The ui , vi , i = 1, 2 are functions of KR, JR, but not explic-
itly of R itself.
We expand these equations about the critical value KR =
KC and about JR = 0, obtaining
Solving this for RC and substituting into the first of equa-
tion (2) gives, as a first iteration of a far more general proce-
dure (e.g. Shirkov and Kovalev, 2001)
H [K, 0] ≈ H Kr20]
f(RC)
Ho
f (Rr )
dKR/dR = (KR-Kr)ydlog(f)/dR+(KR-Kr)z/R
dJR/dR = wJRdlog(f)/dR + xJR/R.
(6)
The terms y = du1 /dKR |KR =KC , z =
du2/dKR|KR=KC, w = v1(Kr, 0), x = v2(Kr, 0) are
constants.
Solving the first of these equations gives
χ(K, 0) ≈ χ(Kc/2, 0)Rr = XoRr
(10)
which are the essential relationships.
Note that a power law of the form f(R) = Rm, m = 3,
which is the direct physical analog, may not be biologically
reasonable, since it says that ‘language richness’ can grow
very rapidly as a function of increased network size. Such
rapid growth is simply not observed.
If we take the biologically realistic example of non-integral
‘fractal’ exponential growth,
KR = Kr + (K - Kr)Rzf(R)y,
(7)
f(R) = Rδ,
(11)
again remembering that K1 = K, J1 = J, f(1) = 1.
Wilson’s essential trick is to iterate on this relation, which
is supposed to converge rapidly (Binney, 1986), assuming that
for KR near Kr , we have
where δ > 0 is a real number which may be quite small, we
can solve equation (8) for Rr, obtaining
Rr =
[KC∕(KC - K)][V(δy+z)]
2V(δy+z)
Kr/2 ≈ Kr + (K - Kr)Rzf (R)y.
(12)
(8)
We iterate in two steps, first solving this for f(R) in terms
of known values, and then solving for R, finding a value Rr
that we then substitute into the first of equations (2) to obtain
an expression for H [K, 0] in terms of known functions and
parameter values.
The first step gives the general result
for K near Kr . Note that, for a given value of y , we might
want to characterize the relation α ≡ δy + z = constant as
a “tunable universality class relation” in the sense of Albert
and Barabasi (2002).
Substituting this value for Rr back into equation (9) gives
a somewhat more complex expression for H than equation
(2), having three parameters, i.e. δ, y, z .
A more biologically interesting choice for f (R) is a loga-
rithmic curve that ‘tops out’, for example
(9)
f(Rr)
[KC∕(KC - K)]1/y
21/y RzRv
f(R) = m log(R) + 1.
(13)