LECTURE 1

Aim of Lecture 1: Coevolutionarily constructable communities of adaptive entities tune the structure and couplings of their "fitness landscapes" to a self-organized critical state.

1.1) COULD THERE BE GENERAL LAWS GOVERNING FAR FROM EQUILIBRIUM SYSTEMS?

1.1.1) Evidence that this is not possible: Universal computation. There is a denumerable infinity of computer programs. In principle, each can be cast in its most compressed form, although it may be undecidable that a given form is maximally compressed. Indefinitely many programs behave in ways which cannot be foretold. An example is the Halting Problem. Here one cannot tell if the program will halt except by running it. In general, then, for such programs, no more compact description of its behavior can be had than running the program. If, by a "law," we mean a compact description of a behavior, then no compact law can be obtained to foretell the behavior of such a program. But, since the program can be realized on a Universal computer, a real non-equilibrium physical system with a supply of electricity can behave in indefinitely many different ways, none of which can be foretold. Thus, no general theory (more compact description) of the behavior of all non-equilibrium systems is possible.

1.1.2) But it is we the programmers who create such programs.

1.1.3) This fact leaves open the possibility that there may be general laws governing the behavior of a particular class of far from equilibrium systems: self constructing, evolving, coevolutionarily constructable systems.

1.1.4) These lectures investigate some of the considerations bearing on such a possible law, or laws.

Preliminary Clues to the Structure of Coevolutionarily Constructable Systems.

1.2) NOT ALL COMPLEX SYSTEMS CAN BE ASSEMBLED BY AN EVOLUTIONARY PROCESS: EXAMPLE: MAXIMALLY COMPRESSED COMPUTER PROGRAM OF N BITS LIVES AS ONE VERTEX ON THE N DIMENSIONAL BOOLEAN HYPERCUBE.

Greg Chaitin has shown that for maximally compressed programs, only of order 1 such N bit sequence, hence one vertex, performs the program. Hence finding the program amounts to identifying one among the 2 to the N vertices as the "correct" one. It is a plausible conjecture that, for most maximally compressed programs, the "fitness landscape" over the N dimensional Boolean hypercube is random. Thus, consider a set of input data, the computation, and a set of output data. Measure the normed distance of the output of an arbitrary N bit program from the "correct" output for that input data. Use this distance as a "fitness." One guesses that, for most maximally compressed programs, ANY single bit change in the N bit program will randomize the resulting fitness measure. If this is true, then to evolve to a maximally compressed program on the N cube is an NP-hard problem. The search process must examine essentially the entire set of 2 to the N vertices to find the correct vertex. Hence, for large N, such a complex system cannot be assembled by an evolutionary process.

1.2.1) Chaitin has shown that for programs longer than the minimal length, N, by q bits, there are of order 2 to the q vertices of the corresponding hypercube which carry out the program. Consider a program twice the minimal length, hence 2N. Presumably the resulting fitness landscape is correlated. Nearby vertices have similar fitness due to the redundancy in the code. Based on the redundancy, suppose it IS possible to evolve to one of the vertices that performs the computation correctly. Now let us ask whether it is expected to be possible to "squeeze" down from 2N, successively to 2N - 1, 2N - 2, ....N+2, N+1, N to evolve the minimal program length N. This seems unlikely. To be useful, each step locating a vertex on the N + q hypercube must help locate biased useful regions to search on the successive N + q - 1 hypercube. As this sequence approaches the minimal sequence, N, (hence as q approaches 1), the vertex found on the N + q cube almost certainly contains less information about where to search on the next smaller cube. At the last step, from N + 1 to N, presumably, the location on the N + 1 cube contains no information about where to search on the N cube. If true, then one cannot evolve to the minimal program by "squeezing down" from more redundant programs.

1.2.2) Finally, consider trying to build up to the minimal program, length N, from a set of smaller minimal programs to carry out "simpler" tasks." Presumably, these cannot in general just be "glued" together end to end to achieve the more complex task. If not, then one cannot build up to a specific minimal program from concatenations of minimal programs for simpler tasks.

