More thoughts on Systems Thinking, Complexity, Chaos and New Community Paradigms

The previous post was a fairly abstract article on applying the logistic equation, from the course Dynamical Systems and Chaos by ComplexityExplorer to New Community Paradigm system structures generally, regarding Causal Loop Diagrams of the currently under construction Community Advocacy patterns specifically. This is taking a satellite distant perspective but still with the possibility for an in-depth inquiry into the parameters of a system. It was more a matter of questions than answers and while jumping to conclusions was hopefully avoided, speculations were stretched. The abstraction, unavoidably, continues.

One conclusion reached that should be viable and understandable but not necessarily fully realized is that systems can, sans mitigating factors and based on deterministic function, exhibit stable and periodic behavior that is both constant and consistent. If it can’t be established and maintained over time, then it is not a system. If it is an established system then it is likely to develop some resiliency to drastic external or internal change. This would mean fundamentally changing a system, particularly an entrenched system would require far more energy than is often appreciated.

Another conclusion, far less intuitive, even cognitively dissonant but just as viable, is that a deterministic function can result in a random output sequence. The alternative to a deterministic function is stochastic, the same input does not always result in the same output. There is some element of chance producing a random result, similar to what happens with a fair coin toss sequence. The behavioral orbits are unstable and aperiodic.

What we have then is a deterministic, rule-based system, that once past a region of undetermined predictability, behaves unpredictably despite being a deterministic system. A system in which the function has the property of being deterministic but the qualities of its output are random.

The Complexity course teaches that it is important to distinguish between the properties of a process or a system that generates an outcome, the cause, and the properties of that outcome, the effect, especially in the long term.

The course demonstrates step by step that the logistic equation with r=4 [where r multiplies x(1-x)] is as random as a fair coin toss series. A statistical test would be unable to distinguish between the results produced by the logistic equation and that produced by a random coin toss.

The idea is that a deterministic dynamical system, is capable of producing random, or another way of saying it is chaotic behavior, regardless of how close the system is to the annihilation population {(1-x) where x is between 0 and 1}. Keeping in mind, we often don’t have any idea what is the annihilation population, just that by mathematical logic that there is one. This is applied in a relatively simple sense to a finite population, consisting of similar elements or units that die off or are eliminated and must be propagated to maintain or to increase that population and will be eradicated if not, over a limited number of time periods.

This is a result of the logistic equation that has been proven by mathematics exactly and rigorously. It can be proven, or deduced, from first principles. The claim has been rigorously established. It is not merely a computer or an experimental result.

The long-term behavior of an aperiodic or chaotic orbit depends very sensitively on its initial conditions. The idea is that a dynamical system featuring the phenomenon known as "sensitive dependence on initial conditions," or SDIC or more popularly as the "Butterfly Effect,” can with even extremely small differences in initial conditions result in a difference that can grow to become exceedingly large. This idea applies to numerous dynamical systems, not just iterated functions. It also has a more formal mathematical definition which is provided in the course.

To predict the behavior of a system with sensitive dependence requires knowing the initial condition with impossible accuracy. An example used in the course demonstrates that a difference of nanometers can result in very different results in a few time steps.

The course provides one example of tremendously improving the precision of a measured number to 15.00000001 when in truth the actual number is still 15.0 but the prediction still becomes worthless after a relatively few more time steps. To help visualize this degree of sensitivity, 15 meters is about as tall as a 5-story building, while 0.00000001 or about 10 nanometers is about 1,000 times smaller than a single red blood cell, 10 times larger than a single glucose molecule.

Something 15 meters versus 15 meters + 10 glucose molecules then will exhibit completely different behavior after just a few more time steps. Practically speaking, the difference between 15.0 and 15.00000001 isn’t simply a matter of not having good enough measuring instruments. A very small error in the initial condition grows extremely rapidly meaning long term prediction and even medium term prediction are impossible. More accurate measurements can lead to more accurate and longer term predictions but we have to work exceedingly harder to get only slightly better results. It is r, the growth factor, that makes the significant difference, not x.

The course has us imagine one version of a path of a hurricane hitting New York City, and another version hitting North Carolina based on the tiny difference of the flapping of a butterflies wings. Phenomena such as this though are essentially unpredictable because one can never measure something like this in a manner in which values are this accurate or are even physically meaningful. The course quotes James Gleick from his book, Chaos, who explains that:

‘Its like giving an extra shuffle to a deck of already well-shuffled cards. You know that it will change your luck, but there's no way of knowing how it will change it.’
Even computers are limited by finite precision and having to round off numbers can't calculate the true orbit with we thought we were dealing. The orbit a computer gives us is never the actual true orbit for a particular initial condition. The course explains that the computed orbit "shadows" the other true orbit, also known as the "shadowing lemma.”

