Scalar Thresholds

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In this essay and the attached chapter fragment, I deal with “scalar thresholds:” how big can human groups get before X or Y happens?  And why?  Scalar thresholds have been recognized for decades; they are of interest to a wide range of disciplines from philosophy to AI.   I think Southwestern archaeology can ally with evolutionary cognitive science and complexity science to resolve scalar threshold issues – “resolve” as: to make clearer; not as: to solve.

The ancient Southwest offers remarkably useful data for the study of scalar thresholds, particularly within communities or settlements (I’ll use the two terms interchangeably).  “Community” means the settlement of potential daily, face-to-face interaction.  That can be a massed, compact “pueblo” or a scattered Chacoan “community” (another use of the word, but proper I think).  Community does not mean everyone must interact with everyone, every day; rather, there’s the strong possibility of interaction between and among everyone, every day.  In this summary essay: first, the problem and its solution; then, how that solution connects with larger intellectual issues; then, back to the Southwest to apply the solution.  For a much longer, rougher presentation, see the chapter fragment.

Here’s the specific problem: how big can a permanent community get before it requires governance – specifically, centralized, formal, institutional, hierarchical governance?   Cross-cultural studies by me, Kristina Kosse, and others show that a hard threshold exists at about 2,500 people.  That is, if a permanent settlement or community exceeds (approximately) 2,500 people, it almost always will have permanent, institutional, centralized, hierarchical governance: a chieftain, a mayor, a king, whatever.  (Exceptions are discussed in the chapter fragment.)

The actual value varies of course a few hundred people either side of 2,500 – all figures in this essay must be understood as approximate, not absolute.

I’ve taken to calling [>2,500 = governance] the “Kosse-Lekson rule” or K-L rule because Kosse’s and my analyses converged and agreed, independently, in the early 1980s.  (I published first, she published best.)  At the time, I suggested that some sort of mental “hard wiring” might underlie the K-L rule – social channel capacity?  Rule of Six? – but I didn’t pursue that line of thought.  Kosse also proposed that the 2,500 limit was some sort of cognitive threshold or tipping-point: in a community of that size, people’s brains overloaded and they required new levels of socio-political structure for things to work.  But what was the mechanism or dynamic, exactly?

Kosse explored the emergent order of complexity theory – specifically Stuart Kauffman’s families of mathematical models, which mimicked important aspects of human society.  Kauffman developed K=2 Boolean networks in which everything is “connected” to everything, mathematically.  In K=2 networks, “order” (or, in this case, governance) simply emerges – intrinsically, mathematically – at particular numerical thresholds.  That is, governance might be an emergent property, “order for free.”  Using Kauffman’s networks, Kosse identified likely thresholds or tipping-points at approximately 7, 25, 150, 500 and 2,500.  All correspond to thresholds observed by other researchers in other situations, discussed in the chapter fragment:

7:   the Rule of Six  – a rule of thumb in business for the maximum number of simultaneous interactions – is actually the Rule of 5-7.

25:    Man the Hunter‘s number for hunter-gatherer band size.

150:   Dunbar’s Number (discussed below).

500:   Birdsell’s “magic number”for extended H&G social networks  and minimum for governance in cross-cultural studies.

2,500:   the K-L rule.

Kosse’s numbers are “real” – that is, they are based on empirical observations or projections from empirical data; and they are “theoretical,” derived from Kauffman’s K=2 networks.  Sadly, Kosse was unable to follow up her provocative, intriguing research.  She died in 1995 and we lost a very talented, very smart archaeologist.

Can we generate the K-L rule with Kosse’s numbers?  The Rule of Six is prominent in archaeological thinking (through the work of Gregory Johnson, and more recent developments by Wes Bernardini) but its application to the K-L problem is not immediately obvious.  Nor does the “magic number” for H&G band size – 25 – seem relevant.  (6 and 25 may prove very important; I’m simply saying that I don’t currently see their significance; suppler minds may – and, I hope, will!)  Thresholds at 150 and 500, however, almost certainly have implications – social and numerical – 150 for understanding the K-L rule, and 500 for understanding secondary states (a topic explored in Chapter 4.B, and below).

