Allen’s Evolving Complexity

Peter M Allen Evolving Complexity in the Social Sciences, Systems: New Paradigms for the Human Sciences de Gruyter, 1998

Peter proposes a framework for ‘evolving complexity’ within the human sciences, and provides an extended example based on fisheries.


[The] new paradigm of evolving complexity is not a “biological” metaphor, but simply concerns the basic principles of mathematical modelling and the assumptions that are made in obtaining a reduced description of reality. … The fundamental feature of any mathematical model of course is that it concerns aspects of the system which can be “counted”, and this is perhaps an important limitation on their overzealous use in decision-making, since we may “buy” success in the countable aspects of our system at the expense of the uncountable.

In order to make sensible and effective decisions, however, it is necessary to be able to imagine appropriate options for action, and to have at least some knowledge of the probable consequences of each possible choice. Clearly, this concerns something related to “prediction”, but which would more correctly be called the ability to “explore possible futures”. … The conceptual framework of traditional science, that of mechanical and equilibrium systems, simply was not appropriate for human systems and underlay the emergence of the “two cultures” of western civilization, where the artistic and cultural matters were considered to be quite separate from the scientific. [Science] was thought of as universal and objective, above any particular cultural view.

[The paper shows] a hierarchy of models … . These represent a new domain of organization beyond the “mechanical”, where:

  • the identities and behaviours of the actors are mutually interdependent,
  • the system has many responses to perturbations, and
  • where success is related to the capacity to change, adapt and maintain divers and varied strategies.

System dynamics, self-organisation and evolution

Allen has ‘Figure 1: ‘Data and classification of populations and artefacts leads to the picture of evolutionary tree of some kind, while mathematicasl models have been of fixed taxonomy’. This shows ‘reality’ as a cloud. A process of ‘system boundary classification’ leads to a model of evolving taxonomies and emerging (i.e. new) variables. This complexity is first simplified by a process of averaging to yield a short-term model ‘dynamical system’ with a fixed taxonomy and a fixed set of variables, that change with time. (This may show stationarity and equilibrium, for a while.) Key assumptions are:

    1. Microscopic events occur at their average rate, average densities are correct
    2. Individuals of a given type, say x, are identical, or have a normal diversity around the average type.

(I) Equilibrium models

This is often simplified by modelling just the stationary behaviours, as if the system were in equilibrium.

(II) Non-linear dynamical systems

[Nonlinear] dynamics, System Dynamics, are what results from a modelling exercise when both assumptions are made but equilibrium is not assumed. … They can:

(a) have different possible stationary states. …

(b) have different possible cyclic solutions. …

(c) it may exhibit chaotic motion of various kinds.

[Such] systems cannot of themselves cross a separatrix to a new basin of attraction, and therefore can only continue along trajectories that are within the attractor of their initial condition. Compared to reality then, such systems lack the “vitality” to spontaneously jump to the regime of a different attractor.

(III) Self-organizing systems

This “self-organizing” behaviour is obtained for a system if the first assumption is not made. [Nicolis] and Prigogine (1977) called the phenomenon “Order by Fluctuation”. Mathematically this corresponds to using a deeper, probabilistic dynamics of Markov processes … and leads to the “Master Equation” which while retaining assumption 2 [normally distributed individuals] , assumes that events of different probabilities can and do occur. [What] this does is to destroy the idea of a trajectory, and gives the system a collective adaptive capacity corresponding to the spontaneous spatial reorganization of its structure. That is to say that the presence of “noise” can allow the system itself to cross separatrices and adopt new regimes of collective behaviour, corresponding to spatial or hierarchical organization, and this can be imitated to some degree by simply adding “noise” to the variables of the system. The “noise” probes the stability of any existing configuration and when instability occurs, leads to the emergence of new structures.

(IV) Evolutionary complex systems

The real world is … characterised by microscopic diversity which underlies any classification scheme of variables chosen as representative at any particular time.

Neither of the above assumptions applies. Peter considers a microstructure of characteristics of individuals and a macrostructure concerning selection and payoffs.

Stability, or at least quasi-stability will occur when the microstructures are compatible with the macrostructures they both create and inhabit.

The aim of the model then becomes to explore the different possible regimes of operation of the system, and the probabilities of moving towards these different attractors. [Any] dynamical system that we are running as a model of the system will only be a good description for as long as there is no evolutionary change, and no new variables or mechanisms appear.

[Typically] the different behaviours present grow, split off and gradually fill the possibility space with an “ecology” of activities, each identity and role being performed by the mutual interactions and identities of others.

[The] system evolves in discontinuous steps of instability, separated by periods of taxonomic stability.

[From] observing the population data it would be impossible to observe the “correct” model equations. [It] would be impossible to “untangle” its interactions and infer the equations simply by noting the population’s growth or decline.


It is small wonder that most people probably have a very poor understanding of the real consequences of their actions or policies.

The world will never stop changing, and what sustainability is really about is the capacity to respond, top adapt and to invent new activities. The power to do this lies not in extreme efficiency, nor can it necessarily by allowing [supposedly] free markets to operate unhindered. It lies in creativity. And in turn this is rooted in diversity, cultural richness, openness, and the will and ability to experiment and to take risks. [The] landscape of possible advantage itself is produced by the actors interaction, and that the detailed history of the exploration process itself affects outcome.

Innovation and change occur because of diversity, non-average individuals with their bizarre initiatives, and whenever this leads to an exploration into an area where positive feedback outweighs negative, then growth will occur. Value is assigned afterwards. It is through this process of “post-hoc explanation” that we rationalize events by pretending that some pre-existing “niche” which was revealed by events, although in reality there may have been a million possible niches, and one particular one arose.

The future, then, is not contained in the present … .

Instead of the classical view of science eliminating uncertainty, the new scientific paradigm accepts uncertainty as inevitable. … Evolution is not necessarily progress and neither the future nor the past was preordained. Creativity really exists … .Recognizing this, the first step towards wisdom is the development and use of mathematical tools which capture this truth.


In a similar vein:

Peter’s italicised quote under (IV)bis nonetheless important, and the discussion seems spot-on.

I would, however, note that:

  • There is more to uncertainty than just probability, and that one should seek ‘mathematical tools which capture this truth’. Keynes and Good come to mind. From this viewpoint, it is uncertainty that is key, not the involvement of humans.
  • More generally, mathematics is much broader than Peter supposes in his introduction. Even if uncertainty and value cannot be counted, that does not mean mathematics must be silent, or that appropriate tools could not be developed.

See Also

My notes on science and on uncertainty.

Dave Marsay

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