We formulate and analyze a heterogeneous random hypergraph model, and we provide an achieveability result for recovery of hyperedges from the observed projected graph. We observe a projected graph which combines random hyperedges across all degrees, where a projected edge appears if and only if both vertices appear in at least one hyperedge. Our goal is to reconstruct the original set of hyperedges of degree $d_j$ for some $j$. Our achievability result is based on the idea of selecting maximal cliques of size $d_j$ in the projected graph, and we show that this algorithm succeeds under a natural condition on the densities. This achievability condition generalizes a known threshold for $d$-uniform hypergraphs with noiseless and noisy projections. We conjecture the threshold to be optimal for recovering hyperedges with the largest degree.

翻译：我们提出并分析了一种异构随机超图模型，并给出了从观测到的投影图中恢复超边的可实现性结果。我们观测到的投影图整合了所有阶数的随机超边，其中当且仅当两个顶点同时出现在至少一个超边中时，投影图中才会出现对应的边。我们的目标是重构原始图中某个特定阶数 $d_j$ 的超边集合。我们的可实现性结果基于在投影图中选取大小为 $d_j$ 的极大团的思想，并证明该算法在密度满足自然条件时能够成功恢复。这一可实现性条件推广了已知的关于无噪声及含噪声投影下 $d$-均匀超图的恢复阈值。我们推测该阈值对于恢复最大阶数的超边是最优的。