Computer Simulation of Cultural Drift:

Limitations on Interstellar Colonisation

by

William Sims Bainbridge

Journal of the British Interplanetary Society, Volume 37, pp. 420-429, 1984.

Contents:
Introduction
Cultural Drift
Colonisation in a Sphere of 1,000 Stars
Effect of Ship Range
Effect of Initial Colonisation Probability
The Power of Drift
Conclusions
References

Introduction

Many serious essays have been written about the paradox that extraterrestrial civilisations capable of interstellar colonisation are believed to be numerous, yet none have been observed by earthlings [1-4]. Frank J. Tipler [5-8], convinced that unlimited colonisation will be undertaken by a significant proportion of advanced civilisations, concludes we may be the only intelligent species in the Galaxy. If so, attempts to detect radio signals from extraterrestrial civilisations, a currently popular idea [9 ], are doomed to failure. An alternate possibility, suggested at least as early as A. E. van Vogt's 1942 novelette, "Asylum" [10], is that we are currently surrounded by an interstellar civilisation which prefers to maintain us in our native state as a zoo or reservation for primitive culture. If so, any attempts on our part to detect extraterrestrial signals are superfluous, since ETs observe us constantly and will make themselves known as soon as we have developed sufficient cultural maturity [11]. Tang [12], in contrast, suggests that the apparent absence of extraterrestrials in our Solar System may result from a tendency of higher civilisations to undertake radio communication in preference to physical exploration and colonisation. Thus from the same supposed facts, different authors reach utterly different conclusions about the prevalence of extraterrestrial civilisations and about the value of trying to detect messages from them.

Most authors ignore the contributions which the social sciences might make toward a resolution of the problem. Both Hart [13] and Singer [14] reject "sociological explanations" without citing so much as one standard sociological publication, theory or professional school of thought. To be sure, contemporary sociology is a discipline fraught with disagreement, politicisation and incompetence, but much solid work relevant to the questions at hand has been done. There may exist natural sociological laws which render interstellar societies impossible or which severely limit their expansion. The fact that contemporary social scientists have failed to develop a consistent general model of civilisation may reflect the immaturity of the field rather than the impossibility of such an undertaking. This paper will offer an initial sociocultural model of interstellar colonisation, based in a coherent theory of social behaviour, and use microcomputer simulation to show that it is capable of resolving the paradox.

Several writers have based models of colonisation on standard approaches in population biology and population genetics, often citing The Theory of Island Biogeography by Robert H. MacArthur and Edward O. Wilson [15 ]. This book presents mathematical analyses of species competition in restricted ecological niches as individual members of the species accidentally wander to new environments. While the old theory of panspermia suggests microscopic spores might drift between the stars, interstellar colonisation by multi-cellular organisms cannot happen accidentally, and competition between species will not occur when a single civilisation expands into uninhabited regions of the Galaxy. Wilson, among others, has recently extended quantitative evolutionary theory to the study of cultural development [16-18 ], an approach which is more appropriate for the present questions.

Some analyses have considered interstellar colonisation as a problem in population growth [19-21]. Of course, ordinary colonisation cannot proceed faster than the rate of population increase. But other factors, such as those discussed here, may determine lower rates, and colonisation is not an effective solution to the problem of population explosion on the home world [22]. And the use of concepts from demography has been quite limited in favour of models drawn, as Newman and Sagan comment, "from blast wave physics, soil science, and, especially, population biology" [23, p. 293].

Tipler believes that population biology concepts are appropriate for analysing colonisation by self-reproducing "von Neumann machines," and other authors have suggested that advanced civilisations may produce or evolve into machine cultures capable of unlimited automatic expansion [24, 25]. While Tipler's many far-ranging assumptions have drawn intense criticism, I will merely point out two reasons why we should set his radical scenario aside for the time being. First, his ideas seem designed to eliminate all parameters which might limit colonisation, while we can best advance our understanding of alternative possibilities through theoretical models of colonisation shaped by a variety of restrictive parameters. Second, other writers have not completely succeeded in showing how the paradox of an uncolonised Earth could be resolved in any way other than Tipler's, a task whose accomplishment is sketched below.

