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We revisit the problem of counting the number of copies of a fixed graph in arandom graph or multigraph, including the case of constrained degrees. Ourapproach relies heavily on analytic combinatorics and on the notion ofpatchwork to describe the possible overlapping of copies.  This paper is a version, extended to include proofs, of the paper with thesame title to be presented at the Eurocomb 2017 meeting.

Threshold functions for small subgraphs: an analytic approach