150 is “Dunbar’s number:” the number of people an individual can actually know effectively as individuals.  It is named for Robin I. M. Dunbar, a central figure in human evolutionary cognitive neuroscience.  Dunbar noticed that in primates (apes, monkeys, chimps etc) the size of a social group in the wild was closely correlated with the size of their brain’s neocortex.  Bigger the neocortex, bigger the group.  Humans have really big neocortexes.  Extrapolating from his primate data, Dunbar suggested that humans can know – really know, as individuals – a maximum of 150 people.  (2,000 cyber-friends on Facebook don’t count.)  Beyond that number, we have to categorize, put people into groups based on kinship (real or fictive), social strata, costume clues, linguistic keys, places of residence, or other dimensions that work in our particular society.

There is evidence, I think, for Dunbar’s Number at Chaco.  Consider Casa Rinconada.  Its interior above-bench circumference is 200′.  By ergonomic standards (18″ seat-width), Casa Rinconada can seat about 125-135 people.   Ruth van Dyke (2007:199) calculated that “75 people could stand around the 56 m circumference of an 18 m great kiva”; using my seating standards, 123 people could sit [added to this  post 9/16/11].   Of course those are modern standards; readers who have attended events at Pueblos know that Pueblo proxemics can be tight.   200 people might possibly jam together around Casa Rinconada’s bench; surely that’s an absolute maximum, or very close to it.  Great Kivas were designed with the thought and planning that typified Chacon architecture.  That is, they had a good idea of function, capacity, audience, and so forth before they laid the building out.  Casa Rinconada was intended to seat between 125 and 200 people – simple averaging gives us about 160.  (Most Great Kivas was not quite that large; average above-bench diameter = 16.2m; seating = 112 to 167.)   It’s important to note that we have no idea what went on in Great Kivas.  Wholesale transportation of modern Pueblo kiva functions – a la Aztec Ruins National Monument – is off the table (see “Chaco as Altepetl“).  All we can say is that about 150 people (maybe a few more) could sit around in a big circle and see and hear each other – and see and hear anything going on center-stage.   Dunbar’s number suggests that those people probably knew each other; that is, they represented a community.

How to use Dunbar’s Number?  I begin with two working assumptions (mine, not Dunbar’s) and a conundrum: (1) small groups make decisions through consensus: councils, assemblies, etc., with situational leadership, of course, but without permanent ruler roles; (2) consensus works best, and perhaps requires mutual social knowledge of all actors; it’s hard to reach consensus with strangers.  And here’s the conundrum: Dunbar’s Number suggests that the largest group in which everyone could know everyone else was about 150 people.  Thus we might expect governance to appear at or above 150, but the real threshold is the K-L rule of 2,500.

To get from Dunbar’s Number 150 to the K-L rule 2,500, we need a function or multiplier – or divisor.  Not all 2,500 people in a community “matter,” politically.  Kids, for example, don’t have a lot of political clout.  OK: how many people are actually involved in community decision-making?   How many were “players”?

I will assume that governance is normally a matter for adults (despite recent events in Washington, strongly suggesting the contrary).  And usually, sad to say, governance is very often the business of adult males (which may, in part, explain recent events in Washington).  For those annoyed by my gender assumptions, feel free to substitute “adult female players.”

How many adult male (or female) “players” could there be?   Let’s start with a community of 2,500, at the K-L threshold.  Southwestern populations averaged about 60:40 adults:kids, so we reduce 2,500 to 1,500 adults.  For this exercise, let’s assume 50:50 males:females, so that reduces 1,500 to 750.  Of course it’s not that simple.  At Paquimé, the ratio of men to women was 40:60 (captive women?).   But let’s work with 750: that’s the pool of all adult males.  Dunbar’s Number suggests an assembly of 750 equals would be unworkable.   But all adult males are not equal.  There are honored elders, for example; and at the other end of spectrum, there are young punks who have yet to do anything useful.  We can assume that not all 750 were equal “players.”  The actual number was less, probably far less.  But how many?  Can’t say: we’ve come to the end of this line of analysis.

