The relative <i>f</i>-invariant and non-uniform random sofic approximations

Abstract The f -invariant is an isomorphism invariant of free-group measure-preserving actions introduced by Lewis Bowen, who first used it to show that two finite-entropy Bernoulli shifts over a finitely generated free group can be isomorphic only if their base measures have the same Shannon entropy. Bowen also showed variant sofic entropy; in particular, exponential growth rate expected number good models uniform random homomorphism. In this paper we present analogous formula for relative and use prove approximation which type stochastic block model.