Heterogeneous multidimensional scaling for complex networks

Abstract

Many real-world networks are essentially heterogeneous, where the nodes have di®erent abilities to gain connections. Such networks are di±cult to be embedded into low-dimensional Euclidean space if we ignore the heterogeneity and treat all the nodes equally. In this paper, based on a newly de¯ned heterogeneous distance and a generalized network distance under the constraints of network and triangle inequalities, respectively, we propose a new heterogeneous multidimensional scaling method (HMDS) to embed di®erent networks into proper Euclidean spaces. We¯nd that HMDS behaves much better than the traditional multidimensional scaling method (MDS) in embedding di®erent arti¯cial and real-world networks into Euclidean spaces. Besides, we also propose a method to estimate the appropriate dimensions of Euclidean spaces for di®erent networks, and¯nd that the estimated dimensions are quite close to the real dimensions for those geometrical networks under study. These methods thus can help to better understand the evolution of real-world networks, and have practical importance in network visualization, community detection, link prediction and localization of wireless sensors.

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