Consider the upper tail probability that the homomorphism count of a fixed graph H within a large sparse random graph G_n exceeds its expected value by a fixed factor . Going beyond the Erdős–Rényi model, we establish here explicit, sharp upper tail decay rates for sparse random d_n-regular graphs (provided H has a regular 2-core), and for sparse uniform random graphs. We further deal with joint upper tail probabilities for homomorphism counts of multiple graphs (extending the known results for ), and for inhomogeneous graph ensembles (such as the stochastic block model), we bound the upper tail probability by a variational problem analogous to the one that determines its decay rate in the case of sparse Erdős–Rényi graphs.

中文翻译：

---

约束稀疏随机图中同态计数的上尾

考虑大稀疏随机图G _{n 中}固定图H的同态计数超过其预期值一个固定因子的上尾概率。超越鄂尔多斯-莱利模型，我们在这里建立明确的，锋利的尾上的衰减率稀疏随机d _ñ -regular图（提供^ h有一个普通的2芯），对于稀疏均匀随机图。我们进一步处理多个图的同态计数的联合上尾概率（扩展已知结果)，对于非齐次图集合（例如随机块模型），我们通过一个变分问题来限制上尾概率，该问题类似于在稀疏 Erdős-Rényi 图的情况下确定其衰减率的问题。