{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:4SNQY4DNQCSX5N5CBVMVMGSWJU","short_pith_number":"pith:4SNQY4DN","schema_version":"1.0","canonical_sha256":"e49b0c706d80a57eb7a20d59561a564d309ebcfbeca2ff14c6b3e2ea6d40d1a3","source":{"kind":"arxiv","id":"1807.09846","version":1},"attestation_state":"computed","paper":{"title":"Diffusion and consensus on weakly connected directed graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM","cs.SI"],"primary_cat":"math.CO","authors_text":"E. Kummel, J.J.P. Veerman","submitted_at":"2018-07-25T20:45:02Z","abstract_excerpt":"Let $G$ be a weakly connected directed graph with asymmetric graph Laplacian ${\\cal L}$. Consensus and diffusion are dual dynamical processes defined on $G$ by $\\dot x=-{\\cal L}x$ for consensus and $\\dot p=-p{\\cal L}$ for diffusion. We consider both these processes as well their discrete time analogues. We define a basis of row vectors $\\{\\bar \\gamma_i\\}_{i=1}^k$ of the left null-space of ${\\cal L}$ and a basis of column vectors $\\{\\gamma_i\\}_{i=1}^k$ of the right null-space of ${\\cal L}$ in terms of the partition of $G$ into strongly connected components. This allows for complete characteriza"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1807.09846","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2018-07-25T20:45:02Z","cross_cats_sorted":["cs.DM","cs.SI"],"title_canon_sha256":"2e6cf1906af690221c5b2f11020e07938fd2b0ad69c09269c15def2a31686b51","abstract_canon_sha256":"dbc6b7423d2b3457d475ea239c22e9d2d4a3552880b071a2cabae5c8c3fb993a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:09:46.421462Z","signature_b64":"/DbWsTjG4q0789ptVd61xnHjdb/4d4HEA6kml1zM2KnoU9fVVVzA3kWKgDSqM0IPNVKDbU8cJ0Ht3hohTILLCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e49b0c706d80a57eb7a20d59561a564d309ebcfbeca2ff14c6b3e2ea6d40d1a3","last_reissued_at":"2026-05-18T00:09:46.420743Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:09:46.420743Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Diffusion and consensus on weakly connected directed graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM","cs.SI"],"primary_cat":"math.CO","authors_text":"E. Kummel, J.J.P. Veerman","submitted_at":"2018-07-25T20:45:02Z","abstract_excerpt":"Let $G$ be a weakly connected directed graph with asymmetric graph Laplacian ${\\cal L}$. Consensus and diffusion are dual dynamical processes defined on $G$ by $\\dot x=-{\\cal L}x$ for consensus and $\\dot p=-p{\\cal L}$ for diffusion. We consider both these processes as well their discrete time analogues. We define a basis of row vectors $\\{\\bar \\gamma_i\\}_{i=1}^k$ of the left null-space of ${\\cal L}$ and a basis of column vectors $\\{\\gamma_i\\}_{i=1}^k$ of the right null-space of ${\\cal L}$ in terms of the partition of $G$ into strongly connected components. This allows for complete characteriza"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.09846","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1807.09846","created_at":"2026-05-18T00:09:46.420858+00:00"},{"alias_kind":"arxiv_version","alias_value":"1807.09846v1","created_at":"2026-05-18T00:09:46.420858+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.09846","created_at":"2026-05-18T00:09:46.420858+00:00"},{"alias_kind":"pith_short_12","alias_value":"4SNQY4DNQCSX","created_at":"2026-05-18T12:32:05.422762+00:00"},{"alias_kind":"pith_short_16","alias_value":"4SNQY4DNQCSX5N5C","created_at":"2026-05-18T12:32:05.422762+00:00"},{"alias_kind":"pith_short_8","alias_value":"4SNQY4DN","created_at":"2026-05-18T12:32:05.422762+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/4SNQY4DNQCSX5N5CBVMVMGSWJU","json":"https://pith.science/pith/4SNQY4DNQCSX5N5CBVMVMGSWJU.json","graph_json":"https://pith.science/api/pith-number/4SNQY4DNQCSX5N5CBVMVMGSWJU/graph.json","events_json":"https://pith.science/api/pith-number/4SNQY4DNQCSX5N5CBVMVMGSWJU/events.json","paper":"https://pith.science/paper/4SNQY4DN"},"agent_actions":{"view_html":"https://pith.science/pith/4SNQY4DNQCSX5N5CBVMVMGSWJU","download_json":"https://pith.science/pith/4SNQY4DNQCSX5N5CBVMVMGSWJU.json","view_paper":"https://pith.science/paper/4SNQY4DN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1807.09846&json=true","fetch_graph":"https://pith.science/api/pith-number/4SNQY4DNQCSX5N5CBVMVMGSWJU/graph.json","fetch_events":"https://pith.science/api/pith-number/4SNQY4DNQCSX5N5CBVMVMGSWJU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4SNQY4DNQCSX5N5CBVMVMGSWJU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4SNQY4DNQCSX5N5CBVMVMGSWJU/action/storage_attestation","attest_author":"https://pith.science/pith/4SNQY4DNQCSX5N5CBVMVMGSWJU/action/author_attestation","sign_citation":"https://pith.science/pith/4SNQY4DNQCSX5N5CBVMVMGSWJU/action/citation_signature","submit_replication":"https://pith.science/pith/4SNQY4DNQCSX5N5CBVMVMGSWJU/action/replication_record"}},"created_at":"2026-05-18T00:09:46.420858+00:00","updated_at":"2026-05-18T00:09:46.420858+00:00"}