Testing Dependency of Weighted Random Graphs

In this paper, we study the task of detecting the edge dependency between two weighted random graphs. We formulate this task as a simple hypothesis testing problem, where under the null hypothesis, the two observed graphs are statistically independent, while under the alternative, the edges of one graph are dependent on the edges of a randomly vertex-permuted version of the other graph. For general edge-weights distributions, we establish thresholds at which optimal testing is information-theoretically impossible and possible, as a function of the total number of nodes in the observed graphs and the generative distributions of the weights. Finally, we observe a statistical-computational gap in our problem, and we provide evidence that this is fundamental using the framework of low-degree polynomials.