The Competitiveness of Nations in a Global Knowledge-Based Economy
Stuart A. Kauffman
At Home in the Universe
Chapter 9 Organisms and Artifacts
Oxford
University Press, 1995, 191-206
Organisms arise from the
crafting of natural order and natural selection, artifacts from the crafting of
Homo sapiens. Organisms and
artifacts so different in scale, complexity, and grandeur, so different in the
time scales over which they evolved, yet it is difficult not to see parallels.
Life spreads through time
and space in branching radiations. The
Cambrian explosion is the most famous example. Soon after multicelled forms were invented, a
grand burst of evolutionary novelty thrust itself outward. One almost gets the sense of multicellular
life gleefully trying out all its possible ramifications, in a kind of wild dance
of heedless exploration. As though
filling in the Linnean chart from the top down, from the general to the
specific, species harboring the different major body plans rapidly spring into
existence in a burst of experimentation, then diversify further. The major variations arise swiftly, founding
phyla, followed by ever finer tinkerings to form the so-called lower taxa: the
classes, orders, families, and genera. Later, after the initial burst, after the
frenzied party, many of the initial forms became extinct, many of the new phyla
failed, and life settled down to the dominant designs, the remaining 30 or
so phyla, Vertebrates, Arthropods, and
so forth, which captured and dominated the biosphere.
Is this pattern so
different from technological evolution? Here
human artificers make fundamental inventions.
Here, too, one witnesses, time after time, an early explosion of diverse
forms as the human tinkerers try out the plethora of new possibilities opened
up by the basic innovation. Here, too,
is an almost gleeful exploration of possibilities. And, after the party, we settle down to finer
and finer tinkering among a few dominant designs that command the technological
landscape for some time - until an entire local phylogeny of technologies
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goes extinct. No
one makes Roman siege engines any more. The howitzer and short-range rocket have
driven the siege engine extinct.
Might the same general laws
govern major aspects of biological and technological evolution? Both organisms and artifacts confront conflicting
design constraints. As shown, it is those
constraints that create rugged fitness landscapes. Evolution explores its landscapes without the
benefit of intention. We explore the landscapes
of technological opportunity with intention, under the selective pressure of
market forces. But if the underlying
design problems result in similar rugged landscapes of conflicting constraints,
it would not be astonishing if the same laws governed both biological and
technological evolution. Tissue and
terra-cotta may evolve by deeply similar laws.
In this chapter, I will
begin to explore the parallels between organism and artifact, but the themes
will persist throughout the remainder of the book. I will explore two features of rugged but
correlated landscapes. The first feature
accounts, I believe, for the general fact that fundamental innovations are
followed by rapid, dramatic improvements in a variety of very different
directions, followed by successive improvements that are less and less
dramatic. Let’s call this the “Cambrian”
pattern of diversification. The second
phenomenon I want to explore is that after each improvement the number of
directions for further improvement falls by a constant fraction. As we saw in Chapter 8, this yields an exponential
slowing of the rate of improvement. This
feature, I believe, accounts for the characteristic slowing of improvement
found in many technological “learning curves” as well as in biology itself. Let’s call this the “learning-curve” pattern. Both, I think, are simple consequences of the
statistical features of rugged but correlated landscapes.
In our current efforts, I
will continue to use the NK model of correlated fitness landscapes
introduced in Chapter 8. It is one of
the first mathematical models of tunably rugged fitness landscapes. I believe, but do not know, that the features
we will explore here will turn out to be true of almost any family of rugged
but correlated landscapes. As we have
seen, the NK model generates a family of increasingly rugged landscapes
as K, the number of “epistatic” inputs per “gene,” increases. Recall that increasing K increases the
conflicting constraints. In turn, the
increase in conflicting constraints makes the landscape more rugged and multipeaked.
When K reaches its maximal value (K
= N - 1, in which every gene is dependent on every other), the
landscape becomes fully random.
