{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:NSUQ6D54FFBH5CMLYEF2BP3TJC","short_pith_number":"pith:NSUQ6D54","schema_version":"1.0","canonical_sha256":"6ca90f0fbc29427e898bc10ba0bf73488cece8721dd1c311c277dfe8f06697e7","source":{"kind":"arxiv","id":"1506.00508","version":3},"attestation_state":"computed","paper":{"title":"Isomorphisms in Multilayer Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","cs.SI","math.CO"],"primary_cat":"physics.soc-ph","authors_text":"Mason A. Porter, Mikko Kivel\\\"a","submitted_at":"2015-06-01T14:28:03Z","abstract_excerpt":"We extend the concept of graph isomorphisms to multilayer networks with any number of \"aspects\" (i.e., types of layering). In developing this generalization, we identify multiple types of isomorphisms. For example, in multilayer networks with a single aspect, permuting vertex labels, layer labels, and both vertex labels and layer labels each yield different isomorphism relations between multilayer networks. Multilayer network isomorphisms lead naturally to defining isomorphisms in any of the numerous types of networks that can be represented as a multilayer network, and we thereby obtain isomo"},"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":"1506.00508","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"physics.soc-ph","submitted_at":"2015-06-01T14:28:03Z","cross_cats_sorted":["cs.DM","cs.SI","math.CO"],"title_canon_sha256":"1cb4eec988a4f714ad36ab9b15e4dbddce243bd9d99d72cfbe7c7c8f33ec389f","abstract_canon_sha256":"a51556017cf8fb1e3247751a22a9ef00a88166768048a1cc840f2223f6194bdb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:50:40.603941Z","signature_b64":"PuOjxTZXgpJyX+XRZuElkSK24Qi07AURfH03RmheZMhPF4TnxBOyeeBkOKNahgpIlVhqPenuZKto6eTBPQ/nBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6ca90f0fbc29427e898bc10ba0bf73488cece8721dd1c311c277dfe8f06697e7","last_reissued_at":"2026-05-18T00:50:40.603255Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:50:40.603255Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Isomorphisms in Multilayer Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","cs.SI","math.CO"],"primary_cat":"physics.soc-ph","authors_text":"Mason A. Porter, Mikko Kivel\\\"a","submitted_at":"2015-06-01T14:28:03Z","abstract_excerpt":"We extend the concept of graph isomorphisms to multilayer networks with any number of \"aspects\" (i.e., types of layering). In developing this generalization, we identify multiple types of isomorphisms. For example, in multilayer networks with a single aspect, permuting vertex labels, layer labels, and both vertex labels and layer labels each yield different isomorphism relations between multilayer networks. Multilayer network isomorphisms lead naturally to defining isomorphisms in any of the numerous types of networks that can be represented as a multilayer network, and we thereby obtain isomo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.00508","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":""},"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":"1506.00508","created_at":"2026-05-18T00:50:40.603348+00:00"},{"alias_kind":"arxiv_version","alias_value":"1506.00508v3","created_at":"2026-05-18T00:50:40.603348+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.00508","created_at":"2026-05-18T00:50:40.603348+00:00"},{"alias_kind":"pith_short_12","alias_value":"NSUQ6D54FFBH","created_at":"2026-05-18T12:29:34.919912+00:00"},{"alias_kind":"pith_short_16","alias_value":"NSUQ6D54FFBH5CML","created_at":"2026-05-18T12:29:34.919912+00:00"},{"alias_kind":"pith_short_8","alias_value":"NSUQ6D54","created_at":"2026-05-18T12:29:34.919912+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/NSUQ6D54FFBH5CMLYEF2BP3TJC","json":"https://pith.science/pith/NSUQ6D54FFBH5CMLYEF2BP3TJC.json","graph_json":"https://pith.science/api/pith-number/NSUQ6D54FFBH5CMLYEF2BP3TJC/graph.json","events_json":"https://pith.science/api/pith-number/NSUQ6D54FFBH5CMLYEF2BP3TJC/events.json","paper":"https://pith.science/paper/NSUQ6D54"},"agent_actions":{"view_html":"https://pith.science/pith/NSUQ6D54FFBH5CMLYEF2BP3TJC","download_json":"https://pith.science/pith/NSUQ6D54FFBH5CMLYEF2BP3TJC.json","view_paper":"https://pith.science/paper/NSUQ6D54","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1506.00508&json=true","fetch_graph":"https://pith.science/api/pith-number/NSUQ6D54FFBH5CMLYEF2BP3TJC/graph.json","fetch_events":"https://pith.science/api/pith-number/NSUQ6D54FFBH5CMLYEF2BP3TJC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NSUQ6D54FFBH5CMLYEF2BP3TJC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NSUQ6D54FFBH5CMLYEF2BP3TJC/action/storage_attestation","attest_author":"https://pith.science/pith/NSUQ6D54FFBH5CMLYEF2BP3TJC/action/author_attestation","sign_citation":"https://pith.science/pith/NSUQ6D54FFBH5CMLYEF2BP3TJC/action/citation_signature","submit_replication":"https://pith.science/pith/NSUQ6D54FFBH5CMLYEF2BP3TJC/action/replication_record"}},"created_at":"2026-05-18T00:50:40.603348+00:00","updated_at":"2026-05-18T00:50:40.603348+00:00"}