The distrust of randomness

One ordinary obstacle for using evolution as an outlook on life seems to be the distrust of randomness. It is hard to believe that evolution as a random search process can be of any good. A common view seems to be that it is hopelessly inefficient and unable to create the order and information apparent in the biological sphere. In fact, the main result of randomness is disorder in agreement with a well known law of nature; the second law of thermodynamics (entropy law).

For a start, let us look at the efficient solution of a difficult combinatory problem by simple random search using the evolutionary principles of random variation and selection in cyclic repetition ; the traveling salesman problem. See also Goldberg, 1989, in references.
The traveling salesman problem

The salesman should visit a number of towns, one at a time, and wants to know in what order they should be visited in order to make the tour as short as possible. Suppose that the number of towns is = 60. For a random search process, this is like having a deck of cards numbered 1, 2, 3, ... 59, 60 where the number of permutations is of the same order of magnitude as the total number of atoms in the universe. If the hometown is not counted the number of possible tours becomes 60*59*58*...*4*3 (about 10 raised to 80, 10^80, 1. e. a 1 followed by 80 zeros).

Suppose that the salesman does not have a map showing the location of the towns, but only a deck of numbered cards, which he may permute, put in a card reader - like in the childhood of computers - and let the computer calculate the length of the tour. The probability to find the shortest tour by random permutation is about one in 10^80 so, it will never happen. So, should he give up?

No, by no means, evolution may be of great help to him; at least if it could be simulated on his computer. The natural evolution uses an inversion operator, which - in principle - is extremely well suited for finding good solutions to the problem. A part of the card deck - chosen at random - is taken out, turned in opposite direction and put back in the deck again like in the figure to the left with 6 towns. The hometown (nr 1) is not counted.




If this inversion takes place where the tour happens to have a loop, then the loop is opened and the salesman is guaranteed a shorter tour. The probability that this will happen is greater than 1/(60*60) for any loop if we have 60 towns, so, in a population with one million card decks it might happen 1000000/3600 = 277 times that a loop will disappear.






I have simulated this with a population of 180 card decks, from which 60 decks are selected in every generation (using MATLAB, the language of technical computing). The figure to the left shows a random tour at start.


After about 1500 generations all loops have been removed and the length of the random tour at start has been reduced to 1/5 of the original tour. The human eye can see that some improvements can be made, but probably the random search has found a tour, which is not much longer than the shortest possible. See figure to the left.



In a special case when all towns are equidistantly placed along a circle, the optimal solution is found when all loops have been removed. This means that this simple random search is able to find one optimal tour out of as many as 10^80. This random process is also similar to evolution in the sense that it uses random variation and selection in cyclic repetition. This also means that the cyclic repetition of random variation and selection of individuals is a very important principle for creating a huge amount of information. So generally, there is no reason to distrust random developmental processes.

The given example shows how random search can solve a combinatorial problem efficiently.

Fisher's fundamental theorem does not maximize mean fitness

Creationists have reason to doubt the theory based on Fisher’s fundamental theorem of natural selection published in 1930. In modern terminology (see Wikipedia) Fisher’s theorem has been stated as: “The rate of increase in the mean fitness of any organism at any time ascribable to natural selection acting through changes in gene frequencies is exactly equal to its genic variance in fitness at that time”. (A.W.F. Edwards, 1994).

It relies on the premise that a gene (allele) may have a fitness of its own being a unit of selection. Historically this way of thinking has also influenced the view of egoism as the most important force in evolution; see for instance Hamilton about kin selection, 1963, or Dawkins about the selfish gene, 1976, in references.


A proof as given by Maynard Smith, 1998, shows the theorem to be formally correct. Its formal validity may even be extended to the mean fitness and variance of individual fitness values.

A drawback is that it does not tell us the increase in mean fitness from the offspring in one generation to the offspring in the next (which would be expected), but only from offspring to parents in the same generation. Another drawback is that the variance is a genic variance in fitness (in vertical direction) and not a variance in phenotypes (horizontal direction). Therefore, the structure of a phenotypic landscape – which is of considerable importance to a possible increase in mean fitness - can’t be considered. So, it can’t tell us anything about what happens in phenotypic space.



And there is something dubious in the premise, so let us have a look at the definition of fitness as for instance given by Maynard Smith 1998, in the following way: ”Fitness is a property, not of an individual, but of a class of individuals – for example homozygous for allele A at a particular locus. Thus the phrase ’expected number of offspring’ means the average number, not the number produced by some one individual.”

Even if the definition is useful in breeding programs, it can hardly be of any use as a basis of a theory of an evolution selecting individuals. It seems to me that this definition denies the fitness of the individual. Nevertheless, the individual fitness is needed, because otherwise the “expected average number of offspring” from a certain class of individuals can’t be determined. In addition, if it possible to define the fitness of an allele as an average of individual fitness values, then, it must be possible to define the mean fitness of a whole population from such values. So it can’t be forbidden to use the fitness of the individual.

