What is the probability that a random graph with $n$ vertices and degree sequence $\left(d_i\right)_{i=1..n}$ has a subgraph of $k$ vertices and density $\delta$?
The random graph is typically obtained through the configuration model.
The (simpler) case where the degree sequence is not prescribed but only the number $m$ of edges, leading to an Erdős–Rényi random graph, is also of interest.
Context: I generate random graphs and then add dense subgraphs to them in order to test an algorithm for detecting them, but I need to know if it is reasonable to assume that the initial graph has no such subgraph.