## Graph of the Hemocoel Resembles an Almost Random Graph

How do the dynamics of a hemocoel depend upon the structure of its graph? Graphs can help us understand how the connectivity of a hemocoel functions, as graphs can generalize the connectivity of lattices and trees. The only way to move through a lattice or a tree is to crawl from vertex to vertex. There is no action at a distance, nor are there secret wormholes magically transporting a crawler from one branch tip to another universe. Graphs may show us shortcuts permitting more flexible arrangements or new ways of traveling and do not always rely upon regular step-by-step connections. Graphs let us violate the principle of locality.

The graph of the hemocoel approaches that of an almost random graph. To build a graph of a hemocoel, begin with a collection of n vertices and no edges. The vertices would be the places transported molecules might enter or leave the hemolymph by crossing a membrane. Circulating hemolymph allows pairing of any one vertex to any of the others. We create edges with probability p if a molecule or particle makes or can make the transition from one vertex to another vertex through the hemolymph, but we have an edge being created with a probability of zero if the transition does not occur for any reason.

In the extreme case of a hemocoel empty of hemolymph, our graph remains edgeless as no transitions are possible, but if the hemocoel is filled with hemolymph, the graph becomes a clique. A clique graph has all pairs of vertices adjacent. Between these two extremes at intermediate volumes of hemolymph, we might expect the graph to have all edges placed randomly and independent of each other.

Consider a complete graph, KN, having N vertices or nodes and N(N — 1)/2 undirected edges. To each edge we attach an exponential random variable, Eij having a mean equal to 1. For any two arbitrary nodes we want to know the random number of edges in the shortest path connecting these two nodes. As the hemocoel fills, the size of its graph goes to infinity, so the probability of an edge exiting between two vertices now grows larger than some threshold value, so that the graph of the hemocoel becomes connected. The strength of the random graph of the hemocoel is that the graph can be as dense or sparse as necessary. This change occurs by adding or subtracting edge probability, p. Because each new edge forms or disappears independently from any other, the graph does not form clusters, because neighboring vertices are now no more likely to be linked than would be any other randomly chosen vertices. Unlike a closed tubular circulation, nodes are eliminated in the graph of a hemocoel showing the hemocoel system to be less vulnerable to point defects (Ref: Random Graphs).

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