MacKay’s Mathematical Examples

M.Diakonova and R.S.MacKay Mathematical Examples Of Space-Time Phases International Journal of Bifurcation and Chaos

Diakonova and MacKay exhibit mathematical models of strong emergence and a measure for it.

For deterministic or stochastic dynamics of networks of units … the space-time phases are the limit points of probability distributions for state … which can occur by evolving suitable initial probability distributions for state as a function of space from initial time going to minus infinity.

A system exhibits strong emergence if it has more than one space-time phase, yet is “indecomposable” … . The amount of strong emergence is the diameter of the set of space-time phases, using Dobrushin metric. …

Here, a variety of examples of complex dynamic systems are constructed that we prove to have interesting space-time phases.

These examples are based on the work of Toom. One has a network wrapped around a torus, with binary elements. The base example is to calculate a candidate next state that is the majority of the states of self and those to ‘north’ and ‘east’. The actual next state is then chosen with some characteristic probability, λ, to be either the candidate next state or its opposite. For extreme values of λ there are at least two very different space-time phases, and hence strong emergence.

Gielis & MacKay have previously constructed systems with period one or two ‘ferromagnetic’ or ‘antiferromagnetic’ phases. That is, where at a given time adjacent states are mostly similar, or mostly opposite. This paper shows how to ‘interlace’ examples to get arbitrarily many phases and hence more extreme emergence.

It is noted that an undesirable feature of the Toom method is that it relies on asymmetry in the update rule. (That is, one can’t take the majority across self, east, west, north, south.)

See Also

RC. Ball, M. Diakonova RS. Mackay Quantifying Emergence In Terms Of Persistent Mutual Information Advances in Complex Systems, Vol. 13, No. 3 (2010) 327–338

This gives some different examples of emergence, and some helpful descriptive prose.

Our sharpest measure of Strong Emergence is the Permanently Persistent Mutual Information (PPMI), that is the PMI I(∞) which persists to infinite time. This quantifies the degree of permanent choice spontaneously made by the system, which cannot be anticipated without observation but which persists for all time. A prominent class of example is spontaneous symmetry breaking by ordered phases of matter: here a physical system is destined to order in a state of lower symmetry than the probability distribution of initial conditions, and hence must make a choice (such as direction of magnetization) which (on the scale of microscopic times) endures forever. As a result, Strong Emergence can only be diagnosed by observing multiple independent realizations of the system, not just one long-time history.


The papers above provide some helpful illustrations and note some open questions. This is very much a live area. Here, though, I want to think about the implications.

Where strong emergence holds one has uncertainty, as in the Ellsberg ‘paradox’, matters. For if not one would simply assign a probability distribution over the phases and collapse them to a single effective probability distribution. Equally, the system is not ‘ergodic’ and so nothing like the law of large numbers holds: a single time-history, no matter how long, can be misleading. This phenomena with strong emergence challenge our notions of rationality and empirical reasoning. For example, as Keynes pointed out, strong emergence precludes rational expectations, which are needed for the efficient market hypothesis in finance. Keynes also described how strong emergence corresponds to bubbles. As in the second paper, above, these may endure for a long-time, but not for-ever. Thus in considering emergence in real systems, such as economies, one needs to consider at least three time-scales:

  • In the short-run, one can typically extrapolate on current behaviours, such as the inflating of a bubble.
  • In the mid-term, strong emergence can burst a bubble and lead to many different behaviours that are stable in the mid-term.
  • In the long term there may be different, perhaps more consistent, trends. (E.g., Sornette)

This, while recognizing that these are ‘toy’ examples, it seems to me that they do illustrate the fact that genuine Keynes/Knight/Ellsberg uncertainty can arise in ‘mathematical’ systems, and is not just ’caused by’ awkward humans, and warn against too simplistic approaches to ‘rationality’ 

Dave Marsay

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