{"paper":{"title":"The spectrum of dense kernel-based random graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO","math.FA"],"primary_cat":"math.PR","authors_text":"Alessandra Cipriani, Michele Salvi, Nandan Malhotra, Rajat Subhra Hazra","submitted_at":"2025-02-13T15:39:05Z","abstract_excerpt":"Kernel-based random graphs (KBRGs) are a broad class of random graph models that account for inhomogeneity among vertices. We consider KBRGs on a discrete $d-$dimensional torus $\\mathbf{V}_N$ of size $N^d$. Conditionally on an i.i.d.~sequence of {Pareto} weights $(W_i)_{i\\in \\mathbf{V}_N}$ with tail exponent $\\tau-1>0$, we connect any two points $i$ and $j$ on the torus with probability\n  $$p_{ij}= \\frac{\\kappa_{\\sigma}(W_i,W_j)}{\\|i-j\\|^{\\alpha}} \\wedge 1$$ for some parameter $\\alpha>0$ and $\\kappa_{\\sigma}(u,v)= (u\\vee v)(u \\wedge v)^{\\sigma}$ for some $\\sigma\\in(0,\\tau-1)$.\n  We focus on th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.09415","kind":"arxiv","version":2},"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/2502.09415/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"}