The Evolution of Complexity

In the previous page we addressed how the “Collaborative Interplay” of the Reverse Law of Large Numbers (RLLN) and the Law of Large Numbers (LLN) brings about the Emergence of Complex Structures of Integrated Diversity.  In this post we will address how the “Competitive Interplay” of the RLLN and the LLN brings about The Progressive Evolution to Ever Greater Complexity…

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Part 1  —  Lock-In + Upheaval

The Second Law of Thermodynamics (SLOT) states that left undisturbed all systems gravitate towards “Thermal Equilibrium”.  Not only that, but any system which is disturbed from equilibrium will ultimately be pulled back to equilibrium; so this means that the SLOT actually behaves like a “restoring force”.

The restoring force is a fundamental concept in coarse mathematics; and consequently there are many examples of restoring forces in nature.  In self-balancing systems it is the internal feedback that acts as the restoring force.  In elastic materials it is the strength of the internal bonds.  In a pendulum it is gravity.  And in “fluid-like systems” it is entropic force of the SLOT.

Entropic Limit + Lock-in

The restoring force of the SLOT means that all systems (of any type) are all naturally damped systems.  Complex Adaptive Systems (CAS) however are “Damped-Driven Systems” – naturally damped by entropic decay of the SLOT, but also driven by “symmetry-breaking positive-feedback”.

Positive feedback, in a CAS, can persist or evaporate over time; which means that emergent structures form and dissolve all the time.  In CASs we know that we know that positive feedback can cause the emergence of local imbalance, but such local imbalances are usually offset by wider balancing co-emergent diversity.  Complexity relies on this co-emergence; complexity is local imbalance within global balance.

Unrestrained positive feedback however has the potential to disrupt complexity by causing wider regional and global imbalance.  Excessive positive feedback within a CAS can over-amplify internal imbalances and drive the system beyond its “Entropic Limit”.

The entropic limit is the elastic limit of the restoring force of entropy; it is the point beyond which reinforcement can no longer be restrained; a point beyond which the natural entropic force of the SLOT is overpowered by emergent positive feedback.

This entropic limit can be considered a tipping point for a system-wide “phase transition” to a domain where positive reinforcement can effectively “hold” a structure far away from equilibrium – a domain where emergent structures can become virtually “Locked-in”.

Skewing the Gaussian

We know from our study of materials that when a restoring force is over-stressed (beyond its elastic limit) a permanent deformation can occur; and in a similar vein, in a CAS when the restoring force of entropy is over-stressed by excessive positive feedback, a mathematical deformation can occur in the system’s Gaussian Distribution.

In mathematics we have something called the Central Limit Theorem (CLT).  The CLT deals with distributions of distributions.  More specifically the CLT deals with the probable distribution of an underlying probability distribution.  It is the CLT that enforces the SLOT; for it is the CLT that relies on the LLN to enforce a Gaussian Distribution.

The CLT says if we sample, a distribution of outcomes of a probabilistic event, a large enough number (LLN) of times then the sample distribution (of the underlying distribution of outcomes) will itself ultimately converge on a Gaussian Distribution – [equally balanced between higher-than-the-average-underlying-distribution-of-outcomes (RLLN+) and lower-than-the-average-underlying-distribution-of-outcomes (RLLN-)…]

As we have seen however, at every level of scale in a CAS, excessive positive feedback can reverse the balancing effect of the LLN in the CLT.  The effect of the RLLN on the CLT is to initially widen out a Gaussian distribution up until it reaches the entropic limit – beyond which a “skewing” of the distribution will occur.

It is this potential (in CASs) for the skewing of the balancing Gaussian distribution that can cause the emergence of Power Law Distributions…

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The Emergence of the Power Law

In nature there exists, a mathematical “universality” in the behavior of all sorts of unrelated systems that appears blind to the underlying details of the actual system itself.  This mathematical universality is the “Power Law Distribution”, and the reason for its mysterious universality, is simply because it is driven by the universality of “Positive Feedback and the RLLN”.

Power Laws are ubiquitous in Nature!  In nature a “Power Law Distribution” usually describes a relationship between the size of something and how often it occurs.

Often we only become aware of this universality as a result of so-called “extreme events”.  Power laws are the result of nature’s constant and ubiquitous interplay of formation and decay, growth and decline, build-up and release.   Most of the time a minor reversal goes nowhere, but sometimes a minor event can randomly initiate some degree of positive feedback, which itself can sometimes amplify the disturbance into an “extreme event” – large earthquakes and massive forest fires are obvious examples!  Different reversals/releases/collapses experience different amounts of positive feedback, and as a result, such events often follow a power law distribution.

This mathematical universality is often mimicked in the behavior of CAS’s.  Power laws do not only describe the distribution of positive feedback driven collapse so ubiquitously at play in nature, they can also describe the distribution of positive feedback driven formation causing the emergence of the RLLN in CAS’s

Power laws in CASs, often describe a “skewed distribution of reinforcement”; effectively describing the different degrees of reinforcement (degrees of coarse damping) in different parts of the system.  Consequently, the stronger the power law the more unbalanced is the system, and the greater is the potential for excessive positive feedback and System-Wide Lock-in.

