{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:JMS2EKTKTJJO7UNP5WR4ENGTDQ","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":"4a97ecd1f896cff34c5e4b615ab13c0717eda7361abba2c680f7678e07a40950","cross_cats_sorted":["math.IT"],"license":"","primary_cat":"cs.IT","submitted_at":"2006-01-05T23:40:19Z","title_canon_sha256":"40a2e69f8724502e5c074d87bdd57f134eef323532f68d7e2f0979c7e0ab9d18"},"schema_version":"1.0","source":{"id":"cs/0601012","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"cs/0601012","created_at":"2026-05-18T00:58:26Z"},{"alias_kind":"arxiv_version","alias_value":"cs/0601012v3","created_at":"2026-05-18T00:58:26Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.cs/0601012","created_at":"2026-05-18T00:58:26Z"},{"alias_kind":"pith_short_12","alias_value":"JMS2EKTKTJJO","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"JMS2EKTKTJJO7UNP","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"JMS2EKTK","created_at":"2026-05-18T12:25:54Z"}],"graph_snapshots":[{"event_id":"sha256:9f344a0deb2a1b6a1b381448b96414cc60b0d3b8f7d4c00c79d8a47aca905f90","target":"graph","created_at":"2026-05-18T00:58:26Z","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":"We provide a tight approximate characterization of the $n$-dimensional product multicommodity flow (PMF) region for a wireless network of $n$ nodes. Separate characterizations in terms of the spectral properties of appropriate network graphs are obtained in both an information theoretic sense and for a combinatorial interference model (e.g., Protocol model). These provide an inner approximation to the $n^2$ dimensional capacity region. These results answer the following questions which arise naturally from previous work: (a) What is the significance of $1/\\sqrt{n}$ in the scaling laws for the ","authors_text":"Devavrat Shah, Olivier Leveque, Ritesh Madan","cross_cats":["math.IT"],"headline":"","license":"","primary_cat":"cs.IT","submitted_at":"2006-01-05T23:40:19Z","title":"Product Multicommodity Flow in Wireless Networks"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"cs/0601012","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:bb81cc880962d4dd439cdf518166dbee104328513b30745781825d1b27a3827e","target":"record","created_at":"2026-05-18T00:58:26Z","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":"4a97ecd1f896cff34c5e4b615ab13c0717eda7361abba2c680f7678e07a40950","cross_cats_sorted":["math.IT"],"license":"","primary_cat":"cs.IT","submitted_at":"2006-01-05T23:40:19Z","title_canon_sha256":"40a2e69f8724502e5c074d87bdd57f134eef323532f68d7e2f0979c7e0ab9d18"},"schema_version":"1.0","source":{"id":"cs/0601012","kind":"arxiv","version":3}},"canonical_sha256":"4b25a22a6a9a52efd1afeda3c234d31c3d72052b9b436eee851ace58d09cd55b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4b25a22a6a9a52efd1afeda3c234d31c3d72052b9b436eee851ace58d09cd55b","first_computed_at":"2026-05-18T00:58:26.688879Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:58:26.688879Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6oejc/0nmJrFVCXIZsOAw9fjR/AN9Odg7DbTNz2MbtMbQomR1NUi8E2XOfJVv9THGZq1zy18nx76+86NhoLVAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:58:26.689505Z","signed_message":"canonical_sha256_bytes"},"source_id":"cs/0601012","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bb81cc880962d4dd439cdf518166dbee104328513b30745781825d1b27a3827e","sha256:9f344a0deb2a1b6a1b381448b96414cc60b0d3b8f7d4c00c79d8a47aca905f90"],"state_sha256":"d80c97c18a82c11d48dbff3bb0c7d5dfd4613ed02dc77ddb06c7b27c71884c2d"}