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I begin by describing a
simple, idealized kind of adaptive walk - long-jump adaptation - on a
correlated but rugged landscape. We have
already looked at adaptive walks that proceed by generating and selecting
single mutations that lead to fitter variants. Here, an adaptive walk proceeds step-by-step
in the space of possibilities, marching steadfastly uphill to a local peak. Suppose instead that we consider
simultaneously making a large number of mutations that alter many features at
once, so that the organism takes a “long jump” across its fitness landscape. Suppose we are in the Alps and take a single
normal step. Typically, the altitude
where we land is closely correlated with the altitude from which we started. There are, of course, catastrophic exceptions;
cliffs do occur here and there. But
suppose we jump 50 kilometers away. The
altitude at which we land is essentially uncorrelated with the altitude from
which we began, because we have jumped beyond what is called the correlation
length of the landscape.
Now consider NK landscapes
for modest values of K, say N 1,000 and K = 5-1,000 genes
whose fitness contributions each depends on 5 other genes. The landscape is rugged, but still highly
correlated. Nearby points have quite
similar fitness values. If we flip one,
five, or 10 of the 1,000 genes, we will end up with a combination that is not
radically different in fitness from the one with which we began. We have not exceeded the correlation length.
NK
landscapes have a well-defined correlation length. Basically, that length shows how far apart
points on the landscape can be so that knowing the fitness at one point still
allows us to predict something about the fitness at the second point. On NK landscapes, this correlation
falls off exponentially with distance. Therefore,
if one jumped a long distance away, say changing 500 of the 1,000 allele states
- leaping halfway across the space - one would have jumped so far beyond the
correlation length of the landscape that the fitness value found at the other
end would be totally random with respect to the fitness value from which one
began.
A very simple law governs
such long-jump adaptation. The result,
exactly mimicking adaptive walks via fitter single-mutant variants on random
landscapes is this: every time one finds a fitter long-jump variant, the
expected number of tries to find a still better long-jump variant doubles!
This simple result is shown in Figure
9.1 (HHC: figure not reproduced). Figure 9.la shows the results of long-jump
adaptation on NK landscapes with K = 2 for different values of N.
Each curve shows the fitness
attained on the y-axis plotted against the number of tries. Each curve increases rapidly at first, and
then ever more slowly, strongly suggesting an exponential slowing. (If the slowing is, in fact, exponential, reflecting
the fact that at each im-
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HHC: Figure not reproduced
Figure 9.1 Taking long leaps. NK landscapes can be traversed by
taking long jumps - that is, mutating more than one gene at a time. But on a correlated landscape, each time a
fitter long-jump variant is found, the expected number of tries to find an even
better variant doubles. Fitness
increases rapidly at first, and then slows and levels off. (a) This slowing is shown for a variety
of K = 2 landscapes. “Generation”
is the cumulative number of independent long-jump trials . Each curve is the
mean of 100 walks. (b)
Logarithmic
scales are used to plot the number of improvements against the number of
generations.
provement the number of tries to make the next
improvement really doubles, then if we replot the data from Figure 9.la using
the logarithm of the number of tries, we should get a linear relation. Figure 9.lb shows that this is true. The expected number of improvement steps, S =
1nG.)
The result is simple and
important, and appears nearly universal. In adaptation via long jumps beyond the
correlation lengths of landscapes, the number of times needed to find fitter
variants doubles at each improvement step, hence slowing exponentially. It takes 1,000 tries to find the first 10
fitter variants, then 1 million tries to find the next 10, then 1 billion to
find the next 10.
(Figure 9.la also shows
another important feature: as N increases, enlarging the space of
possibilities, long-jump adaptation attains ever poorer results after the same
number of tries. From other results we
know that the actual heights of peaks on NK landscapes do not change as N
increases. Thus this decrease in
fitness is a further limit to selection that, in my book The Origins of
Order, I called a complexity catastrophe. As the number of genes increases, long-jump
adaptations becomes less and less
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fruitful; the more complex an organism, the more
difficult it is to make and accumulate useful drastic changes through natural
selection.)
