{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:RSHDU3TMLCB6AVCUNOY4HVGGME","short_pith_number":"pith:RSHDU3TM","canonical_record":{"source":{"id":"1607.06042","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-07-20T17:44:49Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"342e9001179708dc128691e5bfed05e16cc238915c85825d8dee9fa24af8beaf","abstract_canon_sha256":"547ecc282e25c296ff71c15ac162730de4a112b8276e3c0e32d2b924537cf521"},"schema_version":"1.0"},"canonical_sha256":"8c8e3a6e6c5883e054546bb1c3d4c6613b52d4860670ec7099a6a098df554766","source":{"kind":"arxiv","id":"1607.06042","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.06042","created_at":"2026-05-18T00:31:31Z"},{"alias_kind":"arxiv_version","alias_value":"1607.06042v1","created_at":"2026-05-18T00:31:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.06042","created_at":"2026-05-18T00:31:31Z"},{"alias_kind":"pith_short_12","alias_value":"RSHDU3TMLCB6","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_16","alias_value":"RSHDU3TMLCB6AVCU","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_8","alias_value":"RSHDU3TM","created_at":"2026-05-18T12:30:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:RSHDU3TMLCB6AVCUNOY4HVGGME","target":"record","payload":{"canonical_record":{"source":{"id":"1607.06042","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-07-20T17:44:49Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"342e9001179708dc128691e5bfed05e16cc238915c85825d8dee9fa24af8beaf","abstract_canon_sha256":"547ecc282e25c296ff71c15ac162730de4a112b8276e3c0e32d2b924537cf521"},"schema_version":"1.0"},"canonical_sha256":"8c8e3a6e6c5883e054546bb1c3d4c6613b52d4860670ec7099a6a098df554766","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:31:31.621572Z","signature_b64":"FYC1WFdhkwsQNpjG1hO4+0truRIwMcbitFo7zNMT/dp01GaIEYVo/m4YToFk/Pn/B85CXht+N12QR0IE+oG2Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8c8e3a6e6c5883e054546bb1c3d4c6613b52d4860670ec7099a6a098df554766","last_reissued_at":"2026-05-18T00:31:31.620928Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:31:31.620928Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1607.06042","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":"2026-05-18T00:31:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XElBip1NwQ5rTyIQtaOfJ8ENCosstkjHNe3dJu1O/IXfeGQyIqpQxoiOnrNMvfqb7x5PwR0s4TlU1NNOLPoUAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T16:14:54.839404Z"},"content_sha256":"c3e65d82b1c75e08592ce6eb32943dd5f73cb04b40cb5a8cc89ad039032beb02","schema_version":"1.0","event_id":"sha256:c3e65d82b1c75e08592ce6eb32943dd5f73cb04b40cb5a8cc89ad039032beb02"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:RSHDU3TMLCB6AVCUNOY4HVGGME","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The DoF of Two-way Butterfly Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Mehdi Ashraphijuo, Vaneet Aggarwal, Xiaodong Wang","submitted_at":"2016-07-20T17:44:49Z","abstract_excerpt":"This paper studies the two-way butterfly network, a class of two-way four-unicast networks. We first show that bidirectional links do not increase the degrees of freedom for this network thus giving the first example for networks, to the best of our knowledge, where bidirectional links do not increase the degrees of freedom. Further, we see that sufficient caching at the relays or increasing the number of antennas in the relays can double the two-way degrees of freedom for butterfly network."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.06042","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":"2026-05-18T00:31:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"q+hpOpmzxcOU61tkuyc358S7U05s6UdS9K0umWtKAp3A82iE8cg6i8ePfUmtffqmz7L+JmxTqgmpUMzYOalQDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T16:14:54.841747Z"},"content_sha256":"74ad5c57a1d1bf8f4ea936d1f6ed2a2e01d47e938ca9691c12e25c264ed46645","schema_version":"1.0","event_id":"sha256:74ad5c57a1d1bf8f4ea936d1f6ed2a2e01d47e938ca9691c12e25c264ed46645"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RSHDU3TMLCB6AVCUNOY4HVGGME/bundle.json","state_url":"https://pith.science/pith