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In our model we assume that each node in one network can function only if it has at least a single support node in the other network. If both networks A and B are Erd\\H{o}s-R\\'enyi networks, A and B, with (i) sizes $N^A$ and $N^B$, (ii) average degrees $a$ and $b$, and (iii) $c^{AB}_0N^B$ support links from network A to B and $c^{BA}_0N^B$ support links from network B to A, we find that unde"},"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":"1011.0234","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"physics.data-an","submitted_at":"2010-11-01T01:24:47Z","cross_cats_sorted":["cs.SI","nlin.CD","physics.soc-ph"],"title_canon_sha256":"261b3a109f2af4a183d9c40a4781e4b02923b74d61ad733dbdcae00a74e02dad","abstract_canon_sha256":"99172103fd3cb44556f553ef632fa3ad23a17d9ccf8faaa3e5a6045b58f4f6a0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:04:54.306105Z","signature_b64":"BOPhEQSRIkqoF1gbI/UaRiP+295hqDeL99oIZiXAIgeHWtknZ+hPtmnbAcbwqYdG1fgy3rtjv6SEkjRV37PtAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5b5d2bda600c028f186cbc615ecc77160a6d6eb8b226f016dd1dd03469ca8fbe","last_reissued_at":"2026-05-18T02:04:54.305470Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:04:54.305470Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cascade of failures in coupled network systems with multiple support-dependent relations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.SI","nlin.CD","physics.soc-ph"],"primary_cat":"physics.data-an","authors_text":"H. 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