The germane issue is this:
the “universal law” governing long-jump adaptation suggests that adaptation on
a correlated landscape should show three time scales - an observation that may
bear on the Cambrian explosion. Suppose
that we are adapting on a correlated, but rugged NK landscape, and begin
evolving at an average fitness value. Since the initial position is of average
fitness, half of all nearby variants will be better. But because of the correlation structure or
shape of the landscape, those nearby variants are only slightly better. In contrast, consider distant variants. Because the initial point is of average
fitness, again half the distant variants are fitter. But because the distant variants are far
beyond the correlation length of the landscape, some of them can be very
much fitter than the initial point. (By the same token, some distant variants can
be very much worse.) Now consider an
adaptive process in which some mutant variants change only a few genes, and
hence search the nearby vicinity, while other variants mutate many genes, and
hence search far away. Suppose that the
fittest of the variants will tend to sweep through the population the fastest. Thus early in such an adaptive process, we
might expect the distant variants, which are very much fitter than the nearby
variants, to dominate the process. If
the adapting population can branch in more than one direction, this should give
rise to a branching process in which distant variants of the initial genotype,
differing in many ways from one another as well, emerge rapidly. Thus early on, dramatically variant forms
should arise from the initial stem. Just
as in the Cambrian explosion, the species exhibiting the different major body
plans, or phyla, are the first to appear.
Now the second time scale:
as distant fitter variants are found, the universal law of long-jump adaptation
should set in. Every time such a distant
fitter variant is found, the number of mutant tries, or waiting time, to find
still another distant variant doubles. The first 10 improvements may take 1,000 tries;
the next 10 may take 1 million tries; the next 10 may take 1 billion tries. As this exponential slowing of the ease and
rate of finding distant fitter variants occurs, then it becomes easier to find
fitter variants on the local hills nearby. Why? Because
the fraction of fitter nearby variants dwindles very much more slowly than in
the long-jump case. In short, in the mid
term of the process, the adaptive branching populations should begin to climb
local hills. Again, this is what
happened in the Cambrian explosion. After species with a number of major body
plans sprang into existence, this radical creativity slowed and then dwindled
to slight tinkering. Evolution
concentrated its sights closer to home, tinkering and adding filigree to its
inventions.
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In the long term, the third
time scale, populations may reach local peaks and stop moving or, as shown in
Chapter 8, may drift along ridges of high fitness if mutation rates are high
enough, or the landscape itself may deform, the locations of peaks may shift,
and the organisms may follow the shifting peaks.
Recently, Bifi Macready and
I decided to explore the “three time scale” issue in more detail using NK landscapes.
Bifi carried out numerical studies
searching at different distances across the landscape as walks proceeded
uphill. Figure 9.2 shows the results (HHC:
figure not reproduced).
What we wanted to know was
this: as one’s fitness changes, what is the “best” distance to explore to
maximize the rate of improvement? Should we look a long way away, beyond the
correlation length when
HHC: Figures 9.2 (a), (b) & (c) not reproduced
Figure 9.2
As fitness increases, search closer to home. On a correlated landscape,
nearby positions have similar fitnesses.
Distant positions can have Fitnesses very much higher and very much
lower. Thus optimal search distance is high when fitness is low and decreases
as fitness increases. (a) to (c). The
results of sending 1,000 explorers with three different initial fitnesses to
each possible search distance across the landscape. The distribution of fitnesses found by each
1,000 explorers is a bell-shaped, Gaussian curve. Crossmarks on the bars show plus or minus one
standard deviation for each set of 1,000 explorers, and hence correspond to the
best or worse one in six fitnesses they find.
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fitness is average, as I argued earlier? And, as fitness improves, should we look
nearby rather than far away? As Figure
9.3 shows, summarizing the results in Figure 9.2, the answers to both questions
are yes. Fitness is plotted on the
x-axis, and the optimal distance to search to improve fitness is plotted on the
y-axis. The implication is this: when
fitness is average, the fittest variants will be found far away. As fitness improves, the fittest variants will
be found closer and closer to the current position. Therefore, at early stages of an adaptive process,
we would expect to find dramatically different variants emerging. Later, the fitter variants that emerge should
be ever less different from the current position of the adaptive walk on the
landscape.
