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  "path": "/abs/2604.08691v1",
  "publishedAt": "2026-04-13T00:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Kalle Alaluusua",
    "B. R. Vinay Kumar"
  ],
  "textContent": "**Authors:** Kalle Alaluusua, B. R. Vinay Kumar\n\nHypergraph data are often projected onto a weighted graph by constructing an adjacency matrix whose $(i,j)$ entry counts the number of hyperedges containing both nodes $i$ and $j$. This reduction is computationally convenient, but it can lose information: distinct hypergraphs may induce the same matrix, and the matrix entries are generally dependent because each hyperedge contributes to multiple pairs. We study the planted clique problem under this matrix-only observation model. For detection, we show that a spectral norm test is asymptotically powerful at the $\\sqrt{n}$ scale, with explicit dependence on the background hyperedge probability. For recovery, we analyze a polynomial-time spectral method based on the leading eigenvector and prove exact recovery at the canonical $\\sqrt{n}$ scale, again with explicit dependence on the background hyperedge probability. We also extend both results to sparse regimes in which the background hyperedge probability may depend on $n$. Our analysis adapts a leave-one-out eigenvector framework to this setting. These results provide rigorous detection and recovery guarantees when only the adjacency matrix is observed.",
  "title": "Planted clique detection and recovery from the hypergraph adjacency matrix"
}