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Each point $x_t^i$ can be active or not in the network with a Bernoulli probability $p$. Each pair being connected by affinity thanks to a step connection function if the interpoint distance $\\|x_t^i-x_t^j\\|\\leq a_\\mathsf{f}^\\star$ for any $i\\neq j$. We prove that when $a_\\mathsf{f}^\\star=\\sqrt{\\frac{(s_t)^{l-1}}{p\\pi}}$ for $l\\in(0,1)$, t"},"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":"1810.06479","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-10-15T15:49:49Z","cross_cats_sorted":[],"title_canon_sha256":"769839b0625e58623644655b5203d333ad2d377401dc0385d41b41a5340cb89a","abstract_canon_sha256":"21729a52908b35ad032aac4efe3d44c949e334a006f13fc460037eca71b3026d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:03:20.383957Z","signature_b64":"xfZMhEAjKOlksmH0rJLkVWzp0sR4t+8m3QGbMHMapwNcO5Y9qruJ0Lyceh+BUj8Pv+7A34jwVnjSTPqI4EQ2Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b0e3755367eacc809121d8911e15caf2e0c87437c3088d378c520a708af99f5b","last_reissued_at":"2026-05-18T00:03:20.383517Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:03:20.383517Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotic adaptive threshold for connectivity in a random geometric social network","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Ahmed Sid-Ali, Khader Khadraoui","submitted_at":"2018-10-15T15:49:49Z","abstract_excerpt":"Consider a dynamic random geometric social network identified by $s_t$ independent points $x_t^1,\\ldots,x_t^{s_t}$ in the unit square $[0,1]^2$ that interact in continuous time $t\\geq 0$. 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