Comparing Graphs

We compared the graphs of a closed tube circulation and an open cavity circulation. The graph of the closed circulation is a directed Eulerian trail. For each stroke of the heart-pump, blood flows in only one direction, from arteries and arterioles through capillaries to veins and back to the heart.

An open circulation or hemocoel exploits the principle that the shorter the path length, the faster, more direct and more economical is transmission through the system. Hemocoels remain the same size, but the volume of hemolymph in them and hence, the network graph for hemolymph within a hemocoel, grows and shrinks.

The graph of the hemocoel resembles a Moore graph in that every vertex potentially connects to every other vertex in the cavity directly. These connecting edges are identical and unweighted. Advantages of a cavity circulation are that nodes in the hemocoel are and remain dimensionless, featureless vertices. Transmission is potentially bidirectional. There are no hubs, as every point in the hemocoel potentially possesses direct 'airline' connectivity with every other point. Routes are lines or edges that are not vulnerable to point blockages as a fixed network of vessels would be. As an exponential network, the hemocoel is more stable than a scale-free network. Circulation is quite independent of the volume of hemolymph in the smallest hemocoels, and shortcuts through the two-dimensional film would still be possible.

Because each new edge forms or disappears independently from any other, the graph of the hemocoel does not form clusters, as 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 making the hemocoel system less vulnerable to point defects.

Randomly connected hemocoels differ from the connected lattices of a closed circulation. Introducing just a few 'shortcuts' reveals characteristic lengths that are closer to random graphs than to lattices. Adding a few shortcuts changes the hemo-coel's dynamics appreciably, but adding more shortcuts is not better. Adding shortcuts means to a bee or device adding an energetic cost of volume and weight to be supported. That this may have been tried many times during evolution seems logical because evolution seems to have worked insects and circulations around to sit at the point where hemocoels are maximally functional.

Our third generalization is about modeling: every model contains an unintended intervention on the part of the model maker. One does not feel the need of so many reservations in the case of this principle. But in the sense of the first two, it is perhaps the one that matters least. It refers to our sense of the world. It also more importantly refers to a style of thought.

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