In this paper, we consider the problem of learning an unknown graph via queries on groups of nodes, with the result indicating whether or not at least one edge is present among those nodes. While learning arbitrary graphs with $n$ nodes and $k$ edges is known to be hard in the sense of requiring $\Omega( \min\{ k^2 \log n, n^2\})$ tests (even when a small probability of error is allowed), we show that learning an Erd\H{o}s-R\'enyi random graph with an average of $\bar{k}$ edges is much easier; namely, one can attain asymptotically vanishing error probability with only $O(\bar{k}\log n)$ tests. We establish such bounds for a variety of algorithms inspired by the group testing problem, with explicit constant factors indicating a near-optimal number of tests, and in some cases asymptotic optimality including constant factors. In addition, we present an alternative design that permits a near-optimal sublinear decoding time of $O(\bar{k} \log^2 \bar{k} + \bar{k} \log n)$.

中文翻译：

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通过边缘检测查询学习 Erd\H{o}sR\'enyi 随机图

在本文中，我们考虑通过对节点组的查询来学习未知图的问题，结果表明这些节点中是否至少存在一条边。虽然从需要 $\Omega( \min\{ k^2 \log n, n^2\})$ 测试（即使当允许出现小概率错误），我们表明学习具有平均 $\bar{k}$ 边的 Erd\H{o}sR\'enyi 随机图要容易得多；也就是说，仅通过 $O(\bar{k}\log n)$ 测试就可以获得渐近消失的错误概率。我们为受组测试问题启发的各种算法建立了这样的界限，显式常数因子表示接近最优的测试数量，在某些情况下，包括常数因子在内的渐近最优性。此外，