It’s nearly 4:30AM and I’m probably too tired & spaced-out to do justice to this at this hour, but here goes:

The equation Sic referred to above can be written as I = (P * (P -1)) / 2, where I = interactions and P = participants. You can set this up in a spreadsheet and plug in different values for P and watch what happens to I. It’s quite interesting, and the implications are even more so:-)

I first encountered this while studying cryptology (codes & ciphers) in the early 1980s (don’t ask:-), as it’s the algorithm governing key material distribution in symmetric-key encryption systems. Given a number of participants, each possible pair needs a unique key, and thus the issue of centralized key management rapidly becomes unmanageably complex. This was said to be an arguement in favor of asymmetric key encryption systems. At the time, I made the counter-arguement in favor of decentralized key management: rather than a central key management entity, each individual participant should manage her/his own keys with respect to other participants, thereby eliminating the problem of combinatorial complexity.

Fast-forward 20-something years, including many years’ experience with groups operating on consensus principles. I had noticed that in such groups, there tends to be an upper limit of size, beyond which decision-making processes bog down and cause the group to become at risk of failure.

For households, the size limit seems to be the range of six to eight members (possibly plus young children), which, if you plug those numbers into the variable P in the equation, produces an output I = 15 to 28 potential interactions.

Long story short, “combinatorial overload” occurs when the amount of time required for decision-making processes (and related communications) in a consensual group expands to the point that it intrudes upon time that is required to perform subsistence tasks.

There is another interesting bit of empirical data that converges, at least as far as households are concerned. In my work designing telephone switching systems, I found that most workers in offices can’t handle more than three “line” buttons on their phone: one for the conversation they are presently having, one for the random call that interrupts it briefly, and one for the important call they are waiting for and can’t let themselves miss. For receptionists, the number of line buttons is 4, because their task is not to engage in lengthy conversations, but to connect callers to other persons in the office.

I refer to this as “load factor,” L, the amont of communications load handled by an individual, expressed as the number of interactions I divided by the number of participants P (each interaction consists of two participants).

For example in the aforementioned household of size 6 to 8, you have P = 6 to 8, and the resulting I = 15 to 28. This turns out to be L = 2.5 to 3.5, and the household size of 7 produces I = 21 and L = exactly 3, the magic number of line buttons on the telephone.

—

If you plug the tight-group number of 12 participants in, you get 66 possible interactions, and a load factor of 5.5, which means that individuals will tend to simplify some of their relationships in the group in order to reduce their load factor.

If you plug the “monkeysphere” number of 150 participants into the equations, you get 11,175 possible interactions, and a load factor of 74.5: clearly too high to sustain at the level of intimate interactions, thus conducive to further simplification of relationships beyond the intimate sphere.

—

I don’t think 12 or 150 are absolutes, but rather, guidelines and possibly reresentations of population averages. That is, maximum group size may vary, and 12 and 150 may represent an approximate average for each type of group (whether mean, median, or modal, remains to be seen).

I strongly disagree that people outside of one’s monkeysphere are represented as “things” e.g. “sanitation worker = thing that makes trash go away.” The distinction of “tribe = people / non-tribe = not people” is ethnocentrism taken to the point of sociopathy, and we see the results in terms of criminal gangs and cultures in which crimes of force or fraud are socially acceptable. In clinical psychology, it is “depersonalization,” a hallmark of psychotic states.

What I believe is more likely, is that the brain sets up a symbolic or representational system that groups people into categories based on their mutual similarity. Think of it as a kind of self-revising emergent cluster analysis function.

When an individual encounters a stranger (i.e. someone outside her/his monkeysphere), s/he will attempt to match that person to one of their existing cognitive clusters, and they will tend to understand and respond to the person accordingly. In addition there is always a baseline of “this is a fellow human being” and a set of cultural norms re. how to treat all humans in general.

For example, I meet a new person, I start by recognizing that they are in fact a fellow human, and my initial behavior toward them is governed by learned cultural baselines: for example basic respect for and manners toward others, non-aggression, curiosity about their lives and views, etc. As I learn more about the person, i.e. through conversation or observation, the unconscious pattern-matching will occur with regard to my existing set of categories or clusters.

In many cases, this only has to go as far as a basic degree of social functionality: during the day you interact with numerous people in the context of your job and their jobs, and this calls for professional courtesy in both directions but not for intimately personal conversations. In this culture you don’t need to know about the deeply-held philosophical beliefs of your customers or of other workers to whom you are a customer; but you do need to treat them with basic respect as fellow humans.

I could go on about the idea that functional hierarchy is not the same thing as status hierarchy, but I’ll leave that for tomorrow or some time when I’m not half asleep at the keyboard:-)

Right, I would replace “exponentially” with “quadratically”.

Granted, the word “exponential” is losing its technical meaning as non-math people use it to mean simply “very fast (growth).” But this informal usage seems out of place in such a carefully written book. “Quadratic” here and in Thesis 13. “Much (more destructive)” in 16. That’s as far as I’ve read so far.

The number of relationships doensn’t grow exponentially, it grows subexponentially. The number of relationships in a group of N people is N times (N-1), which grows O(N^2), much slower than 2^N (which is exponential).

Beyond 150, we can’t keep track of the various relationships at play. We can’t remember who relates to who, and how. We’re forced to find “cheats” to simplify the process to something we can understand, and the most obvious cheat is hierarchy. There’s the mechanism to the old claim by, say, Asimov, that democracy and human dignity cannot survive overpopulation.

I can’t help but find it ever so extreme to say that because a person doesn’t belong in this familial-type group/”monkey-sphere” that they are automatically relegated to ‘things’ (!). Not that I dispute the basic tennet of the theory (or know enough to, anyway), but because I can only think of a person on the other side of the world as an abstract entity doesn’t mean that when I meet a stranger today I don’t recognise them as a human being who I can bond with momentarily, or even so much as respect. So I may not relate to them as I would a close member of my family, but that doesn’t mean I don’t have any kind of relationship with strangers, either.

This is why I don’t see how ‘Dunbar’s number’ can prevent a mass society meeting its needs on a co-operative, deomcratic, egalitarian basis. Not that I couldn’t be wrong–we can only talk with certain amount of ignrance.

http://www.firstmonday.org/issues/issue10_11/noveck/

Gary