  The disorder and a simple convex measure of complexity are studied for rank
ordered power law distributions, indicative of criticality, in the case where
the total number of ranks is large. It is found that a power law distribution
may produce a high level of complexity only for a restricted range of system
size (as measured by the total number of ranks), with the range depending on
the exponent of the distribution. Similar results are found for disorder.
Self-organized criticality thus does not guarantee a high level of complexity,
and when complexity does arise, it is self-organized itself only if
self-organized criticality is reached at an appropriate system size.
