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Consider the Erdős–Renyi random graph on vertices where each edge is present independently with probability , with fixed. For large , a typical random graph locally behaves like a Galton–Watson tree with Poisson offspring distribution with mean . Here, we study large deviations from this typical behavior within the framework of the local weak convergence of finite graph sequences. The associated rate function is expressed in terms of an entropy functional on unimodular measures and takes finite values only at measures supported on trees. We also establish large deviations for other commonly studied random graph ensembles such as the uniform random graph with given number of edges growing linearly with the number of vertices, or the uniform random graph with given degree sequence. To prove our results, we introduce a new configuration model which allows one to sample uniform …