Let’s work from the bottom up: families – the basic social unit of society.  Maybe players didn’t need to know everybody; they needed to know families.  Indeed, players needed to know only heads-of-households, heads-of-families, heads-of-lineages.  (This insight came from my colleague Dr. Catherine M. Cameron, who actually knows astonishing details about many more people than Dunbar would expect…really big neocortex?)

How many heads-of-households in a community of 2,500?   That depends on how big a family was.  Wes Bernardini estimated that a 13th century Unit Pueblo – a household – was, on average, about 13 people: 8 adults and 5 kids.  (That’s at the lower end of the global range for extended families, about 10 to 20 people.)  If families/ households averaged 13 people, then a community of 2,500 would have around 200 heads-of-households – slightly more than one-quarter of the total pool of 750 adult males.

Recall Dunbar’s number: 150.  750 is far beyond the cognitive comfort level; 200 is much closer, but still too high.  A community of 2,000 (with 13-member extended families) had about 150 heads-of-households, and a hope of consensus among players.  At 2,500, that number reached 200 and one-quarter of the assembly were strangers.  When the number of players significantly exceeded 150 – say, pushing 200 – things fell apart.  Time for a king.

Perhaps, as was recently claimed in Science, people are naturally egalitarian.  Influential political philosophies are based on that claim.  But maybe, above a certain community size, people become naturally or necessarily hierarchical – above the K-L rule.  Same people, same nature/nurture, but they’ve crossed a cognitive threshold.  If that’s so, it would be really interesting.

I do not claim that these thought experiments and number games “solve” the K-L rule, but I think they “resolve” it a bit.   The addition of other functions and factors (perhaps a role for the Rule of Six?) may lead us, ultimately, to a workable mathematical model of the K-L rule and the rise of hierarchical governance.  I continue to work on this, but I am hopeful that younger minds will tackle this problem, or some version of it. (Bernadini has done interesting work on scalar stress at Oriabi, building on the Rule of Six; we need more of this, incorporating Dunbar’s Number and other “cognitive constants.”)

The Southwest is particularly well suited for these studies.  The accessibility and clarity of settlement plans – especially later, when settlements got big – gives us a remarkable set of data.  Estimating population is a favorite activity; we do it early and often.  Archaeologists may bemoan the ambiguities and difficulties of estimating population but those gripes (about details) indicate a developed or developing methodology.  We are very good at mapping towns and better-than-fair at estimating their population.

As discussed in the chapter fragment, ancient settlement and modern Pueblo populations consistently topped out under 2,000 people.  A handful bumped up against the K-L rule.  And a very few exceeded it.

Chaco and Paquimé were two Southwestern sites for which we can safely assume centralized, formal, elaborate hierarchical governance.  How big were they?  Decades ago, I estimated Chaco’s peak population, beginning with the assumption that small “kivas” were in fact domestic structures, with one “kiva” per family.  I multiplied the number of Pueblo II “kivas” at Chaco (both Great Houses and small sites) by 6.5 – a family size calculated from the floor area of pit structures.  With those assumptions, I estimated 2,100 to 2,700 permanent residents at Chaco – conveniently straddling the K-L rule, before that rule was discovered!   If we use Bernadini’s extended family figure of 13 – derived from the floor area of the “pueblo” portion of Unit Pueblos (“single kiva sites”) – that figure doubles: 4,200 to 5,400!  (Of course it’s more complicated…etc.).  It would take special pleading to drop Chaco far below 2,500 – but the Chaco literature is rife with special pleading.

Paquimé, according to Charles Di Peso (who excavated the site) had a peak population of about 4,700 – well over the K-L rule.  Michael Whalen and others (American Antiquity 75(3), 2010) recently argued that Paquimé was, in fact, only half as big as Di Peso claimed.   I tend to trust the excavator; but maybe Di Peso was wrong and Whalen is right.   That could drop Paquimé’s population to around 2,350 – bumping up against the K-L rule threshold!   In fact, Whalen published an estimate of 2,500 people at Paquimé; but that number was offered only as an approximation.   Whalen simply halved Di Peso’s estimate, to illustrate the effect of halving Paquimé’s size; but I like it a lot!

