Tommaso Bolognesi is working on dynamic trivalent networks, along the lines suggested by Wolfram in the NKS book. He has presented on them in the past, and in this talk he first reviewed how they work, then got to his recent questions. Basically he distinguishes between a global time for the model as viewed from outside, from the sense of time “experienced” (or perhaps more properly, undergone) at a specific location within one of his graphs.
Global time simply means the model steps themselves – at each of those one local subgraph is rewritten according to a definite list of rules with a preferred order of application. The update specifies the location of the next “active site” or the next updated node. He tracks where this occurs by node number and notices that those rules that produce instrinsic randomness tend to revisit all nodes “fairly”, in an average or statistical sense. While globally the network may show exponential growth, for example, if the spacings out of revisits keep pace, the size of the graph at each (global) step on which a given node is updated, appears to grow linearly. Similarly for a rule system that showed square root growth globally – again the “internal” sense of time is of linear growth. It is a feature of fair revisitation that local updates on a graph growing at any speed appear as linear growth in a frame that regards each self-update as a clock-tick, and ignores all other updates.
He also discussed constructing the causal network of site updates, by tracking the faces shared by a given node-update. I suggested linking those backward through time at the update time-slices given by the “effected” node, to get an internal sense of locality, as well.