Connectivity Reviewed

We consider all points in a hemocoel, the vertices or nodes, to be identical, un-weighted and featureless, so connections of one node with another through the hemolymph become our graph's edges. The degree of a vertex is the number of other vertices with which it connects. We consider only the transmission possibilities through the hemolymph as forming the network. For example, transmission of a particle from one side of an organ to its other side within the organ, and, therefore, outside the hemocoel, does not constitute travel within the hemocoel. Travel through the hemolymph from one vertex to another is bidirectional as transmission is equally likely in both directions. The graph of the hemocoel is sparse. O measures the number of shortcuts or the average range if the graph is large, so that the hemocoel graph exhibits logarithmic length scaling with respect to n, its number of vertices. One might presume a measure of O should be close to the characteristic path length for the graph in that edges do not correlate with each other (they are merely paths through fluid in two or three dimensions). Because two vertices are connected does not imply that their second shortest path length should be any shorter than the average path length. Gamma, indicating an undirected graph, and L, indicating a line graph, are larger than for a random graph. Gamma refers to length, so clustering approximations fall apart because any Moore graph approximation of local scale diverges.

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