Efficient identifications of structural similarities for graphs

Measuring and detecting graph similarities is an important topic with numerous applications. Early algorithms often incur quadratic time or higher, making them unsuitable for graphs of very large scales. Motivated by the cooling process of an object in a thermodynamic system, we devise a new method for measuring graph similarities that can be carried out in linear time. Our algorithm, called Random Walker Termination (RWT), employs a large number of random walkers to capture the structure of a given graph using termination rates in a time sequence. To verify the effectiveness of the RWT algorithm, we use three major graph models, namely, the Erdős-Rényi random graphs, the Watts-Strogatz small-world graphs, and the Barabási-Albert preferential-attachment graphs, to generate graphs of different sizes. We show that the RWT algorithm performs well for graphs generated by these models. Our experiment results agree with the actual similarities of generated graphs. Using self-similarity tests, we show that RWT is sufficiently stable to generate consistent results. We use the graph edge rerouting test and the cross model test to demonstrate that RWT can effectively identify structural similarities between graphs.