Regular Networks
"...change in state involves no alteration of the network elements themselves, but rather a transformation in the suble organization of the network of their interactions."
-- Marck Buchanan, paraphrased from Nexus

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Regular networks are "regular" because each node has exactly the same number of links. Regular networks are highly ordered.

A square "regular" lattice is a non-random regular network where each node connects to all of its nearest neighbors. Lattices can also be represented as rings, trees, and stars. The lattice can characterize swarms, schools, herds, and flocks where the behavior of each individual (node) depends upon the behavior of its nearest neighbors. In the lattice network shown on the left, the topological rule is that each node is linked to all of its four nearest neighbors. No statistical rule is needed to define the lattice's degree distribution because the number of degrees (4) is the same for each node.

The circular (ring) lattice shown on the right is exactly the same regular non-random model as the square lattice. It is simply transposed into a circular form by bending the square lattice and joining its ends. Notice that, like its square counterpart, each node is "clustered" (connected) to four near neighbor nodes. For example, nodes 1,2,3,10, and 9 form a cluster because they all connect to node #1. The circular form makes it easy to demonstrate the effects of short path lengths and clustering on connectivity.

Two nodes placed on opposite sides of the circle lattice (see nodes #1 and #6) are members of separate clusters that do not intersect. There are relatively long path lengths (low connectivity) between these remote nodes. Two nodes that are within three nodes of each other on the circle (nodes #1 and #4) belong to separate but intersecting clusters. This is equivalent to two groups of friends where some people belong to both groups. This kind of intersecting clustering enhances connectivity within the network.

A "regular" network architecture, such as the lattice, does not offer high connectivity in many cases because a long and circuitous route is required to reach many nodes. In practical terms, for example, the regular circular lattice network may produce a cumbersome chain of chemical reactions to transform node 5 to a compound at node 10.

You are invited to explore more detail by looking at random networks, small world networks, and scale-free networks.



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