Zipf's Law, Hierarchical Structure, and Cards-Shuffling Model for Urban Development

Table 1

Comparison between the exponential laws of cities and those of rivers and earthquake energy.

Exponential law	Hierarch of cities	Network of rivers	Energy of earthquake
The first law (number law)	𝑁𝑚=𝑁1𝑟𝑛^1−𝑚	𝐵𝑚=𝐵1𝑟𝑏^1−𝑚	𝑓𝑚=𝑓1𝑟𝑓^1−𝑚
The second law (size law)	𝑃𝑚=𝑃1𝑟𝑝^𝑚−1	𝐿𝑚=𝐿1𝑟𝑙^𝑚−1	𝐸𝑚=𝐸1𝑟𝑒^𝑚−1
The third law (area law)	𝐴𝑚=𝐴1𝑟𝑎^𝑚−1	𝑆𝑚=𝑆1𝑟𝑠^𝑚−1	𝑈𝑚=𝑈1𝑟𝑠^𝑚−1

Note: These exponential laws correspond to the visual models displayed in Figure 1. In Horton-Strahler’s law, the ratios are defined as 𝑟_𝑏=𝐵_𝑚/𝐵_𝑚+1,𝑟_𝑙=𝐿_𝑚+1/𝐿_𝑚, and 𝑟_𝑠=𝑆_𝑚+1/𝑆_𝑚; in Gutenberg-Richter’s laws, the ratios are given by 𝑟_𝑓=𝑓_𝑚/𝑓_𝑚+1,𝑟_𝑒=𝐸_𝑚+1/𝐸_𝑚, and 𝑟_𝑢=𝑈_𝑚+1/𝑈_𝑚.