We need to recall one
further point. When fitness is low,
there are many directions uphill. As
fitness improves, the number of directions uphill dwindles. Thus we expect the branching process to be
bushy initially, branching widely at its base, and then branching less and less
profusely as fitness increases.
Uniting these two. features
of rugged but correlated landscapes, we
HHC: Figures 9.3 not reproduced
Figure 9.3 The best distance to search. As one’s fitness changes, what is the best
distance to explore to maximize the rate of improvement? In this graph, as fitness increases, optimal
search distance shrinks from halfway across the space to the immediate
vicinity. When fitness is average, it is
best to look a long way, as fitness improves, it is better to search nearby.
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should find radiation that initially both is bushy and
occurs among dramatically different variants, and then quiets to scant
branching among similar variants later on as fitness increases.
I believe that these
features are just what we see in biological and technological evolution.
From the first chapter of
this book I have regaled you with images of the Cambrian explosion and the
profound asymmetry of that burst of biological creativity compared with that
following the later Permian extinction. In the Cambrian, over a relatively short
period of time, according to most workers in the field, a vast diversity of
fundamentally different morphological forms appeared. Since Linnean taxonomy has been with us, we
have categorized organisms hierarchically. The highest categories, kingdoms and phyla,
capture the most general features of a very large group of organisms. Thus the phylum of vertebrates - fish, fowl, and
human - all have a vertebral column forming an internal skeleton. There are 32 phyla today, the same phyla that
have been around since the Ordovician, the period after the Cambrian. But the best accounts of the Cambrian suggest
that as many as 100 phyla may have existed then, most of which rapidly became
extinct. And as we have seen, the accepted
view is that during the Cambrian, the higher taxonomic groups filled in from
the top down: the species that founded phyla emerged first. These radically different creatures then
branched into daughter species, which were slightly more similar to one another
yet distinct enough to become founders of what we now call classes. These in turn branched into daughter species,
which were somewhat more similar to one another yet distinct enough to warrant
classifying them as founders of orders. They in turn branched and gave off daughter
species distinct enough to warrant being called founders of families, which
branched to found genera. So the early
pattern in the Cambrian shows explosive differences among the species that
branch early in the process, and successively less dramatic variation in the
successive branchings.
But in the Permian
extinction some 245 million years ago, about 300 million years after the
Cambrian, a very different progression unfolded. About 96 percent of all species became
extinct, although members of all phyla and many lower taxa survived. In the vast rebound of diversity that
followed, very many new genera and many new families were founded, as was one
new order. But no new classes or phyla
were formed. The higher taxa filled in
from the bottom up. The puzzle is to
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account for the vast explosion of diversity in the
Cambrian, and the profound asymmetry between the Cambrian and the Permian.
A related general
phenomenon is this: during postextinction rebounds, it appears to be the case
that most of the major diversification occurs early in the branching process of
speciation. Paleontologists call such a
branching lineage a clade. They
speak of “bottom-heavy” clades, which are bushy at the base, or oldest time,
and note that genera typically diverge early in the history of their families,
while families diverge early in the history of their orders. In short, the record seems to indicate that
during postextinction rebounds most of the diversity arises rather rapidly, and
then slows. Thus while the Cambrian
filled in from the top down and the Permian from the bottom up, in both cases
the greatest diversification came first, followed by more conservative
experimentation.
Might it be the case that
the general features of rugged fitness landscapes shed light on these apparent
features of the past 550 million years of evolution? As I have suggested, the probable existence of
three time scales in adaptive evolution on correlated rugged landscapes, summarized
in Figure 9.3, sounds a lot like the Cambrian explosion. Early on in the branching process, we find a
variety of long-jump mutations that differ from the stem and from one another
quite dramatically. These species have
sufficient morphological differences to be categorized as founders of distinct
phyla. These founders also branch, but
do so via slightly closer long-jump variants, yielding branches from each
founder of a phylum to dissimilar daughter species, the founders of classes. As the process continues, fitter variants are
found in progressively more nearby neighborhoods, so founders of orders,
families, and genera emerge in succession.