Of all the competing schools of sociological thought, the one closest in approach to the physical sciences is sometimes called Behaviourism. Disliking "isms" and convinced that parallel developments in economics, psychology, anthropology, ethology and sociology should be combined, I prefer the term Behavioural Science. This theory uses not the gene or the reproducing population as its unit of analysis, but the reward-seeking individual animal or person. It seeks to derive the behaviour of all organisms, including all possible species of intelligent, social beings, from abstract "axioms" presumed to be universal in application, although discovered from experiment and observation of terrestrial rats, pigeons and persons. In his monumental Sociobiology, the New Synthesis, Edward 0. Wilson [26, p. 551] says this approach is compatible with theories of behaviour based in evolutionary biology, so it is quite possible that it represents the intellectual seeds from which a definitive science of society will grow, despite the fact that most sociologists currently work from other perspectives.

In 1938, B. F. Skinner stated the key proposition: "If the occurrence of an operant [unit of spontaneous behaviour] is followed by presentation of a reinforcing stimulus, the strength is increased" [27, p. 21]. Applied to human beings, this became the first of five propositions from which George C. Homans believed he could derive all major forms of social behaviour: "For all actions taken by persons, the more often a particular action of a person is rewarded, the more likely the person is to perform that action" [28, p. 16 ]. Upon such simple and familiar foundations, some rather elaborate theoretical structures have been built. Already in 1941, Neil E. Miller and John Dollard [29 ] showed that even the subtle tendency of humans to imitate the behaviour of others could be explained through such ideas, and the same approach has been used by Peter M. Blau [30] to analyse social power which some individuals may gain over others. Recently, working with Rodney Stark, I have shown that Behavioural Science models are quite capable of explaining even the apparently complex and transcendental phenomena of religious behaviour and belief [31-33].

This theoretical tradition suggests two main propositions with implications for interstellar colonisation. First, as George Homans [34, 35] has repeatedly shown, large social structures (such as civilisations) should be precarious constructions, liable to collapse because they are so far removed from the social exchanges between individuals which are postulated to be the basic constituents of society. Competing social theories, such as Marxism or Structural-Functionalism (popular in mid-century sociology and anthropology), postulate the existence of regular laws operating on the level of whole societies, thus capable of providing permanent societal coherence and guiding evolution to higher levels of development such as interstellar civilisation. But if Skinner and Homans are correct, such laws do not exist, and all social behaviour ultimately serves individual needs.

Second, classical Behavioural Science predicts that societies will not undertake collective projects unless these give a high, short-term profit to those individuals who invest in them. Together, these propositions seem to preclude interstellar colonisation, because the first suggests civilisations will collapse before they colonise, while the second says they will have no motive to plant commercially unprofitable colonies at huge expense across the vast gulfs of interstellar space.

However, the recent application of Behaviourial Science principles to the understanding of religious commitment and radical social movements has shown that a wide range of historical developments may sometimes occur even if they appear quite irrational in terms of individual material gain. From this perspective, in 1976, I wrote that space flight and colonisation were unlikely undertakings for a civilisation, but not impossible [36]. I argued that they could happen under unstable social conditions with the cooperation of historical accidents, although under "normal" conditions in societies they would not take place. This brief preface is not the place for an extended theoretical analysis of the Behavioural Science position on these matters, let alone a comparison with alternative viewpoints. But let us hypothesise, for present purposes, that universal laws of social behaviour do exist, and that they ensure that the probability of an advanced civilisation undertaking interstellar colonisation in any historical era is low.

These assumptions may or may not be correct, but they are plausible and can be derived from at least one contemporary social-scientific theory. In what follows, my task is to examine through computer modelling what the implications are for the rate and shape of interstellar expansion. While our approach will be similar to that of work based on population biology, the key concept will concern not biology but culture.

Cultural Drift

In the Behavioural Science model, there will be rare times when a civilisation develops political tensions, social movements, a conquering religion, or some other unstable structure capable of mobilising great social energies for the accomplishment of a project lacking ordinary utilitarian justification. Since my first book was a social history of the development of modern space rocketry written from this perspective, I naturally imagine scenarios of interstellar colonisation involving a space flight social movement able to exploit dangerous political conflict such as that between the US, USSR and Nazi Germany. But other scenarios are possible. All we assume is that a civilisation somehow develops a culture with an unusually high probability of colonising [36-39].