THE STRUCTURE OF RUGGED FITNESS LANDSCAPES

1.3) THE NK MODEL AND ITS FEATURES:

i. N sites. Each has "A" alternative states or "alleles." Each site makes a fitness contribution which depends upon the allele at that site and the alleles at K other sites, which serve as "epistatic inputs" to that site. The K inputs to each site may be chosen at random among the N sites, or in any other way. The fitness contribution of each allele of each site, in the context of the A to the K alleles at its K inputs sites, is chosen at random from the uniform distribution between 0.0 and 1.0. The fitness of a "configuration" or state of the entire NK system, namely the vector of allele values for the N sites, is defined as the average of the fitness contributions of the N sites. The total number of configurations of the system is A to the N. For A = 2 alleles per site, the NK model lives on the N dimensional Boolean hypercube. Each configuration is then a vertex on that cube. The NK model assigns a fitness to each vertex, hence creates a fitness landscape.

- Define adaptive walks: A adaptive move is a step from a point (vertex) on the hypercube to a fitter 1-mutant neighbor. ( For A = 2, each vertex has N 1-mutant neighbors. In general, each point has N(A-1) 1 - mutant neighbors. An adaptive walk is a succession of adaptive moves. Such a walk must terminate on a local or the global peak.

ii. Properties of K = 0 "Fujiyama" landscapes.

a. There is a single peak that is the global optimum.

b. The landscape is "smooth" for the fitness difference of 1-mutant (1 bit) neighbors differs by at most 1/N, hence goes to 0.0 as N increases.

c. The expected length of an adaptive walk to the global peak is N/2.

d. The number of directions uphill decreases by 1 at each step uphill.

iii. Properties of K = N - 1 "random" landscapes.

a. The expected number of peaks is (2 to the N)/(N +1).

b. The mean length of adaptive walks to nearby peaks is ln N.

c. The expected number of direction uphill is HALVED at each adaptive step.

If adaptive walks can branch, taking many or all directions uphill from an initial point, this property leads to branching walks that are bushy at the base, branching ever more sparsely, until single lineages wend their ways towards peaks.

Cambrian explosion. Technological radiations.

Due to the "halving" property, the rate of finding fitter 1- mutant variants, given trials in random directions with replacement, SLOWS EXPONENTIALLY. 1 try, then 2 tries, then 4 tries, .... The 30th improvement step takes a billion tries. (Learning curves in economics. The exponential slowing in the rate of finding fitter variants has now been applied to learning curves in industries ranging from semiconductors to apparel and diamond cutting)

d. Expected number of tries to reach a local peak is N.

e. Any point can climb to a tiny fraction of the local peaks.

f. Only a tiny fraction of initial states can climb to the global peak. Hence finding it is NP-hard.

iv. For K increasing from 0 to N - 1.

a. The number of peaks increases, and scales as A raised to the (alpha(k) N).

b. The lengths of walks to peaks gradually decrease.

c. As K increases, conflicting constraints, the analogue of "frustration" in spin glasses, increases. Thus, while the number of peaks increase, the heights of the peaks above the average in the space, 0.5, DECREASES.

d. For K > 8, the number of directions uphill falls by an almost constant fraction, f(k) <= 0.5, at each step uphill, and reaches f(k) = 0.5 for K = N - 1. Thus, for a wide class of rugged NK landscapes, exponential slowing of the rate of finding fitter 1-mutant variants occurs.

Relation to learning curves in economics. Applications to industries ranging from semiconductor to apparel, diamond cutting, truck manufacturing. Success suggests that "rugged landscape - hard combinatorial optimization picture fits much of current real technological world.

e. As K increases from 0, local peaks are initially clustered in a "massif central" with the highest peaks near one another. As K reaches larger values, the peaks "diffuse" over the landscape.

f. As K increases from 0, initially, the highest peaks can be climbed from the largest number of initial positions on the landscape. This property fails for large K.

