Dr. Ecco Solution
Solution to "Fractal Biology," DDJ January, 2006.
- With eight nodes, you need only 16 links; see Figure 1.
- With 12 nodes, you could design three sets of four nodes that are completely connected (requiring 3×6=18 nodes), then add-in four links between every pair of four, leading to an additional 12 links for a total of altogether.
- If there is no limit on the number of links per protein node, then try Figure 2. You would need only 21 links. We call this the "two-fan design," because each hub creates a fan. You need two fans in case one hub is wounded.
- Divide the nodes into 96 nodes (the base nodes) that will have six links
each and 12 nodes (the switchboard nodes) that will have 59 links each. This
results in a total of 642 links. Number the 96 base nodes from 1 to 96. Call
1 through 24 the A nodes, 25 through 48 the B nodes, 49 through
72 the C nodes, and 73 through 96 the D nodes. The 12 switchboard
nodes are divided into two groups of six. The first group is called AB,
AC, AD, BC, BD, and CD. The second group is
called AB', AC', AD', BC', BD', and CD'.
Figure 3 shows all the nodes and all the interconnections except the complete
graph among all the switchboard nodes (for example, every switchboard node
is connected to every other one). Note that XY' has the same connections
as XY (both to the base nodes and to the hub nodes).
Why does this work? Any two base nodes, say B and D, are connected through two switchboard nodes (in this case, BD and BD'). Two base nodes are connected through six switchboard nodes if the base nodes are in the same letter group. Any base node is connected to any switchboard node either directly or by connecting to any of the six switchboard nodes that the base node is directly connected to, plus a direct link to the other switchboard node.
DDJ
