{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:IJEQK6LO4DMZ2Z5V5RTCV7FANX","short_pith_number":"pith:IJEQK6LO","canonical_record":{"source":{"id":"1011.4394","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"physics.data-an","submitted_at":"2010-11-19T11:06:02Z","cross_cats_sorted":["cond-mat.dis-nn","cs.SI","nlin.AO","physics.soc-ph"],"title_canon_sha256":"3b7d8976a471cd63bcb4fcb027d2d9c6b9c53a0a5a4c745a9d6137f88b74a2df","abstract_canon_sha256":"2222486237072139295d85ad1a764208d6db0c56655caf78d1cd7eeb9f44d92d"},"schema_version":"1.0"},"canonical_sha256":"424905796ee0d99d67b5ec662afca06de32dad2015d6c1c636feb59d88d3c426","source":{"kind":"arxiv","id":"1011.4394","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.4394","created_at":"2026-05-18T04:26:02Z"},{"alias_kind":"arxiv_version","alias_value":"1011.4394v2","created_at":"2026-05-18T04:26:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.4394","created_at":"2026-05-18T04:26:02Z"},{"alias_kind":"pith_short_12","alias_value":"IJEQK6LO4DMZ","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_16","alias_value":"IJEQK6LO4DMZ2Z5V","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_8","alias_value":"IJEQK6LO","created_at":"2026-05-18T12:26:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:IJEQK6LO4DMZ2Z5V5RTCV7FANX","target":"record","payload":{"canonical_record":{"source":{"id":"1011.4394","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"physics.data-an","submitted_at":"2010-11-19T11:06:02Z","cross_cats_sorted":["cond-mat.dis-nn","cs.SI","nlin.AO","physics.soc-ph"],"title_canon_sha256":"3b7d8976a471cd63bcb4fcb027d2d9c6b9c53a0a5a4c745a9d6137f88b74a2df","abstract_canon_sha256":"2222486237072139295d85ad1a764208d6db0c56655caf78d1cd7eeb9f44d92d"},"schema_version":"1.0"},"canonical_sha256":"424905796ee0d99d67b5ec662afca06de32dad2015d6c1c636feb59d88d3c426","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:26:02.315724Z","signature_b64":"bSN/ehNWGzi0qhK3IEv74DG/Djp0tBefCcLOR1TXsXF02NljrWxI55NEYcViVhCDIysXtpyulcT0noiawHtBAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"424905796ee0d99d67b5ec662afca06de32dad2015d6c1c636feb59d88d3c426","last_reissued_at":"2026-05-18T04:26:02.315303Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:26:02.315303Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1011.4394","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":"2026-05-18T04:26:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mczi2T7u5KAjvB1s7S6T+2IlOkFRPZGj3gVvpJms6crmWzR/KvFn/1X9u/OwteZOjWfMHvb8MpmZ4nv3zhbuCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T06:50:56.929592Z"},"content_sha256":"4e4ff51f85953393acd8c37f4a13a83a5795ceb24ce01f3b363461c42a9329fa","schema_version":"1.0","event_id":"sha256:4e4ff51f85953393acd8c37f4a13a83a5795ceb24ce01f3b363461c42a9329fa"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:IJEQK6LO4DMZ2Z5V5RTCV7FANX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Measuring the Hierarchy of Feedforward Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.dis-nn","cs.SI","nlin.AO","physics.soc-ph"],"primary_cat":"physics.data-an","authors_text":"Bernat Corominas-Murtra, Carlos Rodr\\'iguez-Caso, Joaqu\\'in Go\\~ni, Ricard Sol\\'e","submitted_at":"2010-11-19T11:06:02Z","abstract_excerpt":"In this paper we explore the concept of hierarchy as a quantifiable descriptor of ordered structures, departing from the definition of three conditions to be satisfied for a hierarchical structure: {\\em order}, {\\em predictability} and {\\em pyramidal structure}. According to these principles we define a hierarchical index taking concepts from graph and information theory. This estimator allows to quantify the hierarchical character of any system susceptible to be abstracted in a feedforward causal graph, i.e., a directed acyclic graph defined in a single connected structure. Our hierarchical i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.4394","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":"2026-05-18T04:26:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NHUMMtkKsuunMI/l4bmcXoeo4BaPIqBkmEdbzN/6vQwXZ9ft+Ucb57YMSzPgNiPY7teR7elDnQsdFbqmn9nVAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T06:50:56.930349Z"},"content_sha256":"ad1a044fdf222c39b7dd99c61c0b89ff5cb47b071a65d9a2459f20c7a9ae4168","schema_version":"1.0","event_id":"sha256:ad1a044fdf222c39b7dd99c61c0b89ff5cb47b071a65d9a2459f20c7a9ae4168"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IJEQK6LO4DMZ2Z5V5RTCV7FANX/bundle.json","state_url":"https://pith.science/pith/IJEQK6LO4DMZ2Z5V5RTCV7FANX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IJEQK6LO4DMZ2Z5V5RTCV7FANX/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":"2026-06-12T06:50:56Z","links":{"resolver":"https://pith.science/pith/IJEQK6LO4DMZ2Z5V5RTCV7FANX","bundle":"https://pith.science/pith/IJEQK6LO4DMZ2Z5V5RTCV7FANX/bundle.json","state":"https://pith.science/pith/IJEQK6LO4DMZ2Z5V5RTCV7FANX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IJEQK6LO4DMZ2Z5V5RTCV7FANX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:IJEQK6LO4DMZ2Z5V5RTCV7FANX","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":"2222486237072139295d85ad1a764208d6db0c56655caf78d1cd7eeb9f44d92d","cross_cats_sorted":["cond-mat.dis-nn","cs.SI","nlin.AO","physics.soc-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"physics.data-an","submitted_at":"2010-11-19T11:06:02Z","title_canon_sha256":"3b7d8976a471cd63bcb4fcb027d2d9c6b9c53a0a5a4c745a9d6137f88b74a2df"},"schema_version":"1.0","source":{"id":"1011.4394","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.4394","created_at":"2026-05-18T04:26:02Z"},{"alias_kind":"arxiv_version","alias_value":"1011.4394v2","created_at":"2026-05-18T04:26:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.4394","created_at":"2026-05-18T04:26:02Z"},{"alias_kind":"pith_short_12","alias_value":"IJEQK6LO4DMZ","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_16","alias_value":"IJEQK6LO4DMZ2Z5V","created_at":"2026-05-18T12:26:09Z"},{"alias_kind":"pith_short_8","alias_value":"IJEQK6LO","created_at":"2026-05-18T12:26:09Z"}],"graph_snapshots":[{"event_id":"sha256:ad1a044fdf222c39b7dd99c61c0b89ff5cb47b071a65d9a2459f20c7a9ae4168","target":"graph","created_at":"2026-05-18T04:26:02Z","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":"In this paper we explore the concept of hierarchy as a quantifiable descriptor of ordered structures, departing from the definition of three conditions to be satisfied for a hierarchical structure: {\\em order}, {\\em predictability} and {\\em pyramidal structure}. According to these principles we define a hierarchical index taking concepts from graph and information theory. This estimator allows to quantify the hierarchical character of any system susceptible to be abstracted in a feedforward causal graph, i.e., a directed acyclic graph defined in a single connected structure. Our hierarchical i","authors_text":"Bernat Corominas-Murtra, Carlos Rodr\\'iguez-Caso, Joaqu\\'in Go\\~ni, Ricard Sol\\'e","cross_cats":["cond-mat.dis-nn","cs.SI","nlin.AO","physics.soc-ph"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"physics.data-an","submitted_at":"2010-11-19T11:06:02Z","title":"Measuring the Hierarchy of Feedforward Networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.4394","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":"sha256:4e4ff51f85953393acd8c37f4a13a83a5795ceb24ce01f3b363461c42a9329fa","target":"record","created_at":"2026-05-18T04:26:02Z","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":"2222486237072139295d85ad1a764208d6db0c56655caf78d1cd7eeb9f44d92d","cross_cats_sorted":["cond-mat.dis-nn","cs.SI","nlin.AO","physics.soc-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"physics.data-an","submitted_at":"2010-11-19T11:06:02Z","title_canon_sha256":"3b7d8976a471cd63bcb4fcb027d2d9c6b9c53a0a5a4c745a9d6137f88b74a2df"},"schema_version":"1.0","source":{"id":"1011.4394","kind":"arxiv","version":2}},"canonical_sha256":"424905796ee0d99d67b5ec662afca06de32dad2015d6c1c636feb59d88d3c426","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"424905796ee0d99d67b5ec662afca06de32dad2015d6c1c636feb59d88d3c426","first_computed_at":"2026-05-18T04:26:02.315303Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:26:02.315303Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bSN/ehNWGzi0qhK3IEv74DG/Djp0tBefCcLOR1TXsXF02NljrWxI55NEYcViVhCDIysXtpyulcT0noiawHtBAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:26:02.315724Z","signed_message":"canonical_sha256_bytes"},"source_id":"1011.4394","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4e4ff51f85953393acd8c37f4a13a83a5795ceb24ce01f3b363461c42a9329fa","sha256:ad1a044fdf222c39b7dd99c61c0b89ff5cb47b071a65d9a2459f20c7a9ae4168"],"state_sha256":"5d4c95fa3d7b2dda2789d3f712a64887229752ca0054f96ff591e19146704b5b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FuNMf2sqsnzDURWgVvi497w+2vbSTLQAlTTnu+v9rx3rqlpGBm/ni0ueXaMwwtnC4wqNsj49d5fruZvB6w36CA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T06:50:56.934300Z","bundle_sha256":"d51ff3a541de788e0d6e1e6e7a9656c76c7f5428d990ae2760aadbaee8502948"}}