Chapter 9: Fundamental Physics

Section 7: Space as a Network

Properties of networks

Over the past century or so a variety of global properties of networks have been studied. Typical ones include:

• Edge connectivity: the minimum number of connections that must be removed to make the network disconnected.

• Diameter: the maximum distance between any two nodes in the network. The pictures below show the largest planar trivalent networks with diameters 1, 2 and 3, and the largest known ones with diameters 4, 5 and 6.

• Circumference: the length of the longest cycle in the network. Although difficult to determine in particular cases, many networks allow so-called Hamiltonian cycles that include every node. (Up to 8 nodes, all 8 trivalent networks have this property; up to 10 nodes 25 of 27 do.)

• Girth: the length of the shortest cycle in the network. The pictures below show the smallest trivalent networks with girths 3 through 8 (so-called cages). Girth can be relevant in seeing whether a particular cluster can ever occur in network.

• Chromatic number: the minimum of colors that can be assigned to nodes so that no adjacent nodes end up the same color. It follows from the Four-Color Theorem that the maximum for planar networks is 4. It turns out that for all trivalent networks the maximum is also 4, and is almost always 3.

Image Source Notebooks:

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