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.