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When $q_{ab}$ is constant in $a,b$, the random graph is simply the Erd\\H{o}s-R\\'{e}ny random graph. We evaluate the log partition function of the Ising model on sparse SBM with two communities.\n  As an application, we give consistent parameter estimation of the sparse SBM with two communities in a special case. More specifically, let $d_0,d_1$ be the average degree of the tw"},"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":"1710.05287","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ME","submitted_at":"2017-10-15T06:46:37Z","cross_cats_sorted":[],"title_canon_sha256":"2a65f3179f345bfbf301d2f1953bc1c9e7af895349138735b548c8a213be13c6","abstract_canon_sha256":"12c011fcbf59296beed4b59789b1af8524c9777d25c80315a2077ea3a70b6b06"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:32:53.744639Z","signature_b64":"KP7s2dzGEiU51nJAZ5UVASEjeyzKLWmoggeMHBEvcAht69FuX6gigjaDH1/HMviiENjYC5rJWfUuujcXNhD5Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ece12576073a51f79349291c3fac60d16b9d014bdf86aa8e18a7f643081af432","last_reissued_at":"2026-05-18T00:32:53.743989Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:32:53.743989Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Log Partition Function of Ising Model on Stochastic Block Model","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"stat.ME","authors_text":"Lu Liu","submitted_at":"2017-10-15T06:46:37Z","abstract_excerpt":"A sparse stochastic block model (SBM) with two communities is defined by the community probability $\\pi_0,\\pi_1$, and the connection probability between communities $a,b\\in\\{0,1\\}$, namely $q_{ab} = \\frac{\\alpha_{ab}}{n}$. 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