Expected transient time and damage spreading for the NER automaton on geometrically connected graphs

The extremal rules automata (ER) were introduced as a generalization of an earlier method for elementary image enhancement: the nearest extremum rule automaton (NER). The ER dynamical behavior was characterized for the sequential iteration by a Lyapunov functional which allows proving fixed point steady state behavior together with an exponential bound for the maximal transient time. For the parallel iteration the fixed point steady state behavior were determined by direct proof, but the maximal transient time has not been yet characterized. In this work a numerical study is performed to determine the expected transient time and damage spreading of the NER parallel iteration on geometrically connected graphs. The results can be interpreted as a generalization of [Hernandez, Herrmann, Goles, Extremal automata for image sharpening, Int. J. Modern Phys. C 5(6) (1994) 923-932] for non regular graphs.