The initial state of a network system consists of a network with two connections per node, an "above" connection (shown with a directed arc above the connected nodes) and a "below" connection (shown with an arc below the connected nodes in the opposite direction). Network systems then evolve based on rules that specify how the "above" and "below" connections should be rerouted. [NKS, p. 199]

This Demonstration investigates network systems in which the rule depends on the number of distinct nodes reached by going up to a distance 2 from each node. [NKS, p. 202]

"Depth" here is "distance from". The rule icons for depth-1 systems each contain two cases, depending on whether both connections from a particular node lead to the same or different nodes. [NKS, p. 201]

There are a total of rules of depth 1. The rule space for depth 2 is so large that only 7 rules are sampled.

Each node is denoted by a pair , where is the node reached by traveling the "above" connection and is the node reached by traveling the "below" connection. So, for instance, a cyclic network of five nodes can be denoted by the array .

The rules describe how each "above" and "below" connection should be rerouted on the next step. 1 denotes an "above" connection and 2 denotes a "below" connection.

For more information on the implementation of these systems, please see S. Wolfram, A New Kind of Science, Champaign, IL: Wolfram Media, Inc., 2002 p. 935.