Societies change, often at random in areas which are not crucial to the survival or economic prosperity of their members. Random cultural change, likely in areas which are not of great practical value to members of the society, is called cultural drift. Perhaps the most obvious example is in language. Over time, pronunciation, spelling, vocabulary and even grammar will change. While some of this is evolution in response to changes in technology and natural conditions, much of it apparently represents a "drunkard's walk" chance wandering of the language among an infinite number of equally-functional alternatives. While this random process will produce even unlikely outcomes, so long as they are possible, these outcomes will typically be rare not only because they are hard to find but also because they are easily lost. A familiar statistical phenomenon known as "regression toward the mean" represents the tendency of a case (here a society) with an extreme value on some variable (here the probability of colonising) to change in the direction of the typical value of the variable. That is, a culture which does beat the odds and begins to colonise will tend to drift back away from colonisation.

How can we represent a cultural disposition to colonise and cultural drift away from this characteristic? I have already presented the simple idea of probability of colonising within an historical era. How long an "era" is in years matters not very much for present purposes, because a low probability in a single year can be represented by a higher probability in a century and a still higher probability in a millenium. Here we will let "era" be some significant period of time, comparable to that required for a colony to be established and grow to the point that it would be capable of sending out colonies of its own. The probability of actually colonising in such a period we represent by C, which, since it is a probability, is a real number between zero and one.

At this stage in our theorising, we cannot give a definite law for cultural drift. However, for the purposes of simulation, we can suggest an abstract approximation of what such a law should look like. We want a formula which can be applied to C to produce a new value for the probability of colonising, that is, one which will generate Cl as a function of C0. Obviously, Cl, and all possible descendants of it resulting from repeated application of the formula, must, like C0, range between zero and one. The simplest function which accomplishes this lets Cl equal C0 taken to some positive exponent, X.

What factors should we put into X? Since we want to use the formula in computer simulations, we should have some parameter, P, which will be a constant in any given run but can be varied across different simulations and which can be used to fine-tune the program. Also, a key concept in this approach is that chance plays a crucial role in cultural drift. Therefore, X should also include a random factor, R. As it happens, the language in which my simulations were written, like many others, includes a function which produces essentially random numbers ranging between zero and one. Consideration of a few alternatives convinced me that the best way to combine P and R into X was simply to add them (X = P + R). Or, to state the cultural drift transformation function in BASIC:

Cl =C0^(P+RND(l)).

This formula incorporates the assumptions we have drawn from Behavioural Science theory in the simplest way which will work in a computer simulation. Of course, one can imagine quite other formulae, and sociology is not yet able to specify the version providing the closest approximation to reality. In demography, as Nathan Keyfitz [40] has pointed out, a simple formula often provides the best approximation to reality, although one cannot be sure, of course, in the absence of data. The reader will see how the formula actually operates below, and I provided the full BASIC program in the appendix of the original journal article (but not reproduced here), so that interested readers could experiment with alternative formulae, as well as with different values for the parameters and extensions of the model to include other ideas as well. Table 1 shows how the formula operates with a variety of values for the parameter P.

TABLE 1. Distribution of Drift Effects Over 1,000 Trials.

P (Drift
Formula
Parameter
  After 5 Generations:     After 10 Generations:  
  Mean C     Percent .5+     Mean C     Percent .5+  
.5.54761.2.59265.5
.55.46346.4.44645.4
.6.37530.2.28623.6
.65.29917.9.1699.3
.7.2136.2.0751.8
.75.1493.1.0280.6
.8.0960.5.0070.0
--------------------
.618034.35025.3.24816.5

Table 1 does not report results of full simulations, but instead gives the distribution of outcomes after applying the formula 5 and 10 times to an initial probability C = 0.5 of colonising in a given era. We imagine that a home world with initial C = 0.5 plants a colony at another star. In doing so it also plants its culture, but the culture drifts during the colonisation process. Exactly which technical means are used to establish the colony is not crucial. Heinlein has suggested that cultural drift is quite likely in city-sized spaceships which take generations to reach their destinations, while Logan has said the same will happen even in small, fast ships which carry human genetic material from which to build a colony automatically [41, 42]. So when a colony is planted we apply the formula to its parent's culture to determine the new Cl of the first generation colony. After a suitable time, the colony may itself colonise, and we apply the formula to C1 to get C2. From C2 we get C3, and so on to C5 and C10 which are reported in Table 1. Each trial represents a chain of colonisation linking eleven worlds, the home world and ten generations of offspring.