(Properties "e" and "f" appear to be necessary condition for "recombination to be useful. Presaging central point of lecture 1, since most organisms use recombination, where do the "proper" fitness landscapes come from such that recombination is useful?)

v. The "correlation length" of a landscape: Defined by Ed Weinberger. Consider a random walk on a rugged landscape via randomly chosen 1-mutant neighbors. Write down the fitness value at each step for 1000 steps. Consider the autocorrelation function between two points S steps apart along this random walk. The two point autocorrelation function computes the correlation between any two S steps apart, for S = 1, S = 2, S = 3, ..... For the NK model, the autocorrelation falls off EXPONENTIALLY as S increases. This allows definition of a correlation length on the landscape as that distance, S, such that the correlation is decreased to 1/e of its initial value (at S = 0). As K increases landscapes become more rugged, hence the correlation length DECREASES and becomes fully uncorrelated on random K = N - 1 landscapes since 1-mutant neighbors have fitness values that are uncorrelated (random).

vi. "Three time scales in adaptive walks on rugged but correlated landscapes." At average fitness, half of all nearby mutants and half of all distant mutants are fitter than current location. But fitter mutants, constrained by correlation of landscape, cannot be "much" fitter. Distant mutants, beyond correlation length, can be much fitter. This is of no help if only one "scout" is sent out. If many scouts are sent out and the "best" new position is taken, then initially, it is best to search "far away" on the landscape. Such distant search, however, beyond the correlation length of the landscape, samples a random landscape. Thus, the number of "tries" to find a fitter distant variant doubles after each improvement step. After a while, as fitness improves, fitter variants are found more rapidly near the current location, and the process climbs a local hill or region of hills. Ultimately, the adaptive process is trapped on a local peak and must await a long jump variant which lands at a higher point on the flank of a distant peak.

a. Numerical simulations with NK landscapes confirm this.

b. General implications for Cambrian explosion, post extinction radiations and technological radiations. Early in adaptive process, fittest variants will be found "far away," beyond the correlation length of the landscape. Thereafter fittest variants found progressively closer to current location. This leads to filling in of "higher taxa" - Phylum, class, order, family, genera, from the top down via branching speciation in the Cambrian, and the early emergence of marked variations after a fundamental technological innovation, such as the airplane, then settling down to dominant designs. Rough success suggests that these concepts apply to both biosphere and "econosphere" evolutionary processes.

1.4) HINTS THAT ORGANISMS CONTROL THE RUGGEDNESS OF THE FITNESS LANDSCAPES OVER WHICH THEY SEARCH.

i. Considerations above that to be evolutionarily constructable, systems must evolve on smooth landscapes reflecting REDUNDANCY in the way the "organism" is constructed.

ii. Sex and Recombination. Genetic recombination is supposed to be the reason for sex. It is worth paying twofold price in fitness (to have two parents), because recombination is presumed to be a good search strategy to find useful genes or combinations of genes.

But recombination is only a useful search strategy on relatively smooth, correlated landscapes in which the highest peaks cluster near one another and drain the largest "basins"!

Evidence: NK - recombination only helps for K small. Feldman and colleagues also have shown if a selectable outside gene controls the rate of recombination, recombination is only selected for and increases in model populations if landscape is reasonably smooth.

Most species are sexual:

Therefore, either God gave us landscapes which are smooth and correlated, or Organisms themselves must play a role in controlling the landscapes over which they carry out evolutionary search.

One bets the latter: Then organisms control the character of their own search spaces such that evolution by mutation, recombination and selection succeeds! The capacity to evolve is itself an ACHIEVEMENT of evolution! How?

How? Preliminary ways: Control of pleiotropy, polygeny, sizes of proteins, order versus chaos in genetic regulatory networks, robustness of developmental mechanisms viewed in terms of structural stability and location in large bifurcation volumes in the parameter spaces of those mechanisms.

