Lower Bounds on Rate of Convergence of Matrix Products in All Pairs Shortest Path of Social Network

Abstract: With the rapid development of social network applications, social network has become an important medium for people to interact. For the minimum distance computation of all pairs in networks, Alon N[4] proposed an algorithm with matrix multiplication, combining with distance product association law and block matrix multiplication, all pairs shortest path length algorithm on networks has time bound O((2n^3)/B logn). In practical applications, considering the scale-free characteristics of social networks and the precision limitations of floating-point operations on computer hardware, I found that the shortest path algorithm has an improved time bound O((14n^3)/B). Based on the above theory, I propose an all pairs shortest path algorithm that combines sparseness judgment and convergence judgment, leveraging the distance product algorithm with matrix multiplication, distance product association law, block matrix multiplication, scale-free characteristics of social networks, and limitation of floating-point operations on hardware. Testing on a social network dataset with 8508 actors, compared to Alon N algorithm, proposed algorithm has a performance improvement of 39% to 36.2 times on CPU and GPU.