Since our formula includes R, the random factor, each time we calculate out 5 or 10 generations, we will get a different result. Therefore, Table 1 gives the results not of single runs but rather the distribution of outcomes from 1,000 trials for each value of P. In the first row, we look at P = 0.5, for example. The mean value of C5 after 1,000 trials is 0.547, and after 10 generations of colonies, the average C10 is 0.592. Thus, for P = 0.5 the value of C gradually rises, on average. In 61.2 per cent of the trials C5 > 0.5, while in 65.5 per cent, C10 > 0.5. Thus, in a majority of cases, the probability of colonising increases gradually, while in a smaller number of cases, the probability decreases.

But our theory indicates that colonisation is generally unlikely, and the typical C should be low. Therefore, cultural drift should make C drop most of the time, which is the case for the other rows in the table. For P = 0.8, the rate of drop is very steep, and not a single one of the 1,000 trials at that value for P results in C10 > 0.5. There is a reason for preferring the value P = 0.618 (or P = 0.618034 ... ), a justification which has some theoretical force, albeit weak. This happens to be the positive number which differs from its reciprocal by one. Since R ranges between zero and one, this means that X, which is P + R, ranges from 0.618 to 1.618. Since this establishes the range of the exponent, X, and 0.618 = 1/1.618, then regardless of what happens in one step of cultural drift, a second step can exactly reverse it. At P = 0.618, there is a very moderate cultural drift downward in C, entirely caused by the fact that 61.8 per cent of the time X will be greater than 1, and if 0 < C < 1 and X > 1 then C to the X must be less than C. One could consider other values for P, but 0.618 is a very gentle assumption of downward drift to begin with.

Colonisation in a Sphere of 1,000 Stars

The simulations reported here all track the expansion of a single civilisation with its home world at the centre of a sphere of 1,000 colonisable solar systems. What counts as "colonisable" depends both on the technology available to the colonisers and on the social motivations which impel the expansion. If objects like Rhea and Iapetus are colonisable, than many solar systems are open to colonies, since these satellites of Saturn must represent a very common type of body. If, on the other hand, very Earthlike planets are required, colonisable solar systems must be, on average, further apart [43, 44]. With the home world at the centre, colonisable stars are distributed randomly in the volume of a sphere 1,000 distance units across. The size of this unit takes on meaning only in relation to the maximum range of colonising spaceships, and this range will be a parameter varied across simulations [45]. Therefore it is not necessary to decide the absolute diameter of the sphere, although for those who want a definite image of our star globe, I suggest a diameter between 100 light years and 100 parsecs.

In the first simulation, we will let the maximum flight range of spaceships be 100 units, and begin with a home world which colonises with a cultural probability of doing so equal to one half (C = 0.5). The run goes for 15 eras, and in each one every inhabited solar system which has not already colonised will be given a chance to do so equal to its value of C. When a star colonises, it plants offspring on every uncolonised world within the reach of its ships, here a radius of 100 units. Each new colony gets a value of C derived by applying the formula to the C of its parent. If a star misses the opportunity to colonise in a given era, cultural drift still occurs, but through a change in its own culture. That is, the formula is applied to the non-colonising star's old C, to produce a new C value which will be its chance of colonising in the following era. While different runs will give different results, the one reported in Table 2 is quite typical.

There are nine stars in the first generation of colonies, so counting the home world there are ten societies after the first era. The farthest colony is 93.4 distance units from the home world, while the nearest uncolonised star, just beyond the 100-unit range of the ships, is 103.8 units away. In the second generation, 12 societies are added, bringing the total to 22. The shape of the interstellar civilisation has begun to get lopsided, since the farthest colonised star at 146.3 units is now more distant from the home world than the nearest star not colonised, at 114.0. By the fifth generation, when the total number of societies has risen to 99, this irregularity is more pronounced, the farthest colony being 2.3 times the distance of the nearest uncolonised star, 389.2 units compared with 168.9. In the next generation the expanding wave-front of colonisation comes within reach of the surface of the sphere of stars, and in the eleventh achieves its maximum distance from the home world, 499.4 units.

