Sequential network systems

In the network systems discussed in the main text, every node is updated in parallel at each step. It is however also possible to consider systems in which there is only a single active node, and operations are performed only on that node at any particular step. The active node can move by following its above or below connections, in a way that is determined by a rule which depends on the local structure of the network. The pictures below show examples of sequential network systems; the path of the active node is indicated by a thick black line.

It is rather common for the active node eventually to get stuck at a particular position in the network; the picture below shows the effect of this on the total number of nodes in the last case illustrated above. The rule for this system is

{{1, 1}  {{{{}, {1, 1}}, {2}}, 2}, {1, 2}  {{{2, 2}, {{}, {2, 2}}}, 2}, {2, 1}  {{{}, {2, 2}}, 2}, {2, 2}  {{{1, 2} ,{{1}, {2}}}, 1}, {2, 3}  {{{{1, 2}, {1}}, {{2}, {2, 1}}}, 2}, {2, 4}  {{{2, 2}, {{2, 1}, {}}}, 1}}