Tuesday, August 11, 2009

Meme | Chaos & Complexity

For anyone looking for me and not the article, contact samuel.stathakos@hotmail.com

This is an epic article that relates chaos and complexity theory to the cultural evolution of memes. I will go into detail about a number of phenomena that these fields present us, and how they apply to memetics.

Quick Navigation:
1 Chaos Theory
1.1 Chaotic Systems
1.1.1 Sensitive Dependence (The Butterfly Effect)
1.1.2 Deterministic Behavior
1.1.3 Non-linearity
1.2 State space
1.3 Lorenz attractor

2 Complexity Theory
2.1 Complex adaptive systems
2.1.1 Dissipative System
2.1.2 Emergence Cellular Automata
2.1.3 Self-Organization Fractals
2.1.4 Neural Networks

3 Edge of Chaos

4 Final Remarks

Chaos theory describes the nature of systems that evolve over time – interestingly enough, memes and memeplexes fall under this category, so perhaps we may glean something from this field of study. First we need to understand what a chaotic system is. The three major qualities in the aforementioned are sensitive dependence, deterministic interactions, and non-linear dynamics.

The Butterfly Effect is a well-known folklore describing sensitive dependence in a chaotic system. The story goes that if a butterfly flaps its wings in Argentina, it could cause a tornado in Texas three weeks thereafter. In an identical universe without that flap it may not have occurred. The idea is that the significance of one event quickly becomes amplified; among any neighboring states in an initial system, the outcome trajectories will rapidly diverge. A meme, too, may ultimately result in a significantly altered system. For example, you are in an unfamiliar town looking to find a fireworks show on July 4th. You cannot find it, so you give up and sit down on the side of a hill. A family comes and sits on the hill as well, assuming that you are there because it is a vantage point to view the fireworks. Other people begin to come, imitating the behavior of others – passing on the meme as they all sit down along the hillside. The entire organization of the system depended on the initial condition that you sat down. In this way, memes can be seen to exhibit sensitive dependence if we take culture or memeplexes to be a system.

The term for deterministic events within a chaotic system is unique evolution. For any particular state in a system, the following sequence of events will always be the same. In many popular discussions of chaos, the unstable quality and sensitive dependence of such a system is often confused as being non-deterministic. There are, however, no truly indeterminate events involved. Chaos theory actually supports hard determinism. If we were to take the position that a meme is a selfish replicating entity interacting within a complex adaptive system (which I will explain later), it would be easy to imagine a deterministic world. A human mind will express memes as a direct result of physical stimuli, coupled with an internal state of mind. Memes themselves are simply information recorded in a physical medium (and they follow simple rules that lead to an evolutionary progression) – which means that they also possess unique evolution in the sense of a chaotic system.

The third quality of a chaotic system is non-linearity. A linear system is straightforward, in that if you adjust one variable, the other variables are affected by a fixed proportion. This is not the case with non-linear systems. The proportion by which any variable changes, is itself subject to change. This is effectively the kind of system that requires differential equations for a mathematical representation. Within a well-developed memeplexe, removing a particular meme from the system may have little to no affect – or it may severely decrease its memetic fitness. The effects of the alteration of one variable are not confined to a fixed proportionality.

State space is a mathematical three-dimensional model portraying possibilities within a system. Each point represents a possible situation. This model assumes the fidelity of the model to be high – and that it includes all the crucial elements involved. This is problematic, especially in the real-world application of chaos theory, due to its inherent sensitive nature. The picture below is shows a fitness landscape in state space.

In an evolutionary system, the peaks may represent effective or more complex states. Evolution tends to progress replicators towards a peak (hill-climbing), although not necessarily the highest peak – as there are often local maximums.

As an aside, some theoreticians like to throw in a fourth variable into the definition of chaos: aperiodic evolution. This essentially means that an evolving chaotic system never repeats values in a recognizable or regular manner. It may tend to progress towards or be attracted to a specific outcome, although the relations between microscopic agents never repeat themselves.

The Lorenz attractor is one example of a chaotic system, as seen below.

This is what originated the notion of the Butterfly Effect, as the shape of the attractor in state space resembles a butterfly. You will notice that this type of attractor (known as strange), has stretching and folding properties such that neighboring points in the initial state space dramatically diverge, while other points begin to converge. A sequence of existing states is represented by a particular trajectory. The system contains nested feedback loops. In other words, a perturbation in the system may be amplified and begin to alter the outcome in comparison to neighboring states. But the system is attracted and confined to the model above (so there are also trajectories that converge) because some other factors restrict such states. The system is limited by certain variables. A common example of a limiting factor is the S-curve.

This can be observed in bacterial cell growth. The population rapidly increases until they run out of space and food – a limiting factor.

Complexity theory is used to study systems as a whole, looking at how complex behavior emerges from individual components. A complex adaptive system is a term that describes a system which is dissipative, displays emergent properties on a macroscopic scale, and is self-organized.

A dissipative system has a flow of energy that runs through the system (it is an open system), while it maintains an organized structure. A memeplexe does not have a physical organized structure in the same way that a human body does. But as it evolves in an open system, it maintains its interconnections between individual memes. A memeplexe includes memes which are mutually beneficial to the entire system, which hold the memeplexe together.

