above and an ‘external field strength’ analog J, which gives a
‘direction’ to the system. We will, in the limit, set J = 0.
The implicit parameter, which we call r, is an inherent gen-
eralized ‘length’ characteristic of the phenomenon, on which
J and K are defined. That is, we can write J and K as
functions of averages of the parameter r, which may be quite
complex, having nothing at all to do with conventional ideas
of space: For example r may be defined by the degree of niche
partitioning in ecosystems or separation in social structures.
For a given generalized language of interest with a well
defined (piecewise adiabatically memoryless) ergodic source
uncertainty H we write
H[K,J,X]
Imposition of invariance of H under a renormalization
transform in the implicit parameter r leads to expectation
of both a critical point in K, which we call KC , reflecting
a phase transition to or from collective behavior across the
entire array, and of power laws for system behavior near KC .
Addition of other parameters to the system, e.g. some V ,
results in a ‘critical line’ or surface KC(V ).
Let κ ≡ (KC -K)/KC and take χ as the ‘correlation length’
defining the average domain in r-space for which the informa-
tion source is primarily dominated by ‘strong’ ties. We begin
by averaging across r-space in terms of ‘clumps’ of length R.
Then, taking Wilson’s (1971) analysis as a starting point, we
choose the renormalization relations as
H[KR,JR,X] =f(R)H[K,J,X]
X( T χ x(κ,J)
X(KR, JR) = R ,
2)
with f1) = 1 and J1 = J, κ1 = κ. The first of these equa-
tions significantly extends Wilson’s treatment. It states that
‘processing capacity,’ as indexed by the source uncertainty of
the system, representing the ‘richness’ of the generalized lan-
guage, grows monotonically as f (R), which must itself be a di-
mensionless function in R, since both H[κR, JR] and H [κ, J]
are themselves dimensionless. Most simply, this would require
that we replace R by R/R0, where R0 is the ‘characteristic
length’ for the system over which renormalization procedures
are reasonable, then set R0 ≡ 1, i.e. measure length in units
of R0 . Wilson’s original analysis focused on free energy den-
sity. Under ‘clumping’, densities must remain the same, so
that if F [κR, JR] is the free energy of the clumped system,
and F [κ, J] is the free energy density before clumping, then
Wilson’s equation (4) is F [κ, J] = R-3F [κR, JR], i.e.
F[κR,JR] = R3F[κ,J].
Remarkably, the renormalization equations are solvable
for a broad class of functions f (R), or more precisely,
f R/R0), R0 ≡ 1.
The second relation just states that the correlation length
simply scales as R.
Other, very subtle, symmetry relations - not necessarily
based on the elementary physical analog we use here - may
well be possible. For example McCauley, (1993, p.168) de-
scribes the highly counterintuitive renormalization relations
needed to understand phase transition in simple ‘chaotic’ sys-
tems. This is important, since we suspect that biological or
social systems may alter their renormalization properties -
equivalent to tuning their phase transition dynamics - in re-
sponse to external signals. We will make much of this possi-
bility, termed ‘universality class tuning’, below.
To begin, following Wilson, we take f(R) = Rd for some
real number d > 0, and restrict κ to near the ‘critical value’
κC . If J → 0, a simple series expansion and some clever
algebra Wilson, 1971; Binney et al., 1986) gives
H = H0 κα
0
X = —
κs
3)
where α, s are positive constants. We provide more biolog-
ically relevant examples below.
Further from the critical point matters are more compli-
cated, appearing to involve Generalized Onsager Relations
and a kind of thermodynamics associated with a Legendre
transform of H, i.e. S ≡ H — KdH/dK (Wallace, 2002a).
Although this extension is quite important to describing be-
haviors away from criticality, the full mathematical detail is
cumbersome and the reader is referred to the references. A
brief discussion will be given below.
An essential insight is that regardless of the particular
renormalization properties, sudden critical point transition is
possible in the opposite direction for this model. That is, we
go from a number of independent, isolated and fragmented
systems operating individually and more or less at random,
into a single large, interlocked, coherent structure, once the
parameter K , the inverse strength of weak ties, falls below
threshold, or, conversely, once the strength of weak ties pa-
rameter P = 1/K becomes large enough.
Thus, increasing nondisjunctive weak ties between them
can bind several different cognitive ‘language’ functions into a
single, embedding hierarchical metalanguage which contains
each as a linked subdialect, and do so in an inherently punc-
tuated manner. This could be a dynamic process, creating a
shifting, ever-changing, pattern of linked cognitive submod-
ules, according to the challenges or opportunities faced by the
organism.
To reiterate somewhat, this heuristic insight can be made
more exact using a rate distortion argument (or, more gener-
ally, using the Joint Asymptotic Equipartition Theorem) as
follows (Wallace, 2002a, b):
Suppose that two ergodic information sources Y and B
begin to interact, to ‘talk’ to each other, i.e. to influence