{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2023:MZESI6DWX5FHGZRE6OIAQYYAKE","short_pith_number":"pith:MZESI6DW","canonical_record":{"source":{"id":"2303.00724","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2023-03-01T18:29:03Z","cross_cats_sorted":[],"title_canon_sha256":"ffc572d6dae8d0ad6a3deb8a6a1018598869e9734b5847e5f9c617c44a4bccb2","abstract_canon_sha256":"f01d85e48b5706ac5ebd7ed73d807973d744e07773d2e0a8073a5dfed303f679"},"schema_version":"1.0"},"canonical_sha256":"6649247876bf4a736624f39008630051180e675e26571fa220cc01b66c15e61b","source":{"kind":"arxiv","id":"2303.00724","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2303.00724","created_at":"2026-07-05T09:21:35Z"},{"alias_kind":"arxiv_version","alias_value":"2303.00724v3","created_at":"2026-07-05T09:21:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2303.00724","created_at":"2026-07-05T09:21:35Z"},{"alias_kind":"pith_short_12","alias_value":"MZESI6DWX5FH","created_at":"2026-07-05T09:21:35Z"},{"alias_kind":"pith_short_16","alias_value":"MZESI6DWX5FHGZRE","created_at":"2026-07-05T09:21:35Z"},{"alias_kind":"pith_short_8","alias_value":"MZESI6DW","created_at":"2026-07-05T09:21:35Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2023:MZESI6DWX5FHGZRE6OIAQYYAKE","target":"record","payload":{"canonical_record":{"source":{"id":"2303.00724","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2023-03-01T18:29:03Z","cross_cats_sorted":[],"title_canon_sha256":"ffc572d6dae8d0ad6a3deb8a6a1018598869e9734b5847e5f9c617c44a4bccb2","abstract_canon_sha256":"f01d85e48b5706ac5ebd7ed73d807973d744e07773d2e0a8073a5dfed303f679"},"schema_version":"1.0"},"canonical_sha256":"6649247876bf4a736624f39008630051180e675e26571fa220cc01b66c15e61b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:21:35.825324Z","signature_b64":"F4cY4FXZRJrHr5YGAK7p5YJOM3msuMQYrl79PhysnpbVYWh7O0k7sbbJKerPFFSlrloN+la6W+DChytPA8JTDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6649247876bf4a736624f39008630051180e675e26571fa220cc01b66c15e61b","last_reissued_at":"2026-07-05T09:21:35.824855Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:21:35.824855Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2303.00724","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":"2026-07-05T09:21:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"52QXeou8OYggdo0RO04vEAy94SoqRrWG6K6z9IHxR1iB4ciIB6lFLg4COO6Bo26nUmsQIEl9kGO2SZRP5lLICQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T22:10:59.464939Z"},"content_sha256":"2b26dc6958cd979163124d703942a2250733089c687d3263cb3ead6083f70439","schema_version":"1.0","event_id":"sha256:2b26dc6958cd979163124d703942a2250733089c687d3263cb3ead6083f70439"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2023:MZESI6DWX5FHGZRE6OIAQYYAKE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Cluster-size decay in supercritical kernel-based spatial random graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Dieter Mitsche, Joost Jorritsma, J\\'ulia Komj\\'athy","submitted_at":"2023-03-01T18:29:03Z","abstract_excerpt":"We consider a large class of spatially-embedded random graphs that includes among others long-range percolation, continuum scale-free percolation and the age-dependent random connection model. We assume that the model is supercritical: there is an infinite component. We identify the stretch-exponent $\\zeta\\in(0,1)$ of the decay of the cluster-size distribution. That is, with $|\\mathcal{C}(0)|$ denoting the number of vertices in the component of the vertex at $0\\in \\mathbb{R}^d$, we prove \\[\n  \\mathbb{P}(k< |\\mathcal{C}(0)|<\\infty)=\\exp\\big(-\\Theta(k^{\\zeta})\\big), \\qquad \\text{as }k\\to\\infty.\n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2303.00724","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2303.00724/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":"2026-07-05T09:21:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QuaWhgSribYzb8XPSbPE80Gawzq5dgYtYFqOgpDy0ZQXikXEueH8r55UINzbMXz+EBDS1vZ6xgn54C5gEMwICA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-09T22:10:59.465444Z"},"content_sha256":"fb648da02bc732f3b1036d929beecc8ded7b039f296474f493c03534d82516a1","schema_version":"1.0","event_id":"sha256:fb648da02bc732f3b1036d929beecc8ded7b039f296474f493c03534d82516a1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MZESI6DWX5FHGZRE6OIAQYYAKE/bundle.json","state_url":"https://pith.science