When many individual components co-operate in a complex adaptive system, often there are emergent properties of the whole. The entities are connected on a microscopic scale and achieve this emergence by following a simple set of rules. This rule set leads to complex behavior in an evolving system. As they are all governed by simplistic reactions, there is no leader coordinating an effort; there is decentralized control. An emergent property, in the case of complex adaptive systems, is adaptation. The group becomes resilient to perturbation. This is also known as homeostasis – and it allows an evolving system the level of order required to maintain a dissipative system. This ties in with all replicators, because stability increases longevity, heredity, or inheritance. I will return to this notion of a balance between chaos and order later on.

I am going to talk, now, about a specific example of a complex system which displays emergent properties. It is called cellular automata. The idea is that you set up a grid, with each cell on the grid being able to have one of two conditions: on or off. All of the cells are subject to a simple set of rules, based on the parameters set by a user. Each cell has eight neighbors surrounding it. The computer simulation will count how many neighbors are around a cell, and the cell will either switch from off to on (in the next generation) if the number is between the minimum and maximum parameters, or it will switch off if it fails to meet the condition. If you were to look at this system from the perspective of an individual cell, you would note the simplicity of the rules: (1) count neighbors, (2) if neighbors meet parameter conditions turn on, and (3) if neighbors do not meet parameter conditions turn off. No single cell is intelligent, and yet complex patterns emerge as the system evolves. I have made computer software myself, in the form of a java applet, to allow adjustable parameters and change the initial conditions. The video below demonstrates cellular automata that begin with one cell activated, and parameters set to: min = 1, max = 2.


Cellular automata display a number of interesting properties relating to chaos and complexity theory – including sensitive dependence, unique evolution, various attractors, and emergence. Here is one more picture of an evolved system at generation 299 with the parameters: min = 1, max = 5

You will notice in the video, that the evolution of the system grows in a self-similar fashion – which leads me to self-organization. This is the attraction and repulsion by which an internal organization of an open system increases in complexity without guidance. As I said before, there is decentralized control. A complex adaptive system will often display self-similarity. This is because, through evolution, the most efficient method of growth is sought after. How could such simple initial conditions evolve into a complex whole? When you have a computation method that involves recursion, you can have the same code applicable at multiple levels of a system. Recursion allows efficiency. A fern is pictured below.

You can see that it takes the shape of a fractal – the pattern repeats itself on multiple levels. A fern is comprised of leaves which take that shape, and on each leaf there is again the same shape. Fractals are an interesting example of self-similarity – below is a video of a fractal zoom, showing the infinite recursive nature.

For good measure, I’ll throw in a picture of the Mandelbrot set as well.

Fractals are beyond the scope of this article, however, so I will move on to a quick example of a complex adaptive system.

A neural network satisfies all the conditions of the aforementioned complex adaptive system definition. It is a dissipative system that maintains its internal structure, while also adapting to its environment by rewiring neural connections. The brain is a complex of neurons that interact using deterministic, simple rules – out of which emerges consciousness. This is in line with the Epiphenomenalist view that a mental state emerges from the physical structure of the brain, and the mental state (being an emergent property of the whole) does not affect the physical structure.

The edge of chaos is ultimately where complex adaptive systems exist. This is a state in which there is a mix of chaos and order or complexity. If everything were ordered there would be no adaptive qualities, and yet you also need to bring order to the fray (chaos) enough to maintain the structure and complexity. Memetic systems are complex adaptive systems because they exhibit adaptation and complexity, as I mentioned. Lastly, let’s look at some more specific ways in which we can draw parallels between chaos, complexity, and memes.

Some memes reach the, “tipping point” – when a feedback loop initiates and the meme is amplified through feedback. The top websites on the internet, the best-selling novels, and the current fad in society – are all memes that reach a critical mass. Let’s say that a novel reaches best-selling status. Now people will recognize the authority of that meme, and accept it more easily. In turn, this increases the authority of the novel as more people buy it. And the loop continues.

Memes exist on multiple levels and are self-organized. On the smallest level, there are memes and their variations, followed by aggregations of memes that form memeplexes, followed by groups of memeplexes that form sub-cultures, followed by memeplexes that exist cross-culturally. Culture was not created by an intelligent agent; it is the product of individual agents (information), in the form of a physical medium (like the brain), that necessarily follows simple rules (otherwise they wouldn’t have successfully replicated), and formed a complex adaptive system (of its own accord).

Memeplexes sit on the edge of chaos. The chaos created by error-prone meme transmission becomes belittled when distributed through a population by mass representation such that it may do hill-climbing on the fitness landscape. If the effect of variation or chaos in memetic mutations on memeplexes was too great, it would not allow adaptation, or evolution. It would simply be a random walk (it would not progress in complexity). Yet, there is necessarily a chaotic factor in an evolutionary system. How would it adapt if there were only complete order?

Memetics is often described as the art of importing genetic studies to the social sciences. And I think that memetics has something to learn from chaos and complexity science. There are clearly many similarities between them; we stand to gain a better understanding of memes as a system through the perspective of chaos and complexity theory.

For more non-technical information, see Victor Macgill’s site on chaos and complexity theory, and for more information on memes see my other posts.