Apparent Stability

Such “locked-in structures” might on the surface look stable, but there is potential for chaos to be hidden in the order….

In CAS’s structures form and dissolve all the time.  In CAS’s the whole never stays the same because the parts are constantly changing.  In a fast evolving system it is of no surprise therefore, when redundant things gradually disappear, but what is surprising is when something long standing and apparently stable suddenly collapses.

Often things that appear stable are in fact holding themselves away from a more natural fragmented equilibrium by sheer force of their own internal reinforcement. Many structures that have managed to lock themselves in, far away from equilibrium, don’t always stay locked-in.  Many such structures can be vulnerable to collapse should their internal dynamics change.

While self-reinforcing structures may on the surface look stable, this is in reality merely short or medium term stability; for long term sustainability of a far from equilibrium structure is pretty much doomed, thanks to the entropic power of the SLOT.  Ultimately the SLOT ensures that everything eventual dissolves or collapses – even those things that have had the appearance of absolute solidity for a very long time indeed.

The collapse of structures held away from equilibrium can lead to sudden and violent change.  Such upheavals are usually considered shocking and often described as freak events, but in but in truth here are merely an extreme realization of the universal powers laws that govern the distribution of events in all complex systems which have the capacity for internal reinforcement…

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Part 2  —  The Evolution of Complexity

Spectrum of Equilibrium

Complexity is the constant interplay or collaboration between the LLN and the RLLN in the integration of co-emergent diversity.  The “evolution of complexity” on the other hand is a constant tug-of-war between the LLN and the RLLN in the formation and upheaval of complex structures.

Fortunately most CAS’s do not swing wildly between excessive reinforcement and total collapse (except of course Asset Markets – and this is mainly because policy-markets do not understand that these markets are highly vulnerable to “herding” and the RLLN).

In nature the evolution of complexity is a gradual construction, layer upon layer, of integrated diversity; constantly fine-tuned by, co-emergent-adaptation, and entropic-upheaval.  And as a result, of this constant push and pull, every CAS lies somewhere on the grey-scale between the extremes of the LLN and the RLLN.

Chaos Theory has shown us that there is more to equilibrium than a finely-tuned, highly symmetric, but fundamentally bland and featureless uniformity.  Chaos has shown us that there is in fact a “spectrum of equilibrium” on which lie a range of hidden “attractors”, — and each attractor is merely a different evolutionary path, a different version of a previously hidden but now emergent “coarse and complex equilibrium”.

The “Edge of Chaos”

In actuality thermal-equilibrium is itself an attractor; just not a very interesting attractor.  Thermal-equilibrium is a “point attractor” which always pulls the system to the same place, regardless of where it starts from.

Positive feedback is the driving force that progress the system away from a thermal-equilibrium in a step by step fashion; and as a result the emergence of “coarse equilibrium structures” tends to be a bottom-up evolutionary process of co-emergence and upheaval leading to constant change; constant rejuvenation.

The Edge of Chaos is the system-wide entropic limit; the balancing point between the self-stabilizing forces of entropy and the self-reinforcing forces of symmetry-breaking positive feedback.  The Edge of Chaos would appear to be where nature does its best work, the place where nature combines emergence and upheaval to the optimal effect…

Left to itself many a CAS is capable of self-organizing its own way to the optimal balance between formation and upheaval, between stability and instability – a point of optimal complexity – a point often referred to in complexity literature as the “Edge of Chaos”.

The Edge of Chaos is a point of “criticality”, a point midway between the bland featureless attractor of thermal-equilibrium, and the wild strange attractor of chaotic-equilibrium.  It is the point on the spectrum of equilibrium where elements within a complex system tend to optimally combine the traits of being locally reinforced and globally well adapted; a point on the spectrum of equilibrium we could refer to as “The Optimalibrium”…

The Whole is Greater

All emergent (far from equilibrium) structures are self-organized from the bottom-up and as a result can follow “many evolutionary paths”, but the system as a whole has but one evolutionary path; which is an aggregation and integration of the many diverse evolutionary paths taken by the parts of the whole.

Over time these integrated co-emergent entities, constantly fine-tuned by upheaval and rejuvenation, mutually reinforce in a symbiotic relationship; and in so doing the evolutionary whole is forever becoming greater than the sum of its evolutionary parts.

Evolution is the emergent result when a coarsely-reinforcing naturally-damping system constantly tries and re-tries to stabilize to equilibrium.  It is the constant interplay, in Complex Adaptive Systems, of the LLN & RLLN, of upheaval and rejuvenation, repeated over and over, at every level of scale, that drives Spontaneous System Self-Organization and the “Progressive Emergence of Ever Greater Complexity”!…

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