1.5) INTERLUDE 1: INTRODUCTION TO THE "NO FREE LUNCH" THEOREM OF MACREADY AND WOLPERT.

i. Consider a finite dimensional, compact bounded N dimensional space.

ii. Partition this N dimensional "box" into small subvolumes.

iii. Choose a set of integers, say 0 to 1,000,000.

iv. Consider all possible assignments of these integers to the small subvolumes in the N dimensional box.

v. Let each such assignment constitute a "fitness landscape."

vi. Compare any two search algorithms, each of which specifies a procedure to sample M distinct points on a landscape.

The No Free Lunch Theorem states that, averaged over all landscapes, no algorithm outperforms any other algorithm.

vii. By this theorem, evolutionary "search" over fitness landscapes by mutation and selection, or mutation, recombination and selection will not, averaged over all fitness landscapes, do better than random search, or "downhill search."

viii. The No Free Lunch theorem therefore supports and generalizes the point made about recombination. Since recombination only works well on a subclass of landscapes, and since organisms use recombination, where do those "well wrought" landscapes come from?

ix. More generally, the No Free Lunch theorem suggests that landscapes and search algorithms must be somehow tuned to one another.

x. This begins to suggest that organisms and the fitness landscapes they search cocreate one another! We will return to this theme in Lecture 3. In the meantime, since biological evolution requires that the entities carrying out the search survive at each step in the search process, this constraint seems to imply that biologically generated landscapes will tend to be positively correlated such that, at each point, there are typically neighboring points of the same, higher, or at worst only slightly lower fitness. Were landscapes anticorrelated, search via immediate mutant neighbors could not proceed. In turn, the hypothesis that organisms cocreate the fitness landscapes they explore can be tested by examining whether molecular and other landscapes for known functions tend to be positively correlated.

1.6) INTERLUDE 2: SELF ORGANIZED CRITICALITY.

Bak, Teng, and Wiesenfeld proposed a general model of self organized critical systems. The canonical example is a table onto which sand is continuously and slowly poured. As the sand piles up to its rest angle, sand slides, or avalanches, begin to occur. As sand is added at a steady rate, one considers the size distribution of the avalanches. Many small avalanches and few large avalanches occur. If one plots the logarithm of the number of avalanches at each size on the ordinate and the logarithm of the size of the avalanche on the abscissa one obtains a straight line sloping down to the right. Thus, the size distribution is a power-law, frequency as a function of size of avalanche.

Bak and coworkers argue that many phenomena are self-organized critical. Power-law distributions arise at phase transitions, but typically require tuning of parameters to achieve the phase transition. Here, no external tuning of parameters is required. The system self organizes to a critical, poised, state.

Bak and coworkers assemble evidence that earthquakes, forest fire models, the game of life, and other systems exhibit self-organized critical behavior.

1.7) SECOND CLUE ABOUT HOW ORGANISMS CONTROL THE STRUCTURE OF THEIR SEARCH SPACES:

COEVOLUTION: ORDER, CHAOS, AND COEVOLUTION TO THE EDGE OF CHAOS.

i. Coevolution. The frog and the fly. Adaptive moves by the frog - sticky tongue - alter the fitness of the fly and DEFORM its landscape. Fly should develop slippery feet, sticky-stuff desolver, ...

As frog population climbs towards the peaks of the frog landscape, the fly landscape deforms, and vice versa. Coupled deforming landscapes.

Unlike a fixed fitness landscape with a potential function - fitness, hence with local peaks as point attractors, coevolutionary systems are general dynamical systems. Such systems may have point attractors - an Ordered Regime, or a Chaotic Regime.

ii. The two major regimes:

a. Evolutionary stable strategies - the ordered regime. Here, coevolving partners climb to local peaks that are MUTUALLY CONSISTENT. Each species, "player" is better off not moving as long as other "players" do not move. Analog of Nash Equilibrium in game theory, here with constraint on search distance from current position. ESS, Evolutionary Stable Strategies, adds condition of non-invadability by other variants.

b. The Red Queen - the chaotic regime.