TABLE 2. Simulation I: Colonisation with Cultural Drift.

   Distance from Home World to: 
Gener- 
ation 
 Number of 
 Societies 
 Nearest Star 
 not Colonised 
 Farthest Star 
 Colonised 
110 103.893.4
222 114.0146.3
333 118.4222.1
463 166.7299.3
599 168.9389.2
6167 170.5451.2
7251 170.5476.4
8374 247.9495.9
9531 248.6495.9
10674 252.9499.3
11833 290.7499.4
12916 290.7499.4
13948 290.7499.4
14965 290.7499.4
15975 389.6499.4
Initial colonization probability = .5
Maximum flight = 100
Cultural drift coefficient = .618

Two facts stand out in Table 2. First, as just noted, the expansion becomes rather irregular in form, with some near-in stars escaping colonisation until quite late [46, 47]. Second, the expansion takes rather longer than it might without cultural drift. A star only 290.7 units distant from the home world is not colonised until the 15th generation, although ships might have reached it in the third or fourth generation if distance were the only factor. The range of the ships, 100 units, would take them to the surface of the sphere in only five flights, but of course the colonisation process must use stars as stepping stones, and they are distributed randomly. So part of the reduction in speed of the colonisation wave is due to star placement and part, only, to cultural drift.

The fact that our computer simulation includes 1,000 stars, and not a much larger number, means that the ships will reach the edge soon in some direction and be unable to continue colonisation beyond the radius of 500 units from the home star. Of course, to double this radius with the same density of stars would require increasing the total number to 8,000, and increasing the radius by a factor of ten would require a simulation with 1,000,000 stars. The present program, given in the appendix to the original journal article, searches through the entire list of uncolonised stars each time a society colonises, to see which are within the flight distance, a time-consuming procedure. Although this time could be reduced through various programming tricks in cases where the maximum flight distance was a small proportion of the sphere radius, in general the time the program runs is an exponential function of the number of stars. The current generation of microcomputers could not handle more than 2,000 stars, anyway, and before we rent an expensive Cray I supercomputer for our simulations we should work out the basic principles within the limitations imposed here.

Even if our sphere contained one million stars, the tardy society 290.7 units out (not established until generation 15 in Simulation I) could not have been planted much earlier. Imagine a salient or pseudopod of colonisation, carrying culture with a high C out in some direction from the home star, and a tardy section of wave front a few tens of degrees of arc away from it. While the more active salient could reach out and then send an arm of colonisation back into this hole, it would take it a few generations to do so, even if no colonisation opportunities were missed. Of course, in the real Galaxy there are rifts in space, like gaps between spiral arms and areas around gaseous nebulae, where the density of habitable planets is unusually low. We should be aware of the limitation imposed on our simulations by the surface of the star sphere, but can discover the chief principles illuminated by our model quite effectively within it.

Effect of Ship Range

The wave of colonisation appears to slow down in Table 2, adding the largest number of societies in generations 9, 10 and 11, trailing off after that. Of course, it has reached the edge of the sphere and approached the end of the supply of uncolonised stars at this point. But if the wave moved more slowly, perhaps it would die out well within the 1,000 star limit. Simulation II, described in Table 3, explores this possibility by setting a more modest ship range, 80 units rather than 100. This slight difference in range produces a large difference in the number of stars which can be reached from any starting point, however, on average reducing it by almost exactly half. While the farthest edge of the interstellar civilisation reaches 490 units in the 8th generation of Simulation I, in Simulation II it takes 12 generations. At this pace, one would project that the second simulation would achieve before 25 generations what the first did in 15, but Table 3 shows that this is far from the case.

TABLE 3. Simulation II: Reduced Maximum Flight.