But why, then, was the
flowering after the Permian extinction so different from the explosion during
the Cambrian? Can our understanding of
landscapes afford any possible insight? Perhaps.
A few more biological ideas are needed. Biologists think of development from the
fertilized egg to the adult as a process somewhat akin to building a cathedral.
If one gets the foundations wrong,
everything else will be a mess. Thus
there is a common, and probably correct, view that mutants affecting early
stages of development disrupt development more than do mutants affecting late
stages of development. A mutation
disrupting formation of the spinal column and cord is more likely to be lethal than
one affecting the number of fingers that form. Suppose this common view is correct. Another way of saying this is that mutants
affecting early development are adapting on a more rugged landscape than
mutants affecting late development. If
so, then the fraction of fitter neighbors
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dwindles faster for mutants affecting early
development than those affecting late development. Thus it becomes hard to find mutants altering
early development sooner in the evolutionary process than to find mutants
affecting late development. Hence if
this is correct, early development tends to “lock in” before late development. But alterations in early development are just
the ones that would cause sufficient morphological change to count as change at
the phylum or class level. Thus as the
evolutionary process continues and early development locks in, the most rapid
response to ecological opportunity after a mass-extinction event should be a
rebound with massive speciation and radiation, but the mutations should affect
late development. If this is true, no
new phyla or classes will be found. The
radiation that occurs will be at the genus and family level, corresponding to
minor changes that result from mutations affecting late development. Then the higher taxa should fill in from the
bottom up.
In short, if we imagine
that by the Permian early development in the organisms of most phyla and
classes was well locked in, then after 96 percent go extinct, only traits that
were more minor, presumably those caused by mutations affecting later stages of
an organism’s ontogeny, could be found and improved rapidly.
If these views are correct,
then major features of the record, including wide radiation that fills taxa
from the top down in the Cambrian, and the asymmetry seen in the Permian, may
find natural explanations as simple consequences of the structure of fitness
landscapes. In the same vein, notice
that bushy radiation should generally yield the greatest morphological
variation early in the process. Thus one
might expect that during postextinction rebounds, genera would arise early in
the history of their families and families would arise early in the history of
their orders. Such bottom-heavy clades are
just what is observed repeatedly in the evolutionary record.
Technological Evolution on Rugged
Landscapes
At first glance, the
adaptive evolution of organisms and the evolution of human artifacts seem
entirely different. After all, Bishop
Paley urged us to envision a watchmaker to make watches and God the watchmaker
to make organisms, and then Darwin pressed home his vision of a “blind
watchmaker” in his theory of random variation and natural selection. Mutations, biologists believe, are random with
respect to their prospective significance. Man the toolmaker struggles to invent and
improve, from the first unifacial stone tools some 2 or more million years ago,
to
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the bifacial hand axes of the lower Paleolithic, to
the superbly crafted flint blades hammered free from prepared cores and then
pressure-flaked to stunning perfection. What
in the world can the blind process of adaptive evolution in biological
organisms have to do with technological evolution? Perhaps nothing, perhaps a great deal.
Despite the fact that human
crafting of artifacts is guided by intent and intelligence, both processes
often confront problems of conflicting constraints. Furthermore, if Darwin proposed a blind
watchmaker who tinkered without foreknowledge of the prospective significance
of each mutation, I suspect that much of technological evolution results from
tinkering with little real understanding ahead of time of the consequences. We think; biological evolution does not. But when problems are very hard, thinking may
not help that much. We may all be
relatively blind watchmakers.
