The Law: Miller and Page on Emergence, and statistics in social science

I’m working now through Complex Adaptive Systems by Miller and Page and have been deeply impressed with the clarity with which they lay out key scientific principles.

In their chapter on “Emergence”, they discuss the key problem in science of accounting for how some phenomena emerge from lower level phenomena. In the hard sciences, examples include how the laws and properties of chemistry emerge from the laws and properties of particles as determined by physics. It has been suggested that the psychological states of the mind emerge from the physical states of the brain. In social sciences, there is the open question of how social forms emerge from individual behavior.

Miller and Page acknowledge that “unfortunately, emergence is one of those complex systems ideas that exists in a well-trodden, but relatively untracked, bog of discussions”. Epstein’s (2006) treatment of it is particular aggressive, as he takes aim at early emergence theorists who used the term in a kind of mystifying sense and then attempts to replace this usage with his own much more concrete one.

So far in my reading on the subject there has been a lack of mathematical rigor in the treatment of the subject, but I’ve been impressed now with what Miller and Page specifically bring to bear on the problem.

Miller and Page provide two clear criteria for an emergent phenomenon:

  • “Emergence is a phenomenon whereby well-formulated aggregate behavior arises from localized, individual behavior.
  • “Such aggregate behavior should be immune to reasonable variations in the individual behavior.”

Significantly, their first example of such an effect comes from statistics: it’s the Law of Large Numbers and related theorems like the Central Limit Theorem.

These are basic theorems in statistics about the properties of a sample of random variables. The Law of Large Numbers states that the average of a large number of samples will converge on the expected value of the expected value of one sample. The Central Limit Theorem states that the distribution of the sum of many identical and independent random variables will tends towards a normal (or Gaussian) distribution whatever the distribution of the underlying variables are.

Though mathematically statements about random variables and their aggregate value, Miller and Page correctly generalize from this to say that these Laws apply to the relationship between individual behavior and aggregate patterns. The emergent phenomena here (the mean or distribution of outcomes) fulfill their criteria for emergent properties: they are well formed and depend less and less on individual behavior the more individuals there are involved.

These Laws are taught in Statistics 101. What is under-emphasized, in my experience, is the extent to which these Laws are determinative of social phenonema. Miller and Page cite an intriguing short story by Robert Coates, entitled “The Law” (1956), that explores the idea of what would happen if the Law of Large Numbers gave out. Suddenly traffic patterns would be radically unpredictable as the number of people on the road, or in a shopping mall, or outdoors enjoying nature, would be far from average far more often than we’re used to. Absurdly, the short story ends when the statistical law is at last adopted by Congress. This is absurd because of course this is one Law that affects all social and physical reality all the time.

Where this fact crops up less frequently than it should is in discussions of the origins of distributions of wide inequality. Physicists have for a couple decades been promoting the idea that the highly unequal “long tail” distributions found in society are likely power law distributions. Clauset, Shalizi, and Newman have developed a statistical test which, when applied, demonstrates that the empirical support for many of these claims isn’t truly there. Often these distributions are empirically closer to a log normal distribution, which can be explained by the Central Limit Theorem when one combines variables through multiplication rather than addition. My own small and flawed contribution to this long and significant line of research is here.

As far as explanatory hypotheses go, the immutable laws of statistics have advantages and disadvantages. Their advantage is that they are always correct. The disadvantage of these Laws in particular is that they do not lend themselves to narrative explanation, which means they are in principle excluded from those social sciences that hold themselves to argument via narration. Narration, it is argued, is more interesting and compelling for audiences not well-versed in the general science of statistics. Since many social sciences are interested in discussion of inequality in society, this seems to put these disciplines at odds with each other. Some disciplines, the ones converging now into computational social science, will use these Laws and be correct, but uninteresting. Other disciplines will ignore these laws and be incorrect but more compelling to popular audiences.

This is a disturbing conclusion, one that I believe strikes deeply at the heart of the epistemic crisis affecting politics today. No wonder we have “post-truth” media and “fake news” when our social scientists can’t even bring themselves to accept the inconvenience of learning basic statistics. I’m not speaking out of abstract concern here. I’ve encountered this problem personally and quite dramatically myself through my early dissertation work. Trying to make this very point proved so anathema to the way social sciences have been constructed that I had to abandon the project for lack of comprehending faculty support. This is despite The Law, as Coates refers to it whimsically, being well known and “on the books” for a very, very long time.

It is perhaps disconcerting to social scientists that their fields of expertise may be characterized well by the same kind of laws, grounded in mathematics, that determine chemical interactions that the evolution of biological ecosystems. And indeed there is a strong discourse around downward causation in social systems that discusses the ways in which individuals in society may be different from individuals random variables in a large sample. However, a clear understanding of statistical generative processes must be brought to bear on the understanding of social phenomena as a kind of null hypothesis. These statistical laws are due high prior probability, in the Bayesian sense. I hope to discover one day how to formalize this intuitively clear conclusion in more authoritative, mathematical terms.


Benthall, S. “Testing Generative Models of Online Collaboration with BigBang (pp. 182–189).” Proceedings of the 14th Python in Science Conference. Available at https://conference. scipy. org/proceedings/scipy2015/sebastian_benthall. html. 2015.

Benthall, Sebastian. “Philosophy of computational social science.” Cosmos and History: The Journal of Natural and Social Philosophy 12.2 (2016): 13-30.

Coates, Robert M. 1956. “The Law.” In The World of Mathematics, Vol. 4, edited by James R. Newman, 2268-71. New York: Simon and Schuster.

Clauset, Aaron, Cosma Rohilla Shalizi, and Mark EJ Newman. “Power-law distributions in empirical data.” SIAM review 51.4 (2009): 661-703.

Epstein, Joshua M. Generative social science: Studies in agent-based computational modeling. Princeton University Press, 2006.

Miller, John H., and Scott E. Page. Complex adaptive systems: An introduction to computational models of social life. Princeton university press, 2009.

Sawyer, R. Keith. “Simulating emergence and downward causation in small groups.” Multi-agent-based simulation. Springer Berlin Heidelberg, 2000. 49-67.