   Distance from Home World to: 
Gener- 
ation 
 Number of 
 Societies 
 Nearest Star 
 not Colonised 
 Farthest Star 
 Colonised 
17 80.077.8
215 93.4125.0
316 93.4166.7
425 93.4241.8
532 93.4253.7
641 102.3278.8
748 102.3302.1
859 102.3338.2
967 102.3408.4
1077 102.3445.9
11102 102.3486.0
12125 118.4490.3
13163 118.4490.3
14186 159.9490.3
15210 159.9492.5
16229 159.9492.5
17239 159.9492.5
18263 159.9499.9
19276 159.9499.9
20294 159.9499.9
21318 159.9499.9
22334 159.9499.9
23347 159.9499.9
24364 159.9499.9
25377 159.9499.9
Initial colonization probability = .5
Maximum flight = 80
Cultural drift coefficient = .618

After 25 generations, this simulation with a ship range of 80 units has colonised only slightly over a third of the stars, compared with nearly all after 15 generations in the first run. The nearest uncolonised star is only 159.9 units from the home world, ideally just two flights out. The rate of colonisation reaches its maximum between generations 11 and 15, and drops by a third for the last ten generations. While the simulation ends before colonisation has ceased, clearly it will take a very long time to incorporate the whole star cluster in the slowly expanding civilisation.

Because chance plays a major role in these simulations, both in the operation of cultural drift and in the original placement of stars, runs using exactly the same parameters can come to very different results. A second attempt identical in assumptions to Simulation II resulted in 92 societies at generation 25, rather than 377. It stalled completely after planting the 166th colony in generation 68, with a probability of colonising onward (C) of only 0.009. In this case, cultural drift brought expansion completely to a halt. Of course a ship range of only 80 units may mean that many stars cannot be reached at all, and it is worth trying runs in which C is always 1 and drift downward does not occur, to determine the maximum likely rate of expansion. In one such, ships with a range of 80 units built a civilisation of 292 societies after 10 generations, nearly half the rate in Simulation I and about four times that in Simulation II.

Effect of Initial Colonisation Probability

Rather than varying ship range, we can adjust the initial probability of colonisation, C. A high initial C represents a rare culture in which colonisation is highly likely. A low initial C represents a society in which very special political or other historical events of low probability might start colonisation, but in which the culture transmitted to colonies is not especially conducive to further colonisation. Thus, one might think that we can stall colonisation simply by lowering C. All the runs reported in this paper assume the home world will colonise at its first opportunity, rather than missing early chances and letting its C drift downward and downward. Of course, one could set C abysmally low, so that a second generation of colonies would be almost impossible. But I think the most elegant simulations would permit development of a multi-generation interstellar civilisation before expansion stalls, so we will inspect what happens is we lower C only moderately below the 0.5 level.

Table 4 reports results of Simulation III in which the ship range was set back to 100 units but initial C was reduced from 0.5 to 0.4. After 10 generations, the expansion has fallen behind that of Simulation I, producing a civilisation of 321 societies compared with 674. But by generation 15 it has begun to catch up, and at generation 40 achieves what the first simulation did by generation 15, a total of 975 societies. At this point, Simulation III stalls, and even by generation 75, no further stars have been colonised. While 25 stars remained uncolonised, clearly something happened around generation 10 that kept expansion from stalling before a majority of stars were colonised, a factor capable of overcoming the disadvantage of a low initial probability of colonising.

TABLE 4. Simulation III: Reduced Initial Colonisation

   Distance from Home World to: 
Gener- 
ation 
 Number of 
 Societies 
 Nearest Star 
 not Colonised 
 Farthest Star 
 Colonised 
16 102.498.7
212 103.8150.5
320 103.8204.3
435 103.8250.2
558 115.4295.2
6103 130.4348.7
7156 130.4358.1
8196 130.4432.3
9256 130.4488.8
10321 130.4496.6
11403 130.4497.4
12499 148.6497.9
13584 183.1499.3
14687 245.3499.3
15761 298.4499.6
-------- --------
20921 298.4499.6
25951 298.4499.6
30960 298.4499.6
35974 298.4499.6
40975 298.4499.6
-------- --------
75975 298.4499.6
Initial colonization probability = .4
Maximum flight = 100
Cultural drift coefficient = .618

A principle of natural selection is responsible. Recall that cultural drift can increase C as well as reduce it, although on average the majority of colonies will have lower Cs than the societies which established them. But offspring with increased C will have more chance of colonising immediately than their siblings with lower C. If a low-C society fails to colonise in one generation, its C will tend to drift further down, and when chance does let it colonise, all worlds within a 100-unit radius may already be inhabited. Colonising societies with higher than average C will transmit this superiority to their progeny. While cultural drift will tend to reduce C across generations, natural selection will tend to increase it.

