H = H0κα
(4)
where α is a positive constant. Again, more biologically relevant examples
are presented in the Appendix.
Further from the critical point matters are more complicated, involving
‘Generalized Onsager Relations’ and a kind of thermodynamics associated
with a Legendre transform [56].
An essential insight is that regardless of the particular renormalization
properties, sudden critical point transition is possible in the opposite direc-
tion for this model. That is, we also can move 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 parame-
ter K , the inverse strength of weak ties, falls below threshold, or, conversely,
once the strength of weak ties parameter P = 1/K becomes large enough.
Thus, increasing nondisjunctive weak ties between them can bind sev-
eral different ‘languages’ into a single, embedding hierarchical metalanguage
which contains each as a linked subdialect.
This heuristic insight can be made exact using a rate distortion approach
(or, more generally, using the Joint Asymptotic Equipartition Theorem).
The argument goes as follows [56, 57]:
Suppose that two ergodic information sources Y and B begin to interact,
to ‘talk’ to each other, i.e. to influence each other in some way so that it
is possible, for example, to look at the output of B - strings b - and infer
something about the behavior of Y from it - strings y. We suppose it possible
to define a retranslation from the B-language into the Y-language through
a deterministic code book, and call Y the translated information source, as
mirrored by B.
Take some distortion measure d comparing paths y to paths ^, defining
d(y,y) [14]. We invoke the Rate Distortion Theorem’s mutual information
I (Y, Y), which is the splitting criterion between high and low probability
pairs of paths. Impose, now, a parametization by an inverse coupling strength
15