Notes

Chapter 9: Fundamental Physics

Section 8: The Relationship of Space and Time


Network constraint systems

Cases (a), (f) and (p) allow all networks that do not contain respectively cycles of length 1 (self-loops), cycles of length 3 or less, and cycles of length 5 or less. In cases where an infinite sequence of networks is allowed, there are typically particular subnetworks that can occur any number of times, making the sizes of allowed networks form arithmetic progressions. In cases (m), (n) and (o) respectively triangle, pentagon and square subnetworks can be repeated.

The main text excludes templates that have no dangling connections, and are thus themselves already complete networks. There are 5 such templates involving nodes out to distance one, but of these only 3 correspond to networks that satisfy the constraint that around each node the network has the same form as the template. Among templates involving nodes out to distance two there are 106 that have no dangling connections, and of these only 8 satisfy the constraints.

The main text considers only constraints based on a single template. One can also allow each node to have a neighborhood that corresponds to any of a set of templates. For templates involving nodes out to distance one, there are 13 minimal sets in the sense of page 941, of which only 6 contain just one template, 6 contain two and 1 contains three.

If one does allow dangling connections to be joined within a single template, the results are similar to those discussed so far. There are 52 possible templates involving nodes out to distance two, of which 12 allow complete networks to be formed, none forced to be larger than 12 nodes. There are 46 minimal sets, with the largest containing 4 templates, but none forcing a network larger than 16 nodes.


From Stephen Wolfram: A New Kind of Science [citation]