Here, each coevolving partner chases peaks that move away from it faster than it can climb, each clambering forever uphill on deforming landscapes. The total system of species - "players" - flows through large regions of space of possibilities.

iii. NKCS Model of Coevolving Species.

a. Each species "lives" on an NK landscape, and is assumed to be isogenic - hence occupies a single point on the landscape.

b. NK landscapes of each species is coupled to that of S other species. For each coupled pair, say frog and fly, each of the N sites in the frog is affected by K sites within the frog, and by "C" sites within the fly. Reciprocally, each site in the fly is affected by K sites within the fly and C sites within the frog. The effects are modeled by expanding the random fitness function of each site in the frog to look not only at the alleles of its K internal sites but the corresponding alleles of the C sites of the fly (and vice versa). Random fitness values between 0.0 and 1.0 are added to these new combinations. Thus, when the fly, moves on its landscape by changing the allele at one site, that change affects the fitness contributions of C sites in the frog. Adaptive moves by flies deform the landscape of the frog population, and vice versa.

c. The ordered regime: When K is large relative to the product C x S, a model ecosystem of S species reaches a "Nash Equilibrium" where each species attains a local peak and is better off not moving to any 1-mutant variant as long as the others do not move. This is the analogue of an ESS.

d. The chaotic regime: When K is small relative to the product of C x S, all species continue to be able to find fitter 1-mutant variants. The total system flows through the product space of the NK "genotypes" over very long times.

e. The Coevolutionary Edge of Chaos. When the parameters of the NKCS model are tuned, for example, N, C, and S are held constant, and K is varied from 0 to N - 1, the coevolutionary system is initially in the ORDERED REGIME then switches to the CHAOTIC REGIME at a critical value of K, Kcrit.

Thus, the coevolutionary system passes a PHASE TRANSITION BETWEEN ORDER AND CHAOS - the EDGE OF CHAOS.

1) Evidence for the edge of chaos: Time for model ecosystem to "freeze" onto Nash equilibrium very very long (unobservable) for K < Kcrit. Time to freeze is short for K > Kcrit. System is freezing slowly over observed time scale at Kcrit. (Note relevance of time scale of observation here.)

f. The Highest Mean Fitness is Found at the Edge of Chaos!

1) As K increases from 0 to N - 1, fitness increases then decreases. Maximum occurs just when system begins to freeze slowly over observed time scale.

iv. Coevolution to the Edge of Chaos: Evidence of a Self-Organized Critical State as Attractor.

a. Model the evolution of coevolution by generalizing NKCS model to allow species to alter ruggedness of their own landscape by increasing or decreasing K, and to invade one another's "niches."

b. At each moment, for each species, one of four things may happen: 1. Species remains unchanged in genotype. 2. Species makes an adaptive move to a fitter 1-mutant variant. 3. Species increases or decreases K of all N genes by 1. 4. Another species, godzilla, sends a copy, godzilla prime, to try to invade species niche by playing with species S neighbors.

c. At each moment, "fittest thing" happens. Thus, species may remain unchanged, move ON its landscape, Change Ruggedness of landscape. Or, if godzilla prime plays better with S neighbors, the "home" species is declared extinct, and godzilla prime is instantiated as a new species in that niche.

Note that invasion by species allows those with good landscape ruggedness, K, to reproduce - godzilla has a daughter species, godzilla prime. So landscape ruggedness can itself evolve via replicators. Thus, landscape ruggedness can evolve without "group selection" and hence

AS IF BY AN INVISIBLE HAND!

Note too that species number is held fixed. Each extinction event is matched by a corresponding speciation event.

d. Expectation of AVALANCHES OF EXTINCTION EVENTS. When godzilla prime invades a niche, its own fitness in the new niche is likely to be low. In addition, the S partners of the now extinct species were "used to" playing with it, not with godzilla prime. They are likely to be less fit than before when playing with godzilla prime. Thus, godzilla prime and its S partners are likely to be less fit, so more likely to be INVADABLE by other species. Hence we expect avalanches of extinction events propagating from godzilla prime.

e. Results show:

1) That K does evolve to an INTERMEDIATE VALUE, OR RUGGEDNESS then fluctuate in a narrow band. Thus landscape smoothness is itself evolvable, as if by an invisible hand.

