Sparse matrices are extensively used to model interactions between entities and facilitate computations in neural networks. Sparse Matrix Multiplication (SpGEMM) serves as a fundamental operation in graph algorithms, social network analysis, and deep learning, attracting considerable research interest. Among the four primary paradigms for defining sparse matrix multiplication, the Inner Product (IP) method most closely aligns with the standard definition of matrix multiplication. However, due to its limited data reuse and reliance on index matching, the IP method has been rarely explored in the literature. This paper investigates the strong connection between SpGEMM and set intersection computation, introducing a hybrid sparse matrix multiplication algorithm that builds upon the numerical computation of the IP method. By leveraging the IP method's advantages-such as minimal intermediate results and high flexibility-our approach effectively enhances computational efficiency. Experimental evaluations on benchmark datasets demonstrate the superiority of the proposed algorithm, particularly in scenarios where the resulting matrix exhibits high sparsity. Furthermore, our method proves effective in several applications, including self-transpose multiplication and sparse matrix multiplications in graph neural networks.

In many real-world applications such as social network analysis and online advertising/marketing, one of the most important and popular problems is calledinfluence maximization(IM), which finds a set ofkseed users that maximize the expected number of influenced user nodes. In practice, however, maximizing the number of influenced nodes may be far from satisfactory for real applications such as opinion promotion and collective buying. In this paper, we explore the importance ofstabilityandtrianglesin social networks, and formulate a novel problem in the influence spread scenario, namedtriangular stability maximization, over social networks, and generalize it to ageneral triangle influence maximizationproblem, which is proved to be NP-hard. We develop an efficientreverse influence sampling(RIS) based framework for the triangle IM with theoretical guarantees. To enable unbiased estimators, it demands probabilistic sampling of triangles, that is, sampling triangles according to their probabilities. We propose anedge-based triple samplingapproach, which is exactly equivalent to probabilistic sampling and avoids costly triangle enumeration and materialization. We also design several pruning and reduction techniques, as well as a cost-model-guided heuristic algorithm. Extensive experiments and a case study over real-world graphs confirm the effectiveness of our proposed algorithms and the superiority oftriangular stability maximizationand triangle influence maximization.