Chapter 14, Figure
5
Small world
digraphs. (A) Analogous to the examples presented by Watts & Strogatz (1998),
graphs are varied between totally regular (case a, "nearest neighbor") through
intermediate cases (case b) to totally random (case c). A total of 256 connections
are made between 32 vertices. Vertices are arranged in a circle (small blue
circles) with red lines indicating bi-directional connections (edges), and green
lines indicating unidirectional edges. A scatter plot of characteristic path
length (lpath) and cluster index (fclust) is shown on the right. Numerous graphs
exist for which the value of fclust is high, while the value of lpath is low.
(B) Here, graphs are varied between "clustered" (case a) and totally random
(case c). Compared to (A), higher cluster index values result for case a, as
well as numerous intermediate cases. Note that, in both (A) and (B), characteristic
path lengths are relatively short for almost all cases. This is due to the relatively
high degree of connectivity (0.26), chosen to approximate that of cortical connection
matrices. In Watts & Strogatz' original work, the degree of connectivity for
typical small world examples was set to about 0.04 (n=1000, k=10 per vertex).
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