Abstract
Network embedding is a frontier topic in current network science. The scale-free property of complex networks can emerge as a consequence of the exponential expansion of hyperbolic space. Some embedding models have recently been developed to explore hyperbolic geometric properties of complex networks—in particular, symmetric networks. Here, we propose a model for embedding directed networks into hyperbolic space. In accordance with the bipartite structure of directed networks and multiplex node information, the method replays the generation law of asymmetric networks in hyperbolic space, estimating the hyperbolic coordinates of each node in a directed network by the asymmetric popularity-similarity optimization method in the model. Additionally, the experiments in several real networks show that our embedding algorithm has stability and that the model enlarges the application scope of existing methods.
Highlights
Complex networks can largely simplify real systems and preserve the essential information of the interaction structure
To assess how well the asymmetric popularity-similarity optimization (A-PSO) method performs, we examine the embedding accuracy by comparing the topology inferred by the A-PSO method to real-world directed networks
We especially focused on two main issues: (1) how to identify asymmetrical links from topological information, and (2) how to embed directed networks into the hyperbolic space and whether it is feasible to use empirical data to test the model. e results show that the directed links are hidden in the topological information in a nontrivial way
Summary
Complex networks can largely simplify real systems and preserve the essential information of the interaction structure. Complex networks are models with nongeometric properties, which include a large set of tools and methodologies developed in geometry that cannot be applied to complex networks. In this context, there is a wave of studies exploring geometric properties of complex networks, aiming at mapping complex networks into latent variables (that is, hidden variables and Gaussian latent variables) or a low-dimensional metric space (that is, Euclidean space or hyperbolic space) [1,2,3]. Advances in network geometry have shown that structural properties observed in scale-free networks derived from real complex systems can emerge as the geometrical properties [4, 5]. Related studies are becoming increasingly popular, and network models with geometric properties have been used successfully in many fields in network science and other disciplines, including brain science [7, 8], international trade [9], route transfer [10,11,12], and protein formation mechanisms [13, 14]
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