A framework to preserve distance-based graph properties in network embedding

Recently, there has been a significant interest in the research community on the topic of graph representation learning or graph embeddings (Bengio et al. in IEEE Trans Pattern Anal Mach Intell 35(8):1798–1828, 2013). Ideally, it is required for any graph embedding algorithm to preserve the structural aspects of the graph as well as the graph properties in the embedding space. While there exists a large number of works in the literature to preserve certain structural properties, there are only a few attempts to preserve distance-based graph properties such as asymmetric transitivity in directed graphs [Ou et al., in: 22nd ACM SIGKDD international conference on knowledge discovery and data mining (SIGKDD), 2016] and structural balance property in signed networks [Derr et al., in: 27th ACM international conference on information and knowledge management (CIKM), 2018; Wang et al., in: SIAM International conference on data mining (SDM), 2017]. In the context of preserving distance-based graph properties in the embedding space, to the best of our knowledge, there does not exist any notable work in the literature that preserves shortest distance-based graph properties such as closeness centrality, diameter, radius, and relative ordering of node pairs based on shortest distances in the embedding space. We address this important research gap in this paper. In particular, we first present a set of four metrics to measure how effectively any graph embedding algorithm preserves certain shortest distance-based graph properties. Surprisingly, we notice that several well-known graph embedding algorithms do not satisfactorily preserve these important graph distance-based properties. To address this research gap, we present a new graph embedding algorithm, CAscading-based Robust Embedding (CARE) , which is based on a novel idea of cascading embedding vectors through the underlying graph to effectively preserve shortest distance-based graph properties. We then conduct thorough empirical analysis of the proposed CARE algorithm to demonstrate its efficacy using certain well-known graph analysis tasks (such as clustering, node label classification, and visualization) vis-à-vis several state-of-the-art graph embedding algorithms.