A Graph Entropy Measure From Urelement to Higher-Order Graphlets for Network Analysis

Graph entropy measures have recently gained wide attention for identifying and discriminating various networks in biology, society, transportation, etc. However, existing methods cannot sufficiently explore the structural contents by merely considering the elementary invariants of a graph, ignoring the underlying patterns in higher-order features. In this paper, we propose a general entropy-based graph representation framework (Greet) based on four pertinent properties of graphlet topology from urelement to higher-order statistics. Specifically, we introduce an unbiased graphlet estimation strategy for obtaining both urelement and higher-order statistics. Additionally, we define a novel family of information functions based on hierarchical topological features to compute the graph entropy, then construct a graph information entropy (GIE) vector using the obtained local and global structural statistics to facilitate downstream tasks. Furthermore, there are some advantages that our Greet exhibits over other methods: (a) high accuracy with < 1% relative error; (b) scalable for even larger vertex graphlets; (c) efficient calculation procedure with feasible speedup. Extensive experiments show that Greet exhibits superior performance on graph classification and clustering tasks, achieving remarkable improvements compared to several baselines. Altogether these findings pave the way for a wide range of applications of graphlet-based entropy as a complexity metric in graph analysis.