However the cheese is pared, Chaco and Paquimé both approached or exceeded 2,500 – larger than any other settlement of their times, and almost all Pueblo towns that followed.  Ancient and modern Pueblos almost never exceeded 1,500.  Two or three reached or exceeded 2,500, with exceptional causes and shattering results   Awatovi was one; it was destroyed.   Zuni was another, with multiple villages clustering in defense against Spanish colonizers; nevertheless, it was colonized and became a part of the colonial world system, and no longer a stand-alone society.  (More on this in the chapter fragment).

Chaco and Paquimé reached that limit, developed governments, and lasted for many generations.   At both Chaco and Paquimé, governance might have “emerged” as a function of the K=2 networks, Dunbar’s Number, and the K-L rule.  But there is excellent and abundant evidence that both polities were heavily influenced by Mesoamerica; that is, both were almost certainly secondary states (the subject of Chapter 4.B).  They did not invent government.

Governance at Chaco may have developed – or rather, was purposefully instituted, by leaders following southern models – at relatively small population levels.  Here’s a K-L history for Chaco: (1) Chaco represented the last in a string of theretofore unsuccessful attempts by Great House families (nobles manqué) to establish polities in the northern San Juan area; (2) at relatively low population totals – perhaps 500, perhaps 1000 – the polity “took” in Chaco Canyon during the early and middle 900s, with three major noble families (Pueblo Bonito, Penasco Blanco and Una Vida), mimicking the Mesoamerican altepetl form; (3) starting around 1000, more noble families joined Chaco and built their Great Houses (and attached commoners built hundreds of unit pueblos) until Chaco approached and passed the K-L threshold; (4) political life then became locked-in and fixed: governance became both sufficient and necessary.

Recall that 2,500 is not necessary, it is sufficient.  With in situ growth to 2,500, order will emerge and governance becomes necessary.  But 500 people can support (or at least abide) a king.  I think the Chaco polity started – as a secondary state – well below the K-L threshold, in the Pueblo I period (Al Hayes estimated 1,600 for Pueblo I at Chaco, but of course this figure has been contested, downward).  The hierarchies and apparatus of statehood, however, became essential – absolutely necessary – when Chaco reached 2,500.  Thus, the Chaco polity was not “emergent” – it was in every sense artificial.  Chaco (and Paquimé) were secondary states, borrowing from Mesoamerican traditions of governance and rulership.  History trumped emergent complexity.   Chaco succeeded (i.e., persisted), however, because it reached the K-L threshold, at which point governance was no longer optional.  The institutions of governance became traditions.  It persisted even when it hit hard times and diminished scales – dipped back below K-L – with the shift from Chaco to Aztec Ruin in the early 12th century.  The Chaco polity lasted, in one form or another, for almost four centuries.  That’s a long time for a start-up kingdom.

Paquimé was both secondary and big – as far as we can tell, very close to the K-L rule much earlier in its history than Chaco.  There was a local run-up (and a lot of outside help) but when the city itself appeared (about 1300), it rose in a hurry right (see “Black Mountain and Paquimé” on the Sites page).  Again, scale “locked in” the need for political structure, earlier in the particular history of Paquimé than at Chaco.  Paquimé persisted until 1450 – not a bad run for a Postclassic state!

Chapter fragment: 3.A. Settlement Scalar Thresholds

Video: Scalar Thresholds

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3 Responses to Scalar Thresholds

  1. Michael McKernan, Ph.D. says:

    Seems to me it might make a useful test/comparison case to apply your scalar threshold theory to a modern case like Vermont USA towns (townships), which I believe range in population from under 100 to over 20,000. Even though all these polities have “governance” and some/much of the political structure is ordained by state law, there may be some structures which appear at your thresholds, including your K-L point. If your theories truly have cross-cultural basis, shouldn’t we expect to find at least some vestiges in entities like these? And I suppose it might be equally worthwhile to attempt extending your analysis to larger population structures, beyond K-L, for which the 50 USA states might provide an interesting data set.

