Related Terms/Topics Bifurcations; Fitness Landscape; Attractor States; {{critical-point}}; {{catastrophe}}

Most of us are familiar with the phrase 'tipping point'. We tend to associate it with moments of no return: when overfishing crosses a threshold that causes fish stocks to collapse or when social unrest reaches a breaking point resulting in riot or revolution. The concept is often associated with an extreme shift, brought about by what seems to be a slight variance in what had been incremental change. A system that seemed stable is pushed until it reaches a breaking point, at which point a small additional push results in a dramatic shift in outcomes.

Strike a match and drop it in the forest. How big will the resulting fire be? The forest is dry but not overly so... vegetation is relatively thick. Will the fire burn a few trees and then flame out, or will it jump from branch to branch, burning thousands of acres to the ground?

Complex systems tend towards scale-free, nested hierarchies. By 'Scale-free', we mean to say that we can zoom in on the system at any level of magnification, and observe the same kind of structural relations. Thus, if we look at visualizations of the world wide web, we see a few instances of highly connected nodes (youtube), many instances of weakly connected nodes (your mom's cooking blog), as well as a mid-range of intermediate nodes falling somewhere in between. The weakly connected nodes greatly outnumber the highly connected nodes, but the overall statistical distribution of connected vs unconnected nodes follows a power-law distribution. Thus, if we 'zoom in' on any part of the network (at different levels of magnification), we see similar, repeated patterns.

'Scale Free' entities are therefore fractal-like, although scale-free systems generally are about the scaling of connections or flows, rather than scaling of pictoral imagery (which is what we associate with {{fractals}} HANDLEBAR FAIL or objects that exhibit Self Similarity . Accordingly, a pictoral representation of links in the world wide web does not 'look' like a fractal, but its distributions of connections observes mathematical regularities consistent with what we observe in fractals (that is to say, power laws).

A good way to think about this is that, while both scale-free systems and fractals follow power laws distributions, not all power law distributions 'look' like perfect fractals!

At the same time, sometimes the dynamics of scale free networks align with the visuals we consider to be fractal-like. A good example here is the fractal features of a leaf:

See also: {{schemata}}

Simple Rules - Complex Outcomes

One of the intriguing characteristics of complex systems is that highly sophisticated emergent phenomena can be generated by seemingly simple agents. How does one replicate the efficiencies of the Tokyo subway map? Simple - enlist slime mould and let them discover it!  Results such as these are highly counterintuitive: when we see complicated phenomena, we expect the causal structure at work to be similarly complex. However, in complex systems this is not the case. Even if the agents in a complex system are very simple, the interactions generated amongst them can have the capacity to yield highly complex phenomena.

Power laws are particular mathematical distribution that appear in contexts where a very small number of system events or entities exist that, while rare, are highly impactful, alongside of a very large number of system events or entities exist that, while plentiful, have very little impact. Power laws arise in both natural and social system, in contexts as diverse earthquake behaviors, city population sizes, and word frequency use.

'Normal' vs 'Power Law' Distributions

Complex systems are often characterized by power law distributions. A power law is a kind of mathematical distribution that we see in many different kinds of systems. It has different properties from a well known distribution  - a 'bell curve' 'normal' or 'Gaussian' distribution.

Let's look at the two here:

Network theory is a huge topic in and of itself, and can be looked at on its own, or in relation to complex systems. There are various formal, mathematical ways of studying networks, as well as looser, more fluid ways of understanding how networks can serve as a structuring mechanism.

What do we mean when we speak of Fitness? For ants, fitness might be discovering a source of food that is abundant and easy to reach. For a city, fitness might be moving the maximum number of people in the minimum amount of time. But fitness criteria can also vary - what might be fit for one agent isn't necessarily fit for another. For example, what makes a hotel room 'fit'? Is it location, or price, or cleanliness, or amenities, or all of the above?  For different people, these various factors or parameters have different 'weights'. For a backpacker traveling through Europe, maybe the price is the only thing worth worrying about, whereas for a wealthy business person it may not factor in at all.

This coupling between input affecting output affecting input creates unique dynamics and interdependencies between the two.

Attractor States or 'basins of attraction' and can be visualized as part of a fitness landscape.

Complex Adaptive Systems do not obey predictable, linear trajectories. They are Sensitive to Initial Conditions and small changes in these conditions can lead the system to unfold in entirely unexpected ways. That said, some of these 'potential unfoldings' are more likely to occur than others. We can think of these as 'attractor states' to which a system - out of all possible states - will tend to gravitate.  However, these attractor states may also shift over time, and are subject to system disruptions or  what is referred to as a Perturbation. Attractor states can also emerge gradually over time, as the system evolves, but once present can reinforce itself by constraining the actions of the agents forming the system. Thus we can think of Silicon Valley as being an emergent attractor for tech firms, that has, over time, reinforced its position. When a system finds itself 'trapped' in a basin of attraction (such that it cannot explore other potential configurations that may be more fit, it is considered to be in an Enslaved States .