{"paper":{"title":"Subgraph counting estimation for the $\\beta$-model in sparse networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"stat.ME","authors_text":"Jiashun Jin, Qunqiang Feng, Ting Yan, Yaru Tian","submitted_at":"2026-07-06T16:20:50Z","abstract_excerpt":"The $\\beta$-model is popular for characterizing the commonly observed degree heterogeneity phenomenon in real-world networks. In this study, we develop a cycle counting approach to estimate $n$ node-specific parameters in the $\\beta$-model for moderate or extremely sparse networks. Our proposed estimators, called \\emph{Cycle Counting Ratio (CCR) Estimator}, are based on the log-ratios of two network cycle counting statistics with explicit expressions and therefore easy to compute. We focus on conditions to guarantee statistical properties of the single estimator for each node. Under the very w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.05273","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.05273/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}