Notes

Chapter 9: Fundamental Physics

Section 7: Space as a Network


Counting of [network] nodes

The number of nodes reached by going out to network distance r (with r > 1

r > 1) from any node in the networks on page 477 is (a) 4r - 4
4r - 4
, (b) 3r2/2 - 3r/2 + 1
3\!\(\*SuperscriptBox[\(r\),\(2\)]\)/2-3r/2+1
, and (c)

First[Select[4r3/9 + 2r2/3 + {2, 5/3, 5/3} r - {10/9, 1, -4/9}, IntegerQ]]

First[Select[4\!\(\*SuperscriptBox[\(r\),\(3\)]\)/9+2\!\(\*SuperscriptBox[\(r\),\(2\)]\)/3+{2,5/3,5/3}r-{10/9,1,-4/9},IntegerQ]]

In any trivalent network, the quantity f[r]

f[r] obtained by adding up the numbers of nodes reached by going distance r from each node must satisfy f[0] = n
f[0] = n
and f[1] = 3n
f[1] = 3n
, where n is the total number of nodes in the network. In addition, the limit of f[r]
f[r]
for large r must be n2
\!\(\*SuperscriptBox[\(n\),\(2\)]\)
. The values of f[r]
f[r]
for all other r will depend on the pattern of connections in the network.



Image Source Notebooks:

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