"},"schema_version":"1.0"},"canonical_sha256":"d95f5bcf4f61bc2901842476024b8622ad91a3c5fae9ed3dd7f24e1e5e3e2c89","source":{"kind":"arxiv","id":"2607.10997","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.10997","created_at":"2026-07-14T01:21:51Z"},{"alias_kind":"arxiv_version","alias_value":"2607.10997v1","created_at":"2026-07-14T01:21:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.10997","created_at":"2026-07-14T01:21:51Z"},{"alias_kind":"pith_short_12","alias_value":"3FPVXT2PMG6C","created_at":"2026-07-14T01:21:51Z"},{"alias_kind":"pith_short_16","alias_value":"3FPVXT2PMG6CSAME","created_at":"2026-07-14T01:21:51Z"},{"alias_kind":"pith_short_8","alias_value":"3FPVXT2P","created_at":"2026-07-14T01:21:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:3FPVXT2PMG6CSAMEER3AES4GEK","target":"record","payload":{"canonical_record":{"source":{"id":"2607.10997","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"econ.TH","submitted_at":"2026-07-13T01:49:15Z","cross_cats_sorted":[],"title_canon_sha256":"cf86090ce3ee95d91d434456b7f9ae84d3367f05f1cebe4f12f9725567e2208d","abstract_canon_sha256":"c9ec46dcf2bc568072fdcd529161c42b13238a30e04fb6a9437b33decf45333e"},"schema_version":"1.0"},"canonical_sha256":"d95f5bcf4f61bc2901842476024b8622ad91a3c5fae9ed3dd7f24e1e5e3e2c89","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-14T01:21:51.699848Z","signature_b64":"1isYePZaxKlg2ipNYIX72vCK+fCZEvlCuSKNhFns1aWeSlg3rv2SqJukGnTb3WM+tKLozXyDkG/gg8d8OKS5Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d95f5bcf4f61bc2901842476024b8622ad91a3c5fae9ed3dd7f24e1e5e3e2c89","last_reissued_at":"2026-07-14T01:21:51.699024Z","signature_status":"signed_v1","first_computed_at":"2026-07-14T01:21:51.699024Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2607.10997","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":"2026-07-14T01:21:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Yf9OeZjU4V9WEJY74glNn6LP1JveFU1YfZ1lwwJngAlt+/kxo8DJTeP/wm2sMS5LiX72zsSEBHp3nIEcn1BWBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-21T09:42:19.665199Z"},"content_sha256":"7ca2f545d8d2369caf242b0ddd027f67f13608a12490a56944b1747386d6d277","schema_version":"1.0","event_id":"sha256:7ca2f545d8d2369caf242b0ddd027f67f13608a12490a56944b1747386d6d277"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:3FPVXT2PMG6CSAMEER3AES4GEK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Network games with three types of players","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"econ.TH","authors_text":"Boyu Zhang, Shan Pei, Wenjie Cao","submitted_at":"2026-07-13T01:49:15Z","abstract_excerpt":"In this paper, we analyze a multi-strategy network game with three types of players, conformists, rebels, and stubborn agents. Conformists adopt the strategy that is most common among their neighbors, rebels adopt the least common, and stubborn agents adhere to a fixed strategy. We study the existence and structure of pure strategy Nash equilibrium (PNE). On arbitrary networks, we establish sufficient conditions for PNE existence, and we prove that in large random networks PNE almost surely fails to exist. For several specific network architectures, such as complete network, lines, rings, tree"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.10997","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.10997/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-14T01:21:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yRmOcENzl7fwWqwaBO/nwq5+jTJQfHBKDo1LepKUVFwlUnADuX3dUTaX4WvLhJw3lG1boUaxdQ90PSOvhcjYDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-21T09:42:19.665587Z"},"content_sha256":"dd2a53e37b72e8d1bc2ee06796d991dffc692e09b23d05c611aa877d5ce0e1d1","schema_version":"1.0","event_id":"sha256:dd2a53e37b72e8d1bc2ee06796d991dffc692e09b23d05c611aa877d5ce0e1d1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3FPVXT2PMG6CSAMEER3AES4GEK/bundle.json","state_url":"https://pith.science/pith/3FPVXT2PMG6CSAMEER3AES4GEK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3FPVXT2PMG6CSAMEER3AES4GEK/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":"2026-07-21T09:42:19Z","links":{"resolver":"https://pith