/RSHDU3TMLCB6AVCUNOY4HVGGME/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RSHDU3TMLCB6AVCUNOY4HVGGME/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":"2026-06-07T16:14:54Z","links":{"resolver":"https://pith.science/pith/RSHDU3TMLCB6AVCUNOY4HVGGME","bundle":"https://pith.science/pith/RSHDU3TMLCB6AVCUNOY4HVGGME/bundle.json","state":"https://pith.science/pith/RSHDU3TMLCB6AVCUNOY4HVGGME/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RSHDU3TMLCB6AVCUNOY4HVGGME/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:RSHDU3TMLCB6AVCUNOY4HVGGME","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":"547ecc282e25c296ff71c15ac162730de4a112b8276e3c0e32d2b924537cf521","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-07-20T17:44:49Z","title_canon_sha256":"342e9001179708dc128691e5bfed05e16cc238915c85825d8dee9fa24af8beaf"},"schema_version":"1.0","source":{"id":"1607.06042","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.06042","created_at":"2026-05-18T00:31:31Z"},{"alias_kind":"arxiv_version","alias_value":"1607.06042v1","created_at":"2026-05-18T00:31:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.06042","created_at":"2026-05-18T00:31:31Z"},{"alias_kind":"pith_short_12","alias_value":"RSHDU3TMLCB6","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_16","alias_value":"RSHDU3TMLCB6AVCU","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_8","alias_value":"RSHDU3TM","created_at":"2026-05-18T12:30:41Z"}],"graph_snapshots":[{"event_id":"sha256:74ad5c57a1d1bf8f4ea936d1f6ed2a2e01d47e938ca9691c12e25c264ed46645","target":"graph","created_at":"2026-05-18T00:31:31Z","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":"This paper studies the two-way butterfly network, a class of two-way four-unicast networks. We first show that bidirectional links do not increase the degrees of freedom for this network thus giving the first example for networks, to the best of our knowledge, where bidirectional links do not increase the degrees of freedom. Further, we see that sufficient caching at the relays or increasing the number of antennas in the relays can double the two-way degrees of freedom for butterfly network.","authors_text":"Mehdi Ashraphijuo, Vaneet Aggarwal, Xiaodong Wang","cross_cats":["math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-07-20T17:44:49Z","title":"The DoF of Two-way Butterfly Networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.06042","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:c3e65d82b1c75e08592ce6eb32943dd5f73cb04b40cb5a8cc89ad039032beb02","target":"record","created_at":"2026-05-18T00:31:31Z","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":"547ecc282e25c296ff71c15ac162730de4a112b8276e3c0e32d2b924537cf521","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-07-20T17:44:49Z","title_canon_sha256":"342e9001179708dc128691e5bfed05e16cc238915c85825d8dee9fa24af8beaf"},"schema_version":"1.0","source":{"id":"1607.06042","kind":"arxiv","version":1}},"canonical_sha256":"8c8e3a6e6c5883e054546bb1c3d4c6613b52d4860670ec7099a6a098df554766","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8c8e3a6e6c5883e054546bb1c3d4c6613b52d4860670ec7099a6a098df554766","first_computed_at":"2026-05-18T00:31:31.620928Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:31:31.620928Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FYC1WFdhkwsQNpjG1hO4+0truRIwMcbitFo7zNMT/dp01GaIEYVo/m4YToFk/Pn/B85CXht+N12QR0IE+oG2Aw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:31:31.621572Z","signed_message":"canonical_sha256_bytes"},"source_id":"1607.06042","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c3e65d82b1c75e08592ce6eb32943dd5f73cb04b40cb5a8cc89ad039032beb02","sha256:74ad5c57a1d1bf8f4ea936d1f6ed2a2e01d47e938ca9691c12e25c264ed46645"],"state_sha256":"8b97dc727b4b3eef22fb9f9136d58bd25fd66299d6ba1b534699ce205aa7c3c8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"igV02pPHYN2occ82JboU0m6riJqwu6VNQOeW7zDRslIiOuY5r54Gy9XMV+lUiQdzuGZb9hkACYcC88zsPEhDAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T16:14:54.845618Z","bundle_sha256":"467f28ad483ad46b5465d38661c1eea61f6aa9e144c1c5dbe7444736b4b18a4b"}}