Weiqi games as a tree: Zipf's law of openings and beyond

Weiqi is one of the most complex board games played by two persons. The placement strategies adopted by Weiqi players are often used to analog the philosophy of human wars. Contrary to the western chess, Weiqi games are less studied by academics partially because Weiqi is popular only in East Asia, especially in China, Japan and Korea. Here, we propose to construct a directed tree using a database of extensive Weiqi games and perform a quantitative analysis of the Weiqi tree. We find that the popularity distribution of Weiqi openings with a same number of moves is distributed according to a power law and the tail exponent increases with the number of moves. Intriguingly, the superposition of the popularity distributions of Weiqi openings with the number of moves no more than a given number also has a power-law tail in which the tail exponent increases with the number of moves, and the superposed distribution approaches to the Zipf law. These findings are the same as for chess and support the conjecture that the popularity distribution of board game openings follows the Zipf law with a universal exponent. We also find that the distribution of out-degrees has a power-law form, the distribution of branching ratios has a very complicated pattern, and the distribution of uniqueness scores defined by the path lengths from the root vertex to the leaf vertices exhibits a unimodal shape. Our work provides a promising direction for the study of the decision making process of Weiqi playing from the angle of directed branching tree.Comment: 6 Latex pages including 6 figure

Network analysis of the worldwide footballer transfer market

The transfer of football players is an important part in football games. Most studies on the transfer of football players focus on the transfer system and transfer fees but not on the transfer behavior itself. Based on the 470,792 transfer records from 1990 to 2016 among 23,605 football clubs in 206 countries and regions, we construct a directed footballer transfer network (FTN), where the nodes are the football clubs and the links correspond to the footballer transfers. A systemic analysis is conduced on the topological properties of the FTN. We find that the in-degrees, out-degrees, in-strengths and out-strengths of nodes follow bimodal distributions (a power law with exponential decay), while the distribution of link weights has a power-law tail. We further figure out the correlations between node degrees, node strengths and link weights. We also investigate the general characteristics of different measures of network centrality. Our network analysis of the global footballer transfer market sheds new lights into the investigation of the characteristics of transfer activities.Comment: 7 pages, 6 figure

Controlling the Intrinsic Josephson Junction Number in a Bi2Sr2CaCu2O8+δ\mathbf{Bi_2Sr_2CaCu_2O_{8+\delta}} Mesa

In fabricating $\mathrm{Bi_2Sr_2CaCu_2O_{8+\delta}}$ intrinsic Josephson junctions in 4-terminal mesa structures, we modify the conventional fabrication process by markedly reducing the etching rates of argon ion milling. As a result, the junction number in a stack can be controlled quite satisfactorily as long as we carefully adjust those factors such as the etching time and the thickness of the evaporated layers. The error in the junction number is within $\pm 1$. By additional ion etching if necessary, we can controllably decrease the junction number to a rather small value, and even a single intrinsic Josephson junction can be produced.Comment: to bu published in Jpn. J. Appl. Phys., 43(7A) 200