  2. Kellam Throgmorton says:

    First off, let me engage with Jakob for a moment to suggest that if the K-L rule is a constant (as I think Steve is saying), then perhaps where differences in effective environment exist we should be looking for different *spatial* scales of settlement on the landscape. 2,500 people in the San Juan basin might cover a lot more ground than 2,500 people in Cahokia.

    Second, and maybe this is outside the scope of the point you intend to make, but it might be interesting to add a couple of the large sites in the Zuni region into the discussion on the K-L rule in the Southwest. Kluckhohn and Box S probably exceed 1000+ rooms, and Atsinna likely approaches that number. They seem to represent a counterpoint to contemporary developments in the Mesa Verde region.

    I am also thinking about the demographic work that Rich Wilshusen has done concerning early villages. He had a figure incorporating cross-cultural data that shows that “villages” averaged 250 people, with a range between about 100 and 650 (Wilshusen 1991:53). Northern San Juan PI villages fit pretty comfortably in this range (1991:211). This seems to have some bearing on Dunbar’s number. Early villages may make an interesting study, because these first attempts at aggregation may “break” some of the rules; aggregation may outpace the ability of social structures which are more equipped to deal with smaller, dispersed populations to developing coping mechanisms, leading to social stress, village fissioning, and violence.

    This is more likely the result of me just not quite grasping things properly, but I am having a hard time understanding exactly what you mean by “sufficient” versus “necessary”. (Page 10: “at 500, governance may be necessary but not sufficient”; page 11: “2,500 people is sufficient for governance”; page 16: “2,500 is not necessary, it is sufficient. With in situ growth to 2,500 order will emerge and governance becomes necessary.”) As I understand it from the rest of your argument, at 500 hierarchical governance CAN exist, and at 2,500 hierarchical governance practically MUST exist (it becomes a necessity?). So is 2,500 sufficient but not necessary, because governance CAN exist at 500? I can see that it is a phrase you are fond of, but I seem to have gotten a little lost in the terms.

    I am also curious about the qualification “with in situ growth to 2,500″ on page 16. What about settlements such as in the Dolores region in the AD 800s that–while demonstrating population growth suggestive of a Neolithic Demographic Transition–almost certainly grew through immigration as well? The same could be said for many late PIII/early PIV (AD 1275-1325) settlements around the Southwest at this time, too.

    So, with all that said, I actually really enjoyed this chapter, especially the parts on Dunbar’s number because they are more immediately applicable to the early Pueblo settlements I am studying. I also like the fact that you are sticking largely to the population numbers at this point, and not tying these numbers into specific archaeological features (aside from Casa Rinconada). The idea of purely “integrative features” seemed to me one of the problems with early attempts at scalar examinations of Pueblo social organization, first because we can’t agree what features perform an integrative function (e.g. kivas vs pithouses–vs corn cribs!?), and second because the structures we tend to assume are integrative, like great kivas, could equally be arenas of social contestation under the right circumstances. You are correct in that we have a great opportunity to look at this stuff in the Southwest–in places like Mesoamerica these tipping points are overshadowed (in terms of both the archaeology and and academic interest) by the sheer size of the things that followed.

  3. Jakob Sedig says:

    This is interesting stuff! I don’t really have much experience with scalar thresholds, so I probably can’t provide anything too helpful or insightful. My questions (of course) concern the environment and what role it may have played in the K-L threshold. Chaco and Paquime can both be classified as located in “marginal” environments. Do the constraints of living in desert regions, where precipitation, and thus agricultural surplus (which many would argue is key to the emergence of elites and nobility) are limited effect the number? Would the K-L threshold be different at a place like Cahokia/the Southeast US? I think there is probably some good comparison to be made here, which you don’t necessarily have to be concerned with, but are definitely interesting!

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