# Small subgraphs of the random graph

If I look at the distribution of the number of small subgraphs in the random graph isomorphic to a connected graph $$H$$, this is asymptotically Poisson.

What proportion of these small subgraphs intersect another, fixed subgraph?

Also, what if I want them to intersect in a specific way, not just arbitrarily? Do the subgraphs occur on vertices that are selected uniformly at random, so the $$`$$striking' probability is just the probability I select those vertex, chosen uniformly at random?

Given $$G,H$$ two finite connected graphs, let $$H_1,...,H_k$$ be all the connected graphs obtained as a union of $$G$$ and $$H$$ (the connectedness of the $$H_i$$ implies that $$H$$ and $$G$$ intersect). Then making the sum for $$i$$ of the occurences of $$H_i$$ as a subgraph, for $$i$$ between $$1$$ and $$k$$, gives the answer to the first question. For the second question simply count the number of occurences of the $$H_i$$ that you like. Sorry I did not understand the last question.