From ties to coordinates: recovering latent social positions with graph neural networks

Zixuan Wang^1*Can Peng^1,2

• ¹ZHT Lab Co., Ltd., Beijing, China
• ²Beijing Information Science and Technology University, Beijing, China

In social networks the pattern of ties governs how information, influence, and opportunities circulate. Explaining these flows often requires uncovering the hidden geometric space in which network distances acquire meaning, yet inferring that space from the adjacency matrix alone remains a long-standing obstacle in complex-systems research. To overcome it we present Scale-Aware Graph Neural Networks (SA-GNNs), a self-supervised framework that marries the statistical consistency of spectral embeddings with the scalability of message-passing neural networks. By leveraging small-ball return probabilities from random walks, our method estimates vertex-specific diffusion scales and reweights the graph accordingly to approximate an intrinsic Laplacian operator. The learned coordinates arise as minimisers of a Dirichlet energy functional and converge, under mild conditions, to the true latent geometry up to isometry. SA-GNN avoids negative sampling and scales linearly in memory and runtime, enabling applications to multi-million-node networks. Empirical evaluations show that our approach achieves a substantial improvement in alignment error com-pared to classical methods such as Laplacian Eigenmaps, Diffusion Maps, and node2vec. These results position SA-GNN as a theoretically grounded and computationally efficient tool for analyzing large-scale relational systems through the lens of diffusion geometry.