.science/pith/3FPVXT2PMG6CSAMEER3AES4GEK","bundle":"https://pith.science/pith/3FPVXT2PMG6CSAMEER3AES4GEK/bundle.json","state":"https://pith.science/pith/3FPVXT2PMG6CSAMEER3AES4GEK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3FPVXT2PMG6CSAMEER3AES4GEK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:3FPVXT2PMG6CSAMEER3AES4GEK","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":"c9ec46dcf2bc568072fdcd529161c42b13238a30e04fb6a9437b33decf45333e","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"econ.TH","submitted_at":"2026-07-13T01:49:15Z","title_canon_sha256":"cf86090ce3ee95d91d434456b7f9ae84d3367f05f1cebe4f12f9725567e2208d"},"schema_version":"1.0","source":{"id":"2607.10997","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.10997","created_at":"2026-07-14T01:21:51Z"},{"alias_kind":"arxiv_version","alias_value":"2607.10997v1","created_at":"2026-07-14T01:21:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.10997","created_at":"2026-07-14T01:21:51Z"},{"alias_kind":"pith_short_12","alias_value":"3FPVXT2PMG6C","created_at":"2026-07-14T01:21:51Z"},{"alias_kind":"pith_short_16","alias_value":"3FPVXT2PMG6CSAME","created_at":"2026-07-14T01:21:51Z"},{"alias_kind":"pith_short_8","alias_value":"3FPVXT2P","created_at":"2026-07-14T01:21:51Z"}],"graph_snapshots":[{"event_id":"sha256:dd2a53e37b72e8d1bc2ee06796d991dffc692e09b23d05c611aa877d5ce0e1d1","target":"graph","created_at":"2026-07-14T01:21:51Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2607.10997/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we analyze a multi-strategy network game with three types of players, conformists, rebels, and stubborn agents. Conformists adopt the strategy that is most common among their neighbors, rebels adopt the least common, and stubborn agents adhere to a fixed strategy. We study the existence and structure of pure strategy Nash equilibrium (PNE). On arbitrary networks, we establish sufficient conditions for PNE existence, and we prove that in large random networks PNE almost surely fails to exist. For several specific network architectures, such as complete network, lines, rings, tree","authors_text":"Boyu Zhang, Shan Pei, Wenjie Cao","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"econ.TH","submitted_at":"2026-07-13T01:49:15Z","title":"Network games with three types of players"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.10997","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":"sha256:7ca2f545d8d2369caf242b0ddd027f67f13608a12490a56944b1747386d6d277","target":"record","created_at":"2026-07-14T01:21:51Z","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":"c9ec46dcf2bc568072fdcd529161c42b13238a30e04fb6a9437b33decf45333e","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"econ.TH","submitted_at":"2026-07-13T01:49:15Z","title_canon_sha256":"cf86090ce3ee95d91d434456b7f9ae84d3367f05f1cebe4f12f9725567e2208d"},"schema_version":"1.0","source":{"id":"2607.10997","kind":"arxiv","version":1}},"canonical_sha256":"d95f5bcf4f61bc2901842476024b8622ad91a3c5fae9ed3dd7f24e1e5e3e2c89","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d95f5bcf4f61bc2901842476024b8622ad91a3c5fae9ed3dd7f24e1e5e3e2c89","first_computed_at":"2026-07-14T01:21:51.699024Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-14T01:21:51.699024Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1isYePZaxKlg2ipNYIX72vCK+fCZEvlCuSKNhFns1aWeSlg3rv2SqJukGnTb3WM+tKLozXyDkG/gg8d8OKS5Cw==","signature_status":"signed_v1","signed_at":"2026-07-14T01:21:51.699848Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.10997","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7ca2f545d8d2369caf242b0ddd027f67f13608a12490a56944b1747386d6d277","sha256:dd2a53e37b72e8d1bc2ee06796d991dffc692e09b23d05c611aa877d5ce0e1d1"],"state_sha256":"0d006376716198321008b15540499e351b5997dc01d3702bf95fbc00a4b0463e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0W0n7EDv4hzsrEzMIZgA7eIva3FMTSqQzVzRAXRzSymC25m9/LN25OW9y2DdFn8rFk5n7SL3meqALqW7uLJWBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-21T09:42:19.668775Z","bundle_sha256":"61e3692530be732e08200f1337bd54709aa625901f8dab73e29434bfbffcfbb8"}}