The Spectral Zoo of Networks: Embedding and Visualizing Networks with Spectral Moments

Network embedding methods have been widely and successfully used in network-based applications such as node classification and link prediction. However, an ideal network embedding should not only be useful for machine learning, but interpretable. We introduce a spectral embedding method for a network, its Spectral Point, which is basically the first few spectral moments of a network. Spectral moments are interpretable, where we prove their close relationships to network structure (e.g. number of triangles and squares) and various network properties (e.g. degree distribution, clustering coefficient, and network connectivity). Using spectral points, we introduce a visualizable and bounded 3D embedding space for all possible graphs, in which one can characterize various types of graphs (e.g., cycles), or real-world networks from different categories (e.g., social or biological networks). We demonstrate that spectral points can be used for network identification (i.e., what network is this subgraph sampled from?) and that by using just the first few moments one does not lose much predictive power.