which are the essential relationships.
If we take f (R) = Rm, where m > 0 may be non-integral and very small,
representing the geometry of a ‘fractal’ network, we can solve equation (16)
for RC as
RC =
[KC∕(KC - K)]1/(my+z)
21∕(my+z)
(16)
Note that, for given y, m and z could be characterized by a “universality
class relation” of the form α = my + z = constant. Note that nothing in the
development prevents α from being continuously tunable.
Substituting this value for RC back into equation (15) gives a somewhat
more complex expression for H than equation (4), having three parameters,
i.e. m, y, z . Fixing m, z and KC , some exploration shows that tuning y gives
results qualitatively similar to those of equations (6) and (7). The exercise
is best done in a symbolic mathematics program.
If we make the more biologically reasonable assumption of logarithmic
growth, so that
f(R) = m log(R) + 1,
(17)
with f(1) = 1, then plugging in to equation (14) and solving for RC in
Mathematica 4.2 gives
RC = [
_______Q_______
LambertW [Q exp(z/my)]
27
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