2.2. Previous index on surname structure

Isonomy is one of the most commonly used index in the surname researches. The isonomy within a region i is defined as Ii=∑k=1Spki2, where p_ki is the proportion of the population with surname k to the entire population, and S is the total number of surnames. The isonomy between two regions i and j is defined as Iij=∑k=1Spkipkj. The isonomy within a region characterizes the aspect of within‐population structure, while the isonomy between two regions reveals another aspect of population structure, the between‐population similarity.

The difference in population structure between any two regions can be measured by surname distance. There are several definitions of surname distance, such as Lasker's distance (Rodriguez‐Larralde et al., 1998), Euclidean distance, and Nei's distance (Cavalli‐Sforza & Edwards, 1967). Nei's distance, which can also be taken as a specially normalized form of the isonomy between two regions, is commonly used in relevant works and will be adopted in this article. Specifically, Nei's distance between regions i and j is defined as Nij=−logIij/IiIj (Nei, 1972).

The isonomy analysis is helpful for measuring the structure and regional consanguinity of the Chinese population, as shown in previous studies (Du et al., 1992; Yuan, Jin, & Zhang, 1999; Yuan, Zhang, Ma, & Yang, 2000). However, the definition of isonomy implies that the popular surnames have absolute dominance over the less popular ones, so the information contained in the less popular surnames cannot be adequately revealed by isonomy analysis. It is inappropriate above all in the case of China since the 7,000 less popular surnames only account for 6.7% of the population of total 7,184 surnames.

There are several sources of the less popular surnames in a given region. They may be the surnames of local minorities which have maintained at a small size of population for a long time. Maybe they are the relatively new surnames which either mutated recently from the existing surnames of local residents or brought about by foreigners who immigrated from other regions not long ago. Thus, the information contained in the less popular surnames can be especially important for the researches on population structure and population dynamics.

Some other indexes are necessarily required to complement the isonomy analysis in this sense. The ratio of surname to population S/N can be taken as one of these indexes. Suppose that 10 people with a new surname migrate into Beijing with a population of 11.89 million and 1941 surnames. The migration will have no perceptible impact on the index of isonomy in Beijing since these immigrants account for only one millionth of the population, but it will increase S/N by about 0.5‰. Specifically, the less popular surnames possibly increase S/N, while the popular surnames have the opposite impacts. Besides S/N, the proportion of Hapax is another one of these index. Here, Hapax means the surname with only one person in a region, so it focuses on the least popular surnames instead of the most popular ones.