Hyperbolic mapping of complex networks based on community information

• A community intimacy index is proposed to measure the community relationship.
• A community ordering algorithm is proposed based on the community intimacy index.
• Improved hyperbolic mapping method CHM is proposed.
• The new method improves accuracy and running time complexity compared to the state-of-the-art.

To improve the hyperbolic mapping methods both in terms of accuracy and running time, a novel mapping method called Community and Hyperbolic Mapping (CHM) is proposed based on community information in this paper. Firstly, an index called Community Intimacy (CI) is presented to measure the adjacency relationship between the communities, based on which a community ordering algorithm is introduced. According to the proposed Community-Sector hypothesis, which supposes that most nodes of one community gather in a same sector in hyperbolic space, CHM maps the ordered communities into hyperbolic space, and then the angular coordinates of nodes are randomly initialized within the sector that they belong to. Therefore, all the network nodes are so far mapped to hyperbolic space, and then the initialized angular coordinates can be optimized by employing the information of all nodes, which can greatly improve the algorithm precision. By applying the proposed dual-layer angle sampling method in the optimization procedure, CHM   reduces the time complexity to O(n2)O(n2). The experiments show that our algorithm outperforms the state-of-the-art methods.