We can compare the actual behaviour of Simulation III with results from a run like those reported in Table 1. On 1,000 trials, starting with C = 0.4, after ten generations of iterating the drift formula the mean C dropped to 0.177, and in (coincidentally) only 17.7 per cent of the trials did C end up greater than the initial value of 0.4. On the tenth generation of Simulation III, 65 new societies were founded. Their mean C was 0.324, much higher than the 1,000 trials predicted, and 32 per cent of them had Cs greater than 0.4. Moreover, values of C tended to move upward after this, rather than drifting down. On the 15th generation, 74 colonies were established, with a mean C of 0.459, 66 per cent of them having higher probabilities of colonising than the initial 0.4. Thus natural selection overcame downward cultural drift under these conditions.

Obviously, one factor influencing the outcome is how fiercely the formula tends to drive C down. On the other side, a factor encouraging upward movement from natural selection is the number of offspring produced by the typical high-C colonising society. Since such societies will tend to try colonisation before their low-C sibling neighbours, a key determinant of this is simply the range of the space ships or, more precisely, the typical number of uncolonised stars within the range of the ships. Even if the pressure for downward cultural drift is strong, a large number of progeny in each generation means a good chance that at least one will have higher C than its parent and be able to transmit this advantage to at least some of its own offspring.

I did a post-mortem analysis of Simulation III to determine why the 25 holdout stars never got colonised, even given 75 generations. A special program took the positions of the 1,000 stars and determined that only three could not be reached by 100-unit flights from the home-world, two of them in a pair 70 units apart. The remaining 22, which would have been colonised but for cultural drift, included two isolated stars and a pair near the surface of the sphere, as well as the uncolonised star only 298.4 units from the centre. Finally, there were two uncolonised pockets, about 600 units apart, each reaching in from the surface more than 100 units, one with six stars and one with ten.

The star which remained uncolonised, yet was only 298.4 units from the home star, is especially interesting. Since it is 200 units inside the sphere, it could not be colonised from outside unless it could be reached from another uncolonised star, but it was fully 332 units from the nearest of them. However, it was only 88 units from the 163rd colony established in the run, and thus could have been reached from the home world. If the Sun were that lone star, we could be living deep within a huge interstellar empire, untouched by an explosion of colonisation which swept around us many years ago.

It is worth tracing the steps by which the civilisation could have crossed the 298.4 distance units to the lone star, to get a more vivid picture of the process. Colony number 2 was established in the first generation with a C of 0.246, drifted downward from the 0.4 possessed by the home world. In the second generation, only its sibling, Colony 3, was able to plant colonies, and Colony 2's C drifted down from 0.246 to 0.162. In generation 3, the drift was upward to 0.320, while in generation 4 it was down again to 0.228. Finally, in generation 5, Colony 2 was able to produce two offspring, among them Colony 41 with 0.112 for its C. In generation 6, Colony 41 missed a chance to colonise, but drift brought its C up to 0.236. In generation 7, Colony 41 produced four offspring, including Colony 112 with C = 0.301, and in generation 8 Colony 112 had a single offspring, Colony 163 with C = 0.402.

This colony, beginning with a relatively favourable probability of having offspring, was the only world from which the wave of colonisation could reach the lone star. But if it didn't colonise quickly, its C was likely to drift to a prohibitively low level. Generation 9 saw Colony 163's C drop to 0.245. It dropped further to 0.160 in generation 10 and wandered, sometimes rising but mainly dropping until it reached 0.001 in generation 18. In generation 19 it drifted below 0.0005 and was rounded off to zero. Thus it lost its chance to colonise the lone star. But for this to happen on a wide front, the rate of downward drift would have to be much higher, capable of offsetting natural selection.

