The blooming availability of traces for social, biological, and communication networks opens up unprecedented opportunities in analyzing diffusion processes in networks. However, the sheer sizes of the nowadays networks raise serious challenges in computational efficiency and scalability. In this paper, we propose a new hyper-graph sketching framework for inflence dynamics in networks. The central of our sketching framework, called SKIS, is an efficient importance sampling algorithm that returns only non-singular reverse cascades in the network. Comparing to previously developed sketches like RIS and SKIM, our sketch significantly enhances estimation quality while substantially reducing processing time and memory-footprint. Further, we present general strategies of using SKIS to enhance existing algorithms for influence estimation and influence maximization which are motivated by practical applications like viral marketing. Using SKIS, we design high-quality influence oracle for seed sets with average estimation error up to 10x times smaller than those using RIS and 6x times smaller than SKIM. In addition, our influence maximization using SKIS substantially improves the quality of solutions for greedy algorithms. It achieves up to 10x times speed-up and 4x memory reduction for the fastest RIS-based DSSA algorithm, while maintaining the same theoretical guarantees.