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Additional file 1. Additional figures and tables.

The network diameter is an important characteristic parameter of a complex network. Calculation for a large-scale complex network's diameter has been an important subject in the study of complex networks. If the network diameter is calculated directly, the problem mainly exists in efficiency for searching and counting the shortest paths. If the network diameter is calculated indirectly by studying the statistical function about the relationship between the network diameter and parameters affecting the diameter, the problems not only exist in the efficiency of statistic, but also exist in the function which may be not applicable to all kinds of networks. An algorithm for the complex network diameter based on the k order distance matrix is proposed with a matrix multiplication approach, and a mathematical proof for the algorithm correctness is given as well. Furthermore, some relevant propositions and deductions for reducing the complexity of this algorithm are put forward. With a good theoretical basis and a simple calculation process, this algorithm can be used to calculate the diameter of a large-scale complex network with small-world effect more accurately and efficiently. Two cases about the advanced research projects agency (ARPA) network model and the Chinese airline network model are adopted to verify the effect of this algorithm.

In order to evaluate the effect of size and shape of crack defect on steel surface quality and mechanical properties, accurate measurement and characterization should be conducted. Aimed at the high measuring difficulty and low accuracy in the existing manual mode, a new series of characteristic parameters was proposed according to the morphological features of crack defect in steel. Through the quantitative analysis of parameters including linear morphological factor, defect equivalent and defect rate, the automatic measurement and numerical characterization of crack defect were realized. The results show that the above method can completely solve the high measuring difficulty of complex crack defect distributed with blocky inclusion and crack, which provides a reliable basis for the quantitative analysis of steel surface quality.

In consideration of the high tensile strength but low elongation properties of ultra-fine grain microalloyed steels prepared by deformation-induced phase transformation rolling technology, microstructural quantitative analysis was conducted. Results show that the volume ratio of acicular ferrites is mainly influenced by cooling rate, the higher the cooling rate is, the more acicular ferrites are obtained. By increasing the volume ratio of acicular ferrites, the tensile strength increases; On the other hand, the elongation reduces markedly because of the morphological characteristics, needle effect, size factor of acicular ferrites and so on. By adjusting the cooling rate in cooling process, the volume ratio of acicular ferrites can be controlled in the optimum range, so as to improve the comprehensive mechanical properties of steel.

The epidemiological features of lung cancer in Beijing during 1977–1986 were analysed. The data collected showed that lung cancer ncidence and mortality rates ranks first among all other malignant tumor and had been on the increase from year to year. The mortality rate in urban area was higher than that in its suburbs. While the male incidence was higher than that of the female. The sex ratio of the male to female incidence rates was 1.56. The incidence rate rises with age. The lung cancer is one of the lesser prevised cancer and the five-year relative survival rate is 6.5% for both sexes in 1982–1983. The lung cancer mortality rate in Beijing urban area is compared in this report with other countries in the world, and it is found that the female mortality rate of lung cancer in Beijing is among the highest.