Familiar features of
technological evolution appear to bespeak search on rugged landscapes. Indeed, qualitative features of technological
evolution appear rather strikingly like the Cambrian explosion: branching
radiation to create diverse forms is bushy at the base; then the rate of
branching dwindles, extinction sets in, and a few final, major alternative forms,
such as final phyla, persist. Further,
the early diversity of forms appears to be more radical, and then dwindles to
minor tuning of knobs and whistles. The
“taxa” fill in from the top down. That
is, given a fundamental innovation - gun, bicycle, car, airplane - it appears
to be common to find a wide range of dramatic early experimentation with radically
different forms, which branch further and then settle down to a few dominant
lineages. I have already mentioned, in
Chapter 1, the diversity of early bicycles in the nineteenth century: some with
no handlebars, then forms with little back wheels and big front wheels, or
equal-size wheels, or more than two wheels in a line, the early dominant
Pennyfarthing branching further. This
plethora of the class Bicycle (members of the phylum Wheeled Wonders)
eventually settled to the two or three forms dominant today: street, racing,
and mountain bike. Or think of the
highly diverse forms of steam and gasoline flivvers early in the twentieth
century as the automobile took form. Or
of early aircraft design, helicopter design, or motorcycle design. These qualitative impressions are no
substitute for a detailed study; however, a number of my economist colleagues
tell me that the known data show this pattern again and again. After a fundamental innovation is made, people
experiment with radical modifications of that innovation to find ways to improve
it. As better designs are found, it
becomes progressively harder to find further improvements, so variations become
progressively more modest. Insofar as
this is true, it is obviously reminiscent of the claims
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for the Cambrian explosion, where the higher taxa
filled in from the top down. Both may
reflect generic features of branching adaptation on rugged, correlated fitness
landscapes.
A second signature that
technological evolution occurs on rugged fitness landscapes concerns “learning
curves” along technological trajectories. There are two senses in which this occurs. First, the more copies of an item produced by
a given factory, the more efficient production becomes. The general result, as accepted by most
economists, is this: at each doubling of the number of units produced in a
factory, the cost per unit (in inflation-adjusted dollars or in labor hours)
falls by a constant fraction, often about 20 percent. Second, learning curves also arise on what are
called technological trajectories. It
appears common that the rate of improvement of various technologies slows with
total industry expenditure; that is, improvement in performance is rapid at
first, and then slows.
Such learning curves show a
special property called a power-law relation. A simple example of such a power law would be
this: the cost in labor hours of the Nth unit produced is 1/N of the cost of
the first unit produced. So if you make
100 widgets, the last one costs only 1/100 as much as the first. The special character of a power law shows up
when the logarithm of the cost per unit is plotted against the logarithm of the
total number of units produced. The
result is a straight line showing the cost per unit decreasing as the total
number of units, N, increases.
Economists are well aware
of the significance of learning curves. So too are companies, which take them into
account in their decisions on budgets for production runs, projected sales
price per unit, and the expected number of units that must be sold before a
profit is made. In fact, the power-law
shapes of these learning curves are of basic importance to economic growth in
the technological sector of the economy: during the initial phase of rapid
improvements, investment in the new technology yields rapid improvement in
performance. This can yield what
economists call increasing returns, which attract investment and drive further
innovation. Later, when learning slows,
little improvement occurs per investment dollar, and the mature technology is
in a period of what economists call diminishing returns. Attracting capital for further innovation
becomes more difficult. Growth of that
technology sector slows, markets saturate, and further growth awaits a burst of
fundamental innovation in some other sector.
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Despite the ubiquity and
importance of these well-known features of technological evolution and economic
growth, no underlying theory seems to account for the existence of learning
curves. Do our simple insights into
adaptive processes on landscapes offer any help? Again, perhaps, and the story is a typical
Santa Fe Institute adventure. In 1987,
John Reed (chairman of Citicorp) asked Phil Anderson (a Nobel laureate in
physics) and Ken Arrow (a Nobel laureate in economics) to organize a meeting to
bring economists together with physicists, biologists, and others. The institute had its first meeting on
economics and established an economics program, first headed by the Stanford
economist Brian Arthur. I, in turn,
began trying to apply ideas about fitness landscapes to technological
evolution. Several years later, two
young economics graduate students, Phil Auerswald of the University of Washington
and José Lobo from Cornell, were taking the institute’s summer course on
complexity, and asked if they might work with me on applying these new ideas
about landscapes to economics. José
began talking with the Cornell economist Karl Shell, already a friend of the
institute. By the summer of 1994, all
four of us began collaborating, helped by Bill Macready, a solid-state
physicist and postdoctoral fellow working with me at the institute, and Thanos
Siapas, a straight-A computer-science graduate student at MIT. Our preliminary results suggest that the now
familiar NK model may actually account for a number of well-known
features of learning curves: the power-law relationship between cost per unit
and total number of units produced; the fact that after increasingly long
periods with no improvement, sudden improvements often occur; and the fact that
improvement typically reaches a plateau and ceases.