/pith/MZESI6DWX5FHGZRE6OIAQYYAKE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MZESI6DWX5FHGZRE6OIAQYYAKE/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":"2026-08-09T22:10:59Z","links":{"resolver":"https://pith.science/pith/MZESI6DWX5FHGZRE6OIAQYYAKE","bundle":"https://pith.science/pith/MZESI6DWX5FHGZRE6OIAQYYAKE/bundle.json","state":"https://pith.science/pith/MZESI6DWX5FHGZRE6OIAQYYAKE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MZESI6DWX5FHGZRE6OIAQYYAKE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:MZESI6DWX5FHGZRE6OIAQYYAKE","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":"f01d85e48b5706ac5ebd7ed73d807973d744e07773d2e0a8073a5dfed303f679","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2023-03-01T18:29:03Z","title_canon_sha256":"ffc572d6dae8d0ad6a3deb8a6a1018598869e9734b5847e5f9c617c44a4bccb2"},"schema_version":"1.0","source":{"id":"2303.00724","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2303.00724","created_at":"2026-07-05T09:21:35Z"},{"alias_kind":"arxiv_version","alias_value":"2303.00724v3","created_at":"2026-07-05T09:21:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2303.00724","created_at":"2026-07-05T09:21:35Z"},{"alias_kind":"pith_short_12","alias_value":"MZESI6DWX5FH","created_at":"2026-07-05T09:21:35Z"},{"alias_kind":"pith_short_16","alias_value":"MZESI6DWX5FHGZRE","created_at":"2026-07-05T09:21:35Z"},{"alias_kind":"pith_short_8","alias_value":"MZESI6DW","created_at":"2026-07-05T09:21:35Z"}],"graph_snapshots":[{"event_id":"sha256:fb648da02bc732f3b1036d929beecc8ded7b039f296474f493c03534d82516a1","target":"graph","created_at":"2026-07-05T09:21:35Z","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/2303.00724/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We consider a large class of spatially-embedded random graphs that includes among others long-range percolation, continuum scale-free percolation and the age-dependent random connection model. We assume that the model is supercritical: there is an infinite component. We identify the stretch-exponent $\\zeta\\in(0,1)$ of the decay of the cluster-size distribution. That is, with $|\\mathcal{C}(0)|$ denoting the number of vertices in the component of the vertex at $0\\in \\mathbb{R}^d$, we prove \\[\n  \\mathbb{P}(k< |\\mathcal{C}(0)|<\\infty)=\\exp\\big(-\\Theta(k^{\\zeta})\\big), \\qquad \\text{as }k\\to\\infty.\n","authors_text":"Dieter Mitsche, Joost Jorritsma, J\\'ulia Komj\\'athy","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2023-03-01T18:29:03Z","title":"Cluster-size decay in supercritical kernel-based spatial random graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2303.00724","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":"sha256:2b26dc6958cd979163124d703942a2250733089c687d3263cb3ead6083f70439","target":"record","created_at":"2026-07-05T09:21:35Z","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":"f01d85e48b5706ac5ebd7ed73d807973d744e07773d2e0a8073a5dfed303f679","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2023-03-01T18:29:03Z","title_canon_sha256":"ffc572d6dae8d0ad6a3deb8a6a1018598869e9734b5847e5f9c617c44a4bccb2"},"schema_version":"1.0","source":{"id":"2303.00724","kind":"arxiv","version":3}},"canonical_sha256":"6649247876bf4a736624f39008630051180e675e26571fa220cc01b66c15e61b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6649247876bf4a736624f39008630051180e675e26571fa220cc01b66c15e61b","first_computed_at":"2026-07-05T09:21:35.824855Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:21:35.824855Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"F4cY4FXZRJrHr5YGAK7p5YJOM3msuMQYrl79PhysnpbVYWh7O0k7sbbJKerPFFSlrloN+la6W+DChytPA8JTDA==","signature_status":"signed_v1","signed_at":"2026-07-05T09:21:35.825324Z","signed_message":"canonical_sha256_bytes"},"source_id":"2303.00724","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2b26dc6958cd979163124d703942a2250733089c687d3263cb3ead6083f70439","sha256:fb648da02bc732f3b1036d929beecc8ded7b039f296474f493c03534d82516a1"],"state_sha256":"b10855e3dcf0e64048e29c0482e059ec328beeee7206b567a54da9f1ec1770bb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IZPQaYzR5jsVRXGwBCmSOk2IlfRUN9Czxh/nOjDYxuIJZVg4HEbknwaLuFG54Coxezm2n6bhkYaMcyRlHq8mAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-09T22:10:59.470464Z","bundle_sha256":"2c6fe3c33c3599146cb74a8ac90f1772b707f0c8109f6d6c6881630aef63d085"}}