Chaos, like the logistic equation, can then be defined in a mathematical sense. A dynamical system is "chaotic" if the following four criteria are met:

1. A dynamical system has to be deterministic, iterated functions and differential equations are certainly deterministic. A dynamical system is just a deterministic rule, if one knows the rule and one knows (with infinite precision) the initial condition, then the trajectory is unique, it's determined. 
2. The system's orbits are bounded, unable to reach infinity. The logistic equation’s orbits start between zero and one and stay between zero and one. 
3. The orbits also have to be aperiodic, they never repeat and they never follow the exact same path twice. They don't go into a cycle. It is a requirement that the orbits be bounded that eliminates the possibility of orbits going off to infinity. If we then have bounded orbits that are aperiodic then they are confined to stay in a unit interval and yet never repeat. 
4. Has sensitive dependence on initial conditions, as again was demonstrated with r=4 for the logistic equation. 

This, in one sense, extends the George Box principle, "All models are wrong, some models are useful" as discussed in Sailing Complex and Wicked Seas with Icebergs (Systems Thinking). Not only wrong in being incomplete copies or maps of reality but as in being limited perspectives, especially one's own. Wicked problems can be analogous to sailing through a massive storm on the sea. The system surrounding the ship can be overwhelming to the system on the ship so people end up arranging deck chairs.

It also though provides a more in-depth understanding despite an inability to reach infinitely fine precision. For myself, having a mathematical foundation as a basis for understanding provides a great deal of confidence but confidence that can check itself. Not to predict where the storm will turn but the confidence to navigate the best possible course.

As Prof. Feldman advises, we sometimes have to invert our thinking about things with starting with an equation because we don't get handed equations, we get handed life and sometimes we can turn it into data. We can't assume that the world is made up of things that are either orderly or things that are random and that these are separate. That we are wrong in thinking that maybe they get jumbled together but they are separate things and need different types of explanation and requiring different means of managing seems reasonable.

One can get disorder from an orderly system, one can get deterministic randomness. We need then need to think about determinism and randomness in a completely different way that in a sense they are two sides of the same coin. They are not complete opposites and we need to think about them completely differently. The relationship between randomness and order is more subtle than we might have thought.

Systems Thinking, Complexity, Chaos and New Community Paradigms

Currently, while still exploring what was learned through the Digital Advocacy course and experimenting with how it fits into the NCP wiki and systems map, I have been taking a course in Dynamical Systems and Chaos taught by Prof. David P. Feldman through ComplexityExplorer.

I am now questioning what my most recently created Causal Loop Diagram maps are really telling me. What I present here, as a rough summary, is in consideration of my questions and should not be thought of in any way as an even partial substitute for the course. It is simply an attempt to try to apply someone"q new learning. It gets abstract because it involves some mathematical concepts but the ideas in the course are for a general audience and are presented here as general as possible.

A dynamical system is simply a rule for how something changes in time. The NCP systems maps are also intended to provide this type of information though at a different level of precision. The ComplexityExplorer course deals with two types of dynamical systems, iterative functions and differential equations.

f(x) = rx(1-x)

The logistic equation, shown above, is a simple model of population growth. It's an iterated function which might tell us how a population changes from year to year. We do the same thing, apply the same function, this logistic equation with a fixed r value, over and over again, using the output for one year as the input for the next. In the standard form of the logistic equation r is a growth rate parameter, r then is something that could change, and we could then see how the behavior of the equation changes. As an iterated function time is discrete, we are not monitoring the population at every instant. A continuous change would involve a differential equation.

The logistic equation is a second order polynomial, a parabola; a very simple function studied in high school, not an exotic or complicated function. The course offers a couple of simple tools for single iterations and comparisons. It is also pretty simple to create a spreadsheet which can push beyond the parameters set by the web tools. The logistic equation is deterministic. Simply an iterated function, an action repeated again and again which ought to be completely predictable.

The first question is whether the simple circles making up the various loops of a Causal Loop Diagram convey smooth transition rather than the more likely true jumps both positive and negative found with iterative functions, then whether additional loops are sequential or occur more or less simultaneously?

The logistic equation deals with populations, as in a finite collection of items under consideration dealt with as a whole, more specifically for our purposes, a community of living entities in which interbreeding occurs among members and is subject to a growth rate parameter from internal and external forces. There is a natural positive increase in population through procreation that exceeds the forces that work against it by some factor greater than 1. The population is changeable to the point that it can be annihilated, from lack of resources, inability to compete for resources, inability to reproduce and maintain the population, and by destruction from environmental or external forces.