Recall that on random
landscapes, every time a step is taken uphill, the number of directions uphill
falls by a constant fraction, one-half. More
generally, we saw that with the NK landscape model for K larger
than perhaps 8, the number of fitter neighbors dwindles by a constant fraction
at each step toward higher fitness. Conversely, the number of “tries” to find an
improvement increases by a constant fraction after each improvement is found. Thus the rate of finding fitter variants - of
making incremental improvements - shows exponential slowing. The particular rate of exponential slowing
depends, in the NK model, on K. The slowing is faster when the conflicting
constraints, K, are higher and the landscape is more rugged. Finally, recall that adaptive walks on rugged
landscapes eventually reach a local optimum, and then cease further
improvement.
There is something very
familiar about this in the context of technological trajectories and learning
effects: the rate of finding fitter variants
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(that is, making better products or producing them
more cheaply) slows exponentially, and then ceases when a local optimum is
found. This is already almost a
restatement of two of the well-known aspects of learning effects. First, the total number of “tries” between
finding fitter variants increases exponentially; thus we expect that
increasingly long periods will pass with no improvements at all, and then rapid
improvements as a fitter variant is suddenly found. Second, adaptive walks that are restricted to
search the local neighborhood ultimately terminate on local optima. Further improvement ceases.
But does the NK model
yield the observed power-law relationship? To my delight, the answer appears to be yes. We already know that the rate of finding
fitter variants slows exponentially. But
how much improvement occurs at each step? In the NK model, if the “fitness”
values are considered instead as “energy” or “cost per unit,” and adaptive
walks seek to minimize energy or “cost,” then it turns out that with each of
these improvements the reduction in cost per unit is roughly a constant
fraction of the improvement in cost per unit achieved the last time an
improvement was made. Thus the amount of
cost reduction achieved with each step slows exponentially, while the rate of
finding such improvements also slows exponentially. The result, happy for us four, is that cost
per unit decreases as a power-law function of the total number of tries, or
units produced. So if the logarithm of
cost per unit is plotted on the y-axis, and the logarithm of the total number
of tries, or units produced, is plotted on the x-axis, we get our hoped-for
straight-line (or near straight-line) distribution.
Not only that, but to our
surprise - and, at this stage of our work, healthy skepticism - not only does a
power law seem to fall out of the good old NK model, but we find power laws
with about the right slopes to fit actual learning curves.
You should not take these
results as proof that the NK model itself is a proper macroscopic
account of technological evolution. The NK
model is merely a toy world to tune our intuitions. Rather, the rough successes of this first
landscape model suggests that better understanding of technological landscapes
may yield deeper understanding of technological evolution.
I am not an expert on
technological evolution; indeed, I am also not an expert on the Cambrian
explosion. But the parallels are
striking, and it seems worthwhile to consider seriously the possibility that
the patterns of branching radiation in biological and technological evolution
are governed by similar general laws. Not
so surprising, this, for all these forms of adaptive evolution are exploring
vast spaces of possibilities on more or less rugged “fitness” or “cost”
landscapes. If the struc-
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tures of such landscapes are broadly similar, the
branching adaptive processes on them should also be similar.
Tissues and terra-cotta may
indeed evolve in similar ways. General
laws may govern the evolution of complex entities, whether they are works of
nature or works of man.
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The Competitiveness of Nations
in a Global Knowledge-Based Economy
May 2005