# Complex Systems, Emergence, and Multiscale Analysis: A Tutorial and Brief Survey
## 2 Basics of Complex Systems and Emergence
### Power Law and Heavy-Tailed Distributions
#### 2.2.3 Reasons Why the Power Law Is Favored in Modeling  
Two reasons make the power law extremely important in complexity science. One reason is that it embodies the notion of self-similarity, and thus is the natural mathematical tool for describing fractal phenomena. The other reason is that it often signifies great risk, due to infinite variance or even mean. To understand the first reason, imagine a large room with a lot of balls flying around. See Figure 2.  
Assume the size of the balls follows a power law distribution,  
\[p(r)\sim r^{-\alpha}. \tag{17}\]  
When we observe the balls with our naked eyes, we normally will only pay more attention to the balls of certain size ranges--large balls will block our vision, and very small balls cannot be seen. Now assume that our eyes are comfortable with the scales \(r_{0}\), \(2r_{0}\), \(r_{0}/2\), etc. Our perception is determined by the relevant abundance or the ratio of the balls of sizes \(2r_{0}\), \(r_{0}\), and \(r_{0}/2\):  
\[p(2r_{0})/p(r_{0})=p(r_{0})/p(r_{0}/2)=2^{-\alpha}. \tag{18}\]  
Figure 2: Pareto-distributed balls, where \(\alpha=1.8\).  
It is independent of \(r_{0}\). Now suppose we view the balls through a microscope with a magnifying power of 100, so now our eyes will be focusing on the balls with scales \(2r_{0}/100\), \(r_{0}/100\), \(r_{0}/200\), etc. The ratio of the balls on those scales will again be independent of the scale \(r_{0}/100\). A perception independent of the scale is the essence of self-similarity.  
The second reason that the power law is associated with higher risks is easier to understand, since a power law distribution has infinite variance when \(0<\alpha<2\) and even infinite mean when \(0<\alpha\leq 1\). Here, on one hand, one has to have some awe with the power law, as otherwise the cost could be tremendous. For example, during financial crises or economic downturns, the loss of the listed companies follows a power law distribution that is even heavier than the distribution of the gains of all profitable companies [51; 52]. As further examples, the size of forest fires and volcanic eruptions also follow power law distributions (see Figures 3 and 4), which has obvious implications for fire fighting or observation of volcanoes--going too close to the sites could easily lead to casualties. However, on the other hand, one also has to be mindful that having infinite variance or mean is not always associated with the severity of natural disasters. An important counterexample is flooding, as it has been found that stream flow of rivers in dry seasons (especially in desert areas) is better described by power law distributions, while that in wet seasons is better described by log-normal distributions [53]. In deserts, surely flooding does not constitute a major risk.  
Figure 4: Complementary cumulative distribution function (CCDF) for the products of volcanic eruptions in the Holocene: (**a**) tephra volume (km\({}^{3}\)) and dense rock equivalent (DRE) (km\({}^{3}\)), and (**b**) volcanic sulfate (data were from [54]).  
Figure 3: Complementary cumulative distribution function (CCDF) for the forest fires in USA and China, where the size of a fire is measured by its area A. The data for USA are the sizes of individual fires from 1997 to 2018, while those for China are the total annual size of forest fires in the 30 provinces from 1998 to 2017.