The Power of Drift

Table 1 gives us one way to achieve more rapid cultural drift, increasing P above the level of 0.618 we have been using. Partly because 0.618 affords theoretically satisfying symmetry, and partly to show the range of options offered by the simulation approach, I shall use a different method, revision of the drift formula. Since R is a random number between zero and one, we can control the power of cultural drift by changing the distribution of random numbers within this range. The simplest way is to take the square root of R. Since we are dealing in abstract models, and sociology cannot yet specify the form or terms of the correct formula, I cannot offer a strong defense for this move. However, it is a simple change which achieves the desired end without disturbing the range over which C can drift. In BASIC, the formula thus becomes: Cl =C0^(P+SQR(RND(l))).

This revised formula makes C decline rapidly. In a test of 2,000 trials, similar to those in Table 1 but run twice as many times, the mean value of C after five generations was 0.091, and after ten generations was 0.006. In about half a per cent of the cases C rose in the first five generations, but never wound up greater than the initial 0.5 after ten generations. It might seem unfair to the colonising societies to shift to such a harsh rate of drift after the easy-going demands of the formula we have been using. But I must emphasise two things. First, if there exist actual rigid laws of social evolution, applicable to all advanced technical species, they may imply rapid downward drift in colonisation, or they may not, but the suspicion of Behavioural Science that large-scale social phenomena are highly unstable suggests rapid drift. Second, the new formula does not really demand very rapid cultural change. The flight time of an interstellar fleet could well be several hundred years, as could the time required to settle a new world and think about colonising onward, so ten generations could well consume several thousand years.

Table 5 shows the results of our last run, Simulation IV, in which the initial C is 0.5, the ship range is 100, and we use the new formula for drift. The expansion starts out rather vigorously, but stops altogether on the 17th generation, with just a third of the stars colonised and the nearest uninhabited world a mere 139.2 units from the home star, potentially accessible in only two flights. The last star with any chance to plant colonies lost it in generation 26 when its C drifted from 0.002 to zero. The results of this simulation are extremely interesting. If it should happen to model the real Universe at all closely, then there may exist many interstellar civilisations, each encompassing a few hundred star systems, yet none spreading out to fill the entire Galaxy.

TABLE 5. Simulation IV: Revised Drift Formula.

   Distance from Home World to: 
Gener- 
ation 
 Number of 
 Societies 
 Nearest Star 
 not Colonised 
 Farthest Star 
 Colonised 
19 103.098.1
221 106.1171.8
336 106.1259.4
467 106.1350.4
5102 113.1359.4
6164 113.1439.9
7212 113.1449.4
8246 139.2498.7
9276 139.2498.7
10293 139.2499.8
11312 139.2499.8
12323 139.2499.8
13326 139.2499.8
14328 139.2499.8
15330 139.2499.8
16331 139.2499.8
17332 139.2499.8
-------- --------
75332 139.2499.8
Initial colonization probability = .5
Maximum flight = 100
Cultural drift coefficient = .618
Formula: C1 = C0^(P+SQR(RND(l)))

Conclusions

If the general approach of this paper is correct, given values for our parameters which fall in certain wide ranges, then the following propositions would be true:

1. Interstellar civilisations are highly irregular in shape.

2. Uncolonised habitable worlds exist near the home worlds of large interstellar civilisations.

3. The expansion of interstellar civilisations proceeds more slowly than expected by models which do not incorporate the concept of cultural drift.

4. The expansion of interstellar civilisations slows nearly to a halt after a moderate number of worlds have been colonised.

5. A single intelligent species will not quickly colonise an entire galaxy.

6. The observation that the Earth has not been touched by an interstellar civilisation does not imply the non-existence of such civilisations.

7. Models of interstellar colonisation employing the concept of cultural drift strengthen arguments for the attempt to detect extraterrestrial radio signals.

If the premises drawn from Behavioural Science theory are incorrect, some of these propositions are invalid. For some sets of parameters, natural selection overcomes cultural drift, even within the theoretical model offered here. We have assumed that colonisation is unlikely, that some societies nonetheless will develop cultures making colonisation more likely, and that cultural drift will tend to bring these elevated probabilities back down near zero. The fact that we were able to model limited colonisation renders these assumptions especially attractive, since we can now postulate the existence of nearby interstellar societies despite the apparent absence of extraterrestrials on Earth.

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