The logistic equation is not a means of measurement, in the same way, say as using a Newtonian Law to determine the rate of cooling of an object. As the course explained, the logistic equation is more of a caricature than a detailed portrait or photo. The NCP maps then are a few brush strokes but as has been said many times before, while wrong in terms of the totality of information provided can still be useful, in some cases more useful in conveying insights or at least a different level of insights.

If this year's population is larger than last year’s, based on r, the growth factor considered alone, is greater than 1 then next year's will be larger still tending if without bound towards infinity. If r is 1 then the population stays fixed. If r is less than 1 but greater than 0, the population diminishes approaching 0. If r, for example, is 0.5 then next year there will be half as many say rabbits and half again the following year.

The idea that populations grow without bound is though unrealistic. There is some limit to the growth. There is some maximum population beyond which the population can't pass. There will always be some limit to the number of rabbits, or whatever it is being studied. A term is then added to the equation, a term known in this case as the "Annihilation" population or "Apocalypse" population; meaning that if the rabbits ever reach this “A” population then the next year there will be no more rabbits. The rabbits eat all their food, so the following year there are no rabbits left. The maximum possible number of rabbits is determined by this function in which x is measured as a fraction of the annihilation parameter so the equation can be display as:

f(P) = rP(1-P/A)

Which through algebraic manipulation as demonstrated by the course becomes the equation provided at the beginning of this post.  Note that the Annihilation term is not actually a set number like 5,947 is reached and an entire rabbit population disappears. The logistic function does not explain why a population is annihilated. It simply applies an upper bound and defines population growth in terms of that.

For small populations very far away from annihilation, with P much less than A, we should have the potential for rapid growth. When, however, P gets to be large, enough that the rabbits start running out of food, is when population growth starts slowing down approaching its limit. Once the population gets large, the Annihilation term starts to matter more and population growth slows down. There is an absolute upper limit, at the annihilation or apocalypse number, which if reached the population completely crashes.

Nobody should think, however, that the logistic equation actually controls real rabbit populations or fox populations or moose and wolves populations for that matter. It is simply a thought experiment to interpret as the reality, what's actually happening, and what will happen. A rabbit population once established and under normal circumstances would be unlikely to reach either the maximum of the Annihilation Population or total collapse. If an increase in foxes decreases a rabbit population then there will likely be a subsequent decrease in the fox population from a lack of prey and the rabbit population may rebound, explaining the parabolic shape of the curve of the function.

Arguably, many of the elements of the NCP maps could be considered to be populations serving different even opposing functions within a community. There has to be a certain portion of the community population comprising or utilizing the elements making up these maps. I am not sure how to determine an Annihilation population equivalency but the annihilation of elements is possible fundamentally changing the system. The NCP CLD maps denote positive and negative forces or influences by blue and red connections, respectively. The actual nature of Community Advocacy portrayed by the Kumu map then is a result of the net influences making up the particular configuration of the system at that point in time. Causal Loop Diagrams also repeat but adding in more loops, at likely different rates of growth or influence, can create numerous complex outcomes. With the NCP maps, increased calls for greater transparency and open data in government could result in greater pushback by entrenched government institutions. How much of an equivalency is there then with the logistic equation applied to rabbits? Open Data and transparency are ideas made manifest. Can they be propagated and annihilated the same as rabbits or perhaps viral infection would be a better analogy?

The logistic equation is capable of cyclic behavior that is stable or attracting. Different r values can also give rise to cycles of different periodicities, a cycle of period 2 takes two iterations to complete a cycle. The logistic equation with an attracting cycle of period 4 takes four iterations to cycle back then it repeats. It’s attracting because nearby orbits are pulled towards it. If a population is in such a cycle and gets pushed off, it will return back to that cycle.

While the NCP CLD maps are cyclic in nature, they don’t convey periodicities even though subsequent revolutions along the paths could result in far different outcomes with each completion. InsightMaker would seem to be better at this than Kumu. There is though a great probability for interacting cycles to move towards stable orbits regardless if they are desired, particularly when involved with Entrenched Government Institutions. Entrenched Government Institutions have been dealt with before, most recently with Active Citizens in a Digital Age Embracing Organized Complexity. This might raise the question whether the ideas being considered here apply to Warren Weaver's concept of Organized Complexity in "Science and Complexity". I believe it does as the parameters set by the equation don't depend upon the actions or consequences of any individual member but the community as a whole.

However, for a logistic equation with r=4, and other values, the orbit is aperiodic. The orbit doesn't hit some regular cycle, it just keeps bouncing around all over, never repeating. An incredibly repetitious process which always produces something new. The orbit and never settles into periodic behavior. In other words, it is chaotic, a concept which needs to be explored further.