150

12

Systems and Networks

attractors, where upper NN is the number of nodes (i.e., far fewer than the 2 Superscript upper N2^N potentially

accessible states).

More generally, Kauffman considered strings ofupper NN genes, each present in the form

of either of two alleles (0 and 1). ¹⁷ In the simplest case, each gene is independent,

and when a gene is changed from one allele to the other, the total fitness changes by at

most1 divided by upper N1/N. If epistatic interactions (when the action of one gene is modified by others)

are allowed, the fitness contribution depends on the gene plus the contributions from

upper KK other genes (the upper N upper KN K model), ¹⁸ and the fitness function or “landscape” becomes

less correlated and more rugged. ¹⁹

Érdi and Barna (1984) have studied how the pattern of connexions changes when

their evolution is subjected to certain simple rules; the evolution of networks of

automata in which the properties of the automata themselves can change has barely

been touched, although this, the most complex and difficult case, is clearly the one

closest to natural networks within cells and organisms. The study of networks and

their application to real-world problems has, in effect, only just begun.

12.3

Synergetics

General systems theory (Sect. 12.1) can be further generalized and made more pow-

erful by including a diffusion term:

StartFraction partial differential u Subscript i Baseline Over partial differential t EndFraction equals StartFraction 1 Over tau Subscript i Baseline EndFraction upper F Subscript i Baseline left parenthesis u 1 comma u 2 comma ellipsis comma u Subscript n Baseline right parenthesis plus upper D Subscript i Baseline Delta u Subscript i Baseline comma i equals 1 comma 2 comma ellipsis comma n period^∂ui

∂t ^{= 1}

τi

Fi(u1, u2, . . . , un) + DiΔui,

i = 1, 2, . . . , n .

(12.21)

u Subscript iui is a dynamic variable (e.g., the concentration of the iith object at a certain point

in space), upper F Subscript i Baseline left parenthesis u Subscript i Baseline right parenthesisFi(ui) are functions describing the interactions, tau Subscript iτi is the characteristic

time of change, and upper D Subscript iDi is the diffusion coefficient (diffusivity) of the iith object.

Equation (12.21) is thus a reaction–diffusion equation that explicitly describes the

spatial distribution of the objects under consideration. The diffusion term tends to

zero if the diffusion length l Subscript i Baseline greater than upper Lli > L, the spatial extent of the system, where

l Subscript i Baseline equals upper D Subscript i Superscript 1 divided by 2 Baseline tau Subscript i Baseline periodli = D^1/2

i

τi .

(12.22)

Although solutions of Eq. (12.21) might be difficult for any given case under explicit

consideration, in principle we can use it to describe any system of interest. This area

of knowledge is called synergetics. Note that the “unexpected” phenomena often

observed in elaborate systems can be easily understood within this framework, as

we shall see.

17 Cf. Sect. 4.1.2.

18 This is yet another use of the symbolupper KK—see Footnote 4 earlier in this chapter.

19 Note that, as pointed out by Jongeling (1996), fitness landscapes cannot be used to model selection

processes if the entities being selected do not compete.