TOWARD CULTURAL ONCOLOGY: THE EVOLUTIONARY INFORMATION DYNAMICS OF CANCER

and F [K, J] the free energy density independent of renormalization, then
F[K,J] =R^-3F[KR,JR],or

F[KR,JR] =R³F[K,J].

Thus f (R)=R³ for the free energy of a physical system, so that the free
energy density remains constant with increasing R.

Here we assume that, as a network grows in size, language richness grows
as some appropriate function of that size, but not, like free energy, as its
cube. Thus other, (sometimes very subtle), symmetry relations - not nec-
essarily based on the elementary physical analog we use here - may well be
possible. For example [36, p.168] describes the highly counterintuitive renor-
malization relations needed to understand phase transition in simple ‘chaotic’
systems, an example we will revisit below. This is an important subject for
future research, since we suspect that biological or social systems may alter
their renormalization properties wholesale, and not merely the parameters
associated with a particular renormalization.

The ‘richness’ of a biological ‘language’ is not likely to grow exponen-
tially with increase in size of the system, rather it is likely to follow a much
smaller rate of rise, or even approach an asymptotic limit. We explore more
biologically reasonable forms of f (R) in the Appendix, including

f(R) = m log(R) + 1

f(R) = exp[m(R - 1)]

f(R) = exp[m(R - 1)/R].

The latter expression represents a system having a crude upper asymp-
totic limit. More realistic ‘S’-shaped curves take us, unfortunately, beyond
our ability to solve the renormalization equations at this point. Again, we
have written a shorthand for dimensionless adjustment by the ‘characteristic
length’ R0, i.e. f(R) → f(R/R0).

Using the simplest physical analog as a starting point, we take f (R) = R^d,
for some real d > 0. See [9, 60] for details. Limiting K to a region near the
‘critical value’ KC , if J → 0, a simple series expansion and some clever
algebra [9, 60] gives

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