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# Summary on non-linearity

Topic: Non-linearity

by Joe, 2018 Cohort

Note: This entry was created in 2018, when the task was to “summarise a key reading”, and so may not represent a good example to model current primer entries on.

It took about a century for industrialists to realise that a worker is more productive over an eight-hour day than a fifteen-hour one. What did the industrialists fail to understand? What can we learn from them (or fruit, or bacteria) about complexity?

### Definitions#

#### Linearity#

To understand non-linearity, we first need to understand linearity. A linear relationship is a straight line relationship. Consider a simple process with inputs and outputs. Whenever we increase or decrease the input, we get a proportional and consistent increase or decrease in output.

Let us use an (oversimplified) apple tree as an example.[1] We will measure how watering the tree affects its ability to bear fruit. If we water it once per week, we get two apples. Two times, four apples. Three times, six apples and so on. There is a linear relationship (1:2) between the water and the apples; we get two more apples every time we water the tree.

#### Non-linearity#

Non-linear relationships cannot be expressed as a straight line. Subtle changes to input can cause drastic changes to output.

Think of another tree; this time, we will grow oranges. Watering the tree once produces two oranges. Watering the tree three times is the sweet spot, where we get ten oranges. But watering the tree ten times erodes the soil, leaving us with nothing.

Or, we might have a plum tree, which bears ten fruit before it dies. Each fruit bears a seed, causing another plum tree to grow. Even with the previous generation of trees dying off, each new generation will have ten times as many trees as the last.

### Implications for complexity#

Of course, our orange and plum trees are also oversimplifications. Real non-linear systems can be highly unpredictable, and add significant complexity to the problems that we face. The following key points can be drawn from the readings:

1) Almost all systems exhibit non-linearity.

2) Humans are very good at recognising patterns. We often see linear relationships within a small range and assume that the relationship is always linear. Consider our apple tree above: it would be absurd to think we could keep getting the same increase in return forever. Be conscious of what might happen outside of the observed data: non-linearities are usually hiding.

3) Understanding non-linear processes is an ongoing puzzle. Just because a relationship can be simply expressed today doesn’t mean that this will always be so. Changes to initial conditions can fundamentally change the mathematics of a system. Accordingly, humans can forecast behaviour relatively well over the short term, but poorly over the long term.

4) When growth is exponential, imminent danger can seem like a distant threat. Consider Suzukis example: if a colony of bacteria doubles every minute, and overpopulation occurs at minute 60, the system is only half-way to collapse at minute 59.[2] It takes a great deal of foresight to identify snowballing problems, and even more political capital to encourage others to care. Consider a complex system like climate change, where we may have already passed threshold values. Current inputs will cause disproportionately catastrophic damage in the future, but the (apparent) remoteness of the threat limits policy.

### Conclusion#

Non-linearities are pervasive. A systems thinking approach to complex problems requires us to consider the relationships between variables, and to understand the uncertainties that lie beyond the known data.

#### Disclaimer#

This content has been contributed by a student as part of a learning activity.

If there are inaccuracies, or opportunities for significant improvement on this topic, feedback is welcome on how to improve the resource.

You can improve articles on this topic as a student in "Unravelling Complexity", or by including the amendments in an email to: Chris.Browne@anu.edu.au