Degree Sequence

Degree sequence is simple in a lattice. All vertices have the same number of edges, so a plot of the degree sequence reveals a single sharp spike. Any randomness in this graph broadens the peak into a Poisson distribution. Because of the exponential decline, the probability of finding a vertex with k edges grows negligibly small for large k.

Does a power law describe the edges in the hemocoel? That is, are the number of vertices having degree k given not by e-k but by kg, where g is a positive constant? The power law distribution falls off more gradually than an exponential. This gradual decline permits vertices of high degree to exist in the hemocoel.

We can use Barabasi's model of edges in the World Wide Web to study the hemocoel. Begin with n vertices and no edges. At each step, add to the graph a single new vertex and m edges, so that now all new edges link the new vertex to some of the vertices already present. This pattern fits with what happens as volumes of hemolymph in the hemocoel vary as edges disappear from or add to the network depending upon the volume of hemolymph in the cavity.

The probability that a given vertex receives a new edge is proportional to the share of the total set of edges that the vertex already possesses. Hence, a well-connected vertex upon filling of the hemocoel, evolves into a more connected vertex. After t steps, the graph has n + t vertices and m times t edges. Growing by Barbasi's rules, the graph enters a statistical steady state, in which the shape of the distribution of vertex degrees does not alter over time provided the levels of hemolymph remain constant. The probability of finding a vertex having k edges is proportional to kâ3.

At this point, the graph of the hemocoel approaches that of a Moore or random graph. In a Moore graph starting from any vertex, we can reach k vertices at distance 1, then from each of these vertices we can achieve another (k â 1) new vertices at distance 2 and so on without any redundancies until we have filled the hemocoel with its graph. A Moore graph is the most efficient possible k regular graph in the sense that every vertex "reaches" k new vertices. For a perfect Moore graph, a theoretical lower bound that we seldom reach is that for k > 2, the characteristic path length has to grow at least logarithmically with n.

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