Representing the Topology of Complex Networks Based on Graph Embedding

Persistent homology is a multi-scale method to identify robust topological features underlying the structure of high-dimensional data and complex dynamical systems. Due to the large size of complex networks, analyzing complex networks through persistent homology is a challenging research direction. In this paper, we present a graph embedding method and prove that the distance matrix obtained by it preserves the relationship among nodes in the complex network. We can directly evaluate some characteristics of the original graph on the embedded graph instead. And the embedding process can be an iterative process which can reliably summarize the structure of the graph. In addition, one can use topological data analysis (TDA), such as persistence diagrams, to analyze the structure of a graph. However, TDA is a well-known expensive operation, we then propose some sampling algorithms to cope with the situation where the complex network is too large. Our evaluation shows the feasibility of this method and contends that it yields a promising approach to analyze complex networks.