2) That mean fitness increases, (and may be maximized).

3) That the probability of extinction DECREASES, (and may be minimized).

4) Many small and fewer large AVALANCHES of extinction events propagate through the system.

5) The extinction avalanches show a power-law distribution. Frequency of avalanches at a given size (number of species that went extinct in avalanche) decreases as a power of the size of the extinction "event."

1.8) CONCLUSIONS:

This is the first model hinting that a coevolutionary system, in which selection acts only at the level of the INDIVIDUAL, hence, as if by an invisible hand, can tune landscape smoothness to an intermediate value. Organisms can therefore plausibly tune the statistical structure of their search spaces!

This is the first hint that such a system may achieve an analogue of a "self organized critical state."

This is therefore, a first hint that such a self organized critical state may be a GENERAL attractor for complex adaptive systems able to tune the structure and couplings among their landscapes.

This tentative attractor will emerge as part of the candidate "law" for thermodynamically open, self-constructing, coevolutionarily assemblable systems.

vi. Supporting the clue that complex adaptive systems coevolve to the edge of chaos: Tom Ray's Tierra critters, reproducing programs competing in a computer core, give a power-law distribution of sizes of extinction events.

vii. Further supporting clue: Actual size distribution of extinction events in the Record is almost a power law, but bent over at the large avalanche size, perhaps due to finite size effects. Permian extinction eliminated 95% of all species, so near upper limit.

viii. Further supporting clues lie in similarity of life time distributions on generalized NKCS model of species, where most die soon after formation, but a small fraction live a long time before extinction, and a similar distribution at the level of genera in the Record.

ix. Fourth clue supporting coevolution to the edge of chaos may lie in technological coevolution and Schumpeterian gales of creative destruction in technologies. When the car came it, out went the horse, buggy, saddlery, smithy, pony-express; in came gas stations, paved roads, motels, fast food restaurants, traffic lights, tickets, etc. Is there a power-law distribution in technological speciation and extinction avalanches?

1.9.) COEVOLUTION AT THE EDGE OF CHAOS AS AN OPTIMAL PROBLEM SOLVING PROCESS: DIVIDE TO COORDINATE.

i. Selfish patches. Using NK model on a square lattice, partition the lattice into non-overlapping square patches. Minimize energy, or maximize fitness, within each patch by 1-mutant variants - or flipping single spins. If system is partitioned into a single or a few large patches, it rapidly "freezes" on poor local optima. If partitioned into very many tiny patches, the system remains in a chaotic regime. Thus, tuning patch size tunes from an ordered to a chaotic regime. The optimum behavior is found near the phase transition between order and chaos. Here, as if by an invisible hand, the system of selfishly optimizing (hence coevolving) patches, optimizes the optimum obtained.

ii. Parallels of selfishly optimizing patches and organizational "flattening." Optimal decentralization creates adaptive organizations. Trick is to avoid trapping on poor local optima by ignoring some of the constraints some of the time. Optimal systems to be adaptive in fixed, or changing external world may be generically posed between order and chaos.

1.10) The general character of coevolutionary assembly in human ventures: Common Law, Standard Operating Procedures, Design of complex artifacts, Learning on a shop floor. In all cases, the coevolving parts and processes adjust to persistent minor and major changes in adjacent parts, process, precedents. If any part were too compressed, too dramatically altered by any small changes in structure, it would not be possible to assemble the complex system by such a process of persistent mutual adjustment. Thus, as with organisms in ecosystems, coevolutionary assembly of human artifacts and institutions requires that each "part" be robust enough typically to change only slightly when its structure is altered, and connected to other parts in such a way so as to avoid trapping on poor local optima, or lapse into chaos.

Tentative beginning image: Adaptive Agents in ecosystems and econosystems must create coevolutionarily constructable systems. To do so, such agents tune internal redundancy and couplings with one another to achieve this. In doing so, such systems may, as if by an invisible hand, generically tune to a self organized critical state.