VIRGOS: Secure Graph Convolutional Network on Vertically Split Data from Sparse Matrix Decomposition

TL;DR

Virgos addresses the challenge of privacy-preserving GCN training and inference on vertically partitioned data by introducing a sparse-matrix decomposition that converts a sparse adjacency matrix into a sequence of structured linear transformations. It then couples two novel 1-round MPC primitives, Oblivious Permutation (OP) and Oblivious Selection-Multiplication (OSM), to implement a secure (SM)^2 protocol with constant rounds and $O(|E|)$ communication, dramatically reducing both data exchange and memory usage compared with prior dense-MPC approaches. The authors instantiate a complete end-to-end 2-party framework that achieves near-plaintext accuracy on standard datasets (Cora, Citeseer, Pubmed) while delivering substantial speedups and lower memory footprints under various network conditions. The work also provides extensive experiments, ablations, and open-source code, highlighting practical viability for cross-silo collaboration on graph data without exposing private topology or features. Overall, Virgos advances secure graph learning by marrying sparsity-aware decompositions with efficient 2PC primitives, enabling scalable private GCNs in vertical-partition scenarios.

Abstract

Securely computing graph convolutional networks (GCNs) is critical for applying their analytical capabilities to privacy-sensitive data like social/credit networks. Multiplying a sparse yet large adjacency matrix of a graph in GCN--a core operation in training/inference--poses a performance bottleneck in secure GCNs. Consider a GCN with $|V|$ nodes and $|E|$ edges; it incurs a large $O(|V|^2)$ communication overhead. Modeling bipartite graphs and leveraging the monotonicity of non-zero entry locations, we propose a co-design harmonizing secure multi-party computation (MPC) with matrix sparsity. Our sparse matrix decomposition transforms an arbitrary sparse matrix into a product of structured matrices. Specialized MPC protocols for oblivious permutation and selection multiplication are then tailored, enabling our secure sparse matrix multiplication ($(SM)^2$) protocol, optimized for secure multiplication of these structured matrices. Together, these techniques take $O(|E|)$ communication in constant rounds. Supported by $(SM)^2$, we present Virgos, a secure 2-party framework that is communication-efficient and memory-friendly on standard vertically-partitioned graph datasets. Performance of Virgos has been empirically validated across diverse network conditions.

Figures (11)

• Figure 1: Ideal Functionality of $\textnormal{Virgos}\xspace$
• Figure 2: Graph Decomposition through Edges
• Figure 3: Matrix Decomposition Equivalent to Figure \ref{['fig::graph_decom_diff']}
• Figure 4: Graph/Matrix Decomposition with Monotonicity
• Figure 5: Decomposition of $\mathsf{A}_{\mathsf{in}}' = \Sigma \delta'$

Theorems & Definitions (13)

• Theorem 1
• Theorem 2: Sparse Matrix Decomposition
• Theorem 3: Sparse Matrix Multiplication
• Lemma 1
• Definition 1: $\mathsf{Q}$-type matrix
• Definition 2: $\mathsf{P}$-type matrix
• Theorem 4
• Theorem 5
• Definition 3: Semi-honest Security sp/17/Lindell17
• Theorem 6