  The evolution of entanglement entropy in quantum circuits composed of
Haar-random gates and projective measurements shows versatile behavior, with
connections to phase transitions and complexity theory. We reformulate the
problem in terms of a classical Markov process for the dynamics of bipartition
purities and establish a probabilistic cellular-automaton algorithm to compute
entanglement entropy in monitored random circuits on arbitrary graphs. In one
dimension, we further relate the evolution of the entropy to a simple classical
spin model that naturally generalizes a two-dimensional lattice percolation
problem. We also establish a Markov model for the evolution of the zeroth
R\'{e}nyi entropy and demonstrate that, in one dimension and in the limit of
large local dimension, it coincides with the corresponding
second-R\'{e}nyi-entropy model. Finally, we extend the Markovian description to
a more general setting that incorporates continuous-time dynamics, defined by
stochastic Hamiltonians and weak local measurements continuously monitoring the
system.
