## How Do Molecules Know

Shortcut connections between vertices are of little use if a particle in the hemocoel does not know where to go. How does a particle, lacking a nervous system and a topological map of the hemocoel in its brain know the most direct route?

We can modify a two-dimensional square lattice where each vertex joins its four nearest neighbors. To this lattice we add long distance connections, but not merely at random. We rank each edge connecting two vertices to all possible destinations as we would a shortcut edge. We rank each edge based on its distance from the source vertex. Now, the probability of choosing a vertex, d, is proportional to dr where r is an additional parameter of the model. If we set r equal to zero, then we choose destinations at all distances from d with equal probability, so our model just becomes a two-dimensional version of Watts and Strogatz's model. However, if r is large, then our omniscient molecule has appreciable chances of choosing only nearby destinations, so the structure of our starting lattice barely changes. The threshold value is when r = 2. At this point, the probability now obeys an inverse square law (Ref: Routing).

A greedy algorithm can solve such routings. To go from vertex a to vertex b, list all the edges emanating from a. Now choose the one edge taking us closest to b as a measured distance across the lattice. Then repeat the same procedure starting from our new vertex, again proceeding vertex by vertex, until we reach our destination.

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