  Discrete dynamical systems can exhibit complex behaviour from the iterative
application of straightforward local rules. A famous example are cellular
automata whose global dynamics are notoriously challenging to analyze. To
address this, we relax the regular connectivity grid of cellular automata to a
random graph, which gives the class of graph cellular automata. Using the
dynamical cavity method (DCM) and its backtracking version (BDCM), we show that
this relaxation allows us to derive asymptotically exact analytical results on
the global dynamics of these systems on sparse random graphs. Concretely, we
showcase the results on a specific subclass of graph cellular automata with
``conforming non-conformist'' update rules, which exhibit dynamics akin to
opinion formation. Such rules update a node's state according to the majority
within their own neighbourhood. In cases where the majority leads only by a
small margin over the minority, nodes may exhibit non-conformist behaviour.
Instead of following the majority, they either maintain their own state, switch
it, or follow the minority. For configurations with different initial biases
towards one state we identify sharp dynamical phase transitions in terms of the
convergence speed and attractor types. From the perspective of opinion dynamics
this answers when consensus will emerge and when two opinions coexist almost
indefinitely.