Gaussian adaptation is based on the selection of individuals using definition given by Hartl, 1981. The fitness of the individual is the probability s(x) that the individual having the n characteristic parameters x’ = (x1, x2, …, xn) – where x’ is the transpose of x – will survive, i. e. become selected as a parent of new individuals in the progeny. This definition is perhaps less useful in breeding programs, but may be useful in certain philosophical discussions about evolution. If the selection of individuals rules the enrichment of genes, then Gaussian adaptation will perhaps give a more reliable view of evolution. See the preceeding blog “Gaussian adaptation as a model of evolution”.

So, if the selection of individuals rules the enrichment of genes, I am afraid there might be a risk that the classical theory becomes nonsense, and that this is not very well known among biologists.

The image shows two different cases (upper and lower) of individual selection, where the green points with fitness = 1 - between the two lines - will be selected, while the red points outside with fitness = 0 will not. The centre of gravity, m, of the offspring is heavy black and ditto of the parents and offspring in the new generation, m* (according to the Hardy-Weinberg law), is heavy red.

Because the fraction of green feasible points is the same in both cases, Fisher’s theorem states that the increase in mean fitness in vertical direction is equal in both upper and lower case. But the phenotypic variance (not considered by Fisher) in the horizontal direction is larger in the lower case, causing m* to considerably move away from the point of intersection of the lines. Thus, if the lines are pushed towards each other (due to arms races between different species), the risk of getting stuck decreases. This represents a considerable increase in mean fitness (assuming phenotypic variances almost constant). Because this gives room for more phenotypic disorder/entropy/diversity, we may expect diversity to increase according to the entropy law, provided that the mutation is sufficiently high.

So, Fisher’s theorem, the Hardy-Weinberg law or the entropy law does not prove that evolution maximizes mean fitness. On the other hand, Gaussian adaptation obeying the Hardy-Weinberg and entropy laws may perhaps serve as a complement to the classical theory, because it states that evolution may maximize two important collective parameters, namely mean fitness and diversity (average information) in parallel (at least with respect to all Gaussian distributed quantitative traits). This may hopefully show that egoism is not the only important force driving evolution, because any trait beneficial to the collective may evolve by natural selection of individuals.

About my background

In the middle of the 60-ties, I worked at a Swedish telephone company with analysis and optimizations of signal processing systems. Formerly such systems consisted of interconnected components such as resistors, inductors and capacitors.

In the late 60-ties my boss formulated a technical problem: “Try to find system solutions that are insensitive to variations in parameter or component values due to the statistical spread in manufacturing” he said. This means that he wanted the manufacturing yield maximized.




If we have only two components - each having a parameter value – the problem is very simple. Let the first parameter value be the shortest distance to the left edge of a picture (below) while the second value is the distance to the bottom edge. Then, if the interconnection is given, a point in the picture represents the system unambiguously.

Suppose now that all points inside a certain triangle (region of acceptability, marked by red edge) will meet all requirements according to the specification of the system, while all other points does not, and that the spread of parameter values is uniformly distributed over a circle (green). Then, if the circle touches the three sides of the triangle, the centre of the circle would be a perfect solution to the problem.

But if we have 10 or 100 parameters, then the number of possible parameter combinations becomes super-astronomical and the region of acceptability will not possibly be surveyed. I begun to think that the man was not all there.

The problem was almost forgotten until a system designer entered my room about half a year later. He wanted to maximize the manufacturing yield of his system that was able to meet all requirements according to the specification, but with a very poor yield.



Oh, dear! I would not like to get fired immediately. So, we wrote a computer program in a hurry, using a random number generator giving normally Gaussian distributed numbers. The system functions of each randomly chosen system were calculated and compared with the requirements. In this way we got a population (generation) of about 1000 systems from which a certain fraction of approved systems was selected. For the next generation the centre of gravity of the normal distribution was moved to the centre of gravity of the approved systems and this process was repeated for many generations.



After about 100 generations the centres of gravity reached a state of equilibrium. Then the designer said “but this looks very god”. And we were both astonished, because we had only put some things together by chance. A closer look revealed that there is a mathematical theorem (the theorem of normal or Gaussian adaptation), valid for normal distributions only, stating:



If the centre of gravity of the approved systems coincides with the centre of gravity of the normal distribution in a state of selective equilibrium, then the yield is maximal.



This gave an almost religious experience. Here a mathematical theorem solved our problem without our knowledge and independently of the structure of the region of acceptability.Our very simple process was similar to the evolution of natural systems in the sense that it worked with random variation and selection. Later, it turned out that evolution might as well make use of the theorem.