Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hierarchical Multi-Marginal Optimal Transport for Network Alignment

Zhichen Zeng, Boxin Du, Si Zhang, Yinglong Xia, Zhining Liu, Hanghang Tong

TL;DR

This work tackles multi-network alignment by introducing HOT, a hierarchical multi-marginal optimal transport framework. HOT combines a cluster-level FGW barycenter to coarsen the solution space with a node-level multi-marginal FGW (MFGW) distance to capture high-order relationships across networks, all solved efficiently via a proximal point method. Theoretical guarantees include convergence to a local optimum and exponential reduction in space complexity with respect to the number of networks, while experiments show substantial gains in both effectiveness and scalability on plain and attributed graphs. The approach demonstrates strong practical impact by enabling scalable, high-order-consistent alignments across many networks, outperforming state-of-the-art baselines and enabling larger-scale applications.

Abstract

Finding node correspondence across networks, namely multi-network alignment, is an essential prerequisite for joint learning on multiple networks. Despite great success in aligning networks in pairs, the literature on multi-network alignment is sparse due to the exponentially growing solution space and lack of high-order discrepancy measures. To fill this gap, we propose a hierarchical multi-marginal optimal transport framework named HOT for multi-network alignment. To handle the large solution space, multiple networks are decomposed into smaller aligned clusters via the fused Gromov-Wasserstein (FGW) barycenter. To depict high-order relationships across multiple networks, the FGW distance is generalized to the multi-marginal setting, based on which networks can be aligned jointly. A fast proximal point method is further developed with guaranteed convergence to a local optimum. Extensive experiments and analysis show that our proposed HOT achieves significant improvements over the state-of-the-art in both effectiveness and scalability.

Hierarchical Multi-Marginal Optimal Transport for Network Alignment

TL;DR

This work tackles multi-network alignment by introducing HOT, a hierarchical multi-marginal optimal transport framework. HOT combines a cluster-level FGW barycenter to coarsen the solution space with a node-level multi-marginal FGW (MFGW) distance to capture high-order relationships across networks, all solved efficiently via a proximal point method. Theoretical guarantees include convergence to a local optimum and exponential reduction in space complexity with respect to the number of networks, while experiments show substantial gains in both effectiveness and scalability on plain and attributed graphs. The approach demonstrates strong practical impact by enabling scalable, high-order-consistent alignments across many networks, outperforming state-of-the-art baselines and enabling larger-scale applications.

Abstract

Finding node correspondence across networks, namely multi-network alignment, is an essential prerequisite for joint learning on multiple networks. Despite great success in aligning networks in pairs, the literature on multi-network alignment is sparse due to the exponentially growing solution space and lack of high-order discrepancy measures. To fill this gap, we propose a hierarchical multi-marginal optimal transport framework named HOT for multi-network alignment. To handle the large solution space, multiple networks are decomposed into smaller aligned clusters via the fused Gromov-Wasserstein (FGW) barycenter. To depict high-order relationships across multiple networks, the FGW distance is generalized to the multi-marginal setting, based on which networks can be aligned jointly. A fast proximal point method is further developed with guaranteed convergence to a local optimum. Extensive experiments and analysis show that our proposed HOT achieves significant improvements over the state-of-the-art in both effectiveness and scalability.
Paper Structure (38 sections, 8 theorems, 25 equations, 10 figures, 4 tables, 3 algorithms)

This paper contains 38 sections, 8 theorems, 25 equations, 10 figures, 4 tables, 3 algorithms.

Table of Contents

  1. INTRODUCTION
  2. PROBLEM DEFINITION
  3. Notations
  4. Multi-marginal Optimal Transport
  5. Hierarchical Multi-network Alignment
  6. OPTIMIZATION FORMULATION
  7. Position-Aware Cost Tensor
  8. FGW-based Cluster-level Alignment
  9. MFGW-based Node-level Alignment
  10. ALGORITHM AND ANALYSIS
  11. Optimization Algorithm
  12. FGW-based cluster-level alignment
  13. MFGW-based node-level alignment.
  14. Theoretical Analysis
  15. Complexity analysis.
  16. ...and 23 more sections

Key Result

Proposition 1

The MFGW distance in Eq. eq:mfgw1 with $q=2$ can be formulated into a tensor form as: where $\bm{\mathcal{L}}(v_1,...,v_K) = (K-1)\sum_{j=1}^K\mathbf{C}_j(v_j,\cdot)^2 \mathcal{P}_j(\bm{\mathcal{S}}) - 2\sum_{1\leq j<k\leq K}\mathbf{C}_j(v_j,\cdot)\mathcal{P}_{j,k}(\bm{\mathcal{S}})\mathbf{C}_k(v_k,\cdot)^{\mathsf{T}}$.

Figures (10)

  • Figure 1: An overview of Hot. Left: three input networks, where three green nodes connected by the black dash line form an anchor node set. Middle: FGW barycenter co-clusters three graphs into two clusters. Right: the node alignment tensor with blocks $\bm{\mathcal{S}}^1$ for cluster $\mathcal{C}^1$ and $\bm{\mathcal{S}}^2$ for cluster $\mathcal{C}^2$.
  • Figure 2: Alignment results on plain networks: (a) Douban-230; (b) ER-500.
  • Figure 3: Alignment results on attributed networks: (a) ACM(A)-1000; (b) DBLP(A)-1000.
  • Figure 4: Experiments on time complexity w.r.t. (a) number of nodes, and (b) number of graphs: grey points indicate out-of-memory. Note that the running time is in the log scale.
  • Figure 5: Experiments on space complexity w.r.t. (a) number of nodes, and (b) number of graphs: grey points indicate out-of-memory. Note that the memory cost is in the log scale.
  • ...and 5 more figures

Theorems & Definitions (14)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Theorem 1
  • Proposition 3
  • proof
  • Proposition 3
  • proof
  • ...and 4 more