Fast hyperbolic mapping based on the hierarchical community structure in complex networks

Abstract

The hyperbolic geometry of complex networks has recently emerged as a promising framework for the analysis of real and synthetic networks; one significant challenge of the hyperbolic geometry framework is the necessity to map large real networks to latent geometric spaces. Addressing this problem, we present a novel fast hyperbolic mapping algorithm called hyperbolic mapping based on the hierarchical community structure (HMCS), which is based on the network evolution model, the community-sector model in hyperbolic space and the hierarchical community structure (HCS) of complex networks. We present an index called community closeness (CC) to measure the adjacency relationship between the communities. Then we propose a ranking algorithm for first-level and second-level communities based on CC to determine the order of communities on the hyperbolic disc and map the network into hyperbolic space based on the order and the angular range of the corresponding sector of the second-level communities. We find that HMCS greatly reduces the mapping time complexity and achieves linearity () in sparse networks under the premise of ensuring better mapping accuracy compared with existing hyperbolic mapping algorithms. Experiments show that the HMCS algorithm occupies a unique attractive position in the space of tradeoffs between mapping accuracy and computational complexity.