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With the ever-increasing prevalence of graph data in a wide spectrum of applications, it becomes essential to analyze structural trends in dynamic graphs on a continual basis. The shortest cycle is a fundamental pattern in graph analytics. In this paper, we investigate the problem of shortest cycle counting for a given vertex in dynamic graphs in light of its applicability to problems such as fraud detection. To address such queries efficiently, we propose a 2-hop labeling based algorithm called Counting Shortest Cycle (CSC for short). Additionally, techniques for dynamically updating the CSC index are explored. Comprehensive experiments are conducted to demonstrate the efficiency and effectiveness of our method. In particular, CSC enables query evaluation in a few hundreds of microseconds for graphs with millions of edges, and improves query efficiency by two orders of magnitude when compared to the baseline solutions. Also, the update algorithm could efficiently cope with edge insertions (deletions).

The effects of solution treatment temperature from 1120 to 1220 °C on the microstructure of a new nickel‐based powder metallurgy (PM) superalloy have been systematically studied. As the solution temperature increased from 1120 to 1160 °C, the grains of γ matrix grow slowly due to the pinning of the primary γ′ phase. With the solution of γ′ phase and carbides between 1160 and 1180 °C, the γ grains enlarge quickly from 6.7 to 32.1 μm, whereas the grain growth slows down from 1180 to 1220 °C owing to the decrease of interface energy of grain boundaries. The grain growth model has been established, and the activation energy of grain growth is calculated to be 298.5 KJ mol−1. After solution treatment above 1170 °C, both the primary γ′ precipitates at the prior particle cores (PPCs) and prior particle boundaries (PPBs) dissolve completely. The size of secondary γ′ precipitates increases linearly with solution temperature, while the size of MC carbides changes slightly. The hardness variation of the samples under different solution temperatures is mainly due to the combined effects of grain size and the size of γ′ precipitates.

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