  Supervised learning on graphs is a challenging task due to the high
dimensionality and inherent structural dependencies in the data, where each
edge depends on a pair of vertices. Existing conventional methods are designed
for standard Euclidean data and do not account for the structural information
inherent in graphs. In this paper, we propose an iterative vertex screening
method to achieve dimension reduction across multiple graph datasets with
matched vertex sets and associated graph attributes. Our method aims to
identify a signal subgraph to provide a more concise representation of the full
graphs, potentially benefiting subsequent vertex classification tasks. The
method screens the rows and columns of the adjacency matrix concurrently and
stops when the resulting distance correlation is maximized. We establish the
theoretical foundation of our method by proving that it estimates the true
signal subgraph with high probability. Additionally, we establish the
convergence rate of classification error under the Erdos-Renyi random graph
model and prove that the subsequent classification can be asymptotically
optimal, outperforming the entire graph under high-dimensional conditions. Our
method is evaluated on various simulated datasets and real-world human and
murine graphs derived from functional and structural magnetic resonance images.
The results demonstrate its excellent performance in estimating the
ground-truth signal subgraph and achieving superior classification accuracy.
