Is there any field that studies the equivalent of cellular automata, but for arbitrary graphs rather than 1D cells?
Graph dynamical systems is a general term for systems where each vertex of a graph carries a state, and the system evolves in time so that the next state of a vertex only depends on the states in its neighborhood (possibly in a nondeterministic way). Boolean networks are a special case where the graph is finite and each state is either $0$ or $1$. Boolean networks are used e.g. as models of gene expression in cells, and a lot of practical and theoretical research exists on them.