Gromov-Wasserstein Learning for Graph Matching and Node Embedding Hongteng Xu 1 , 2 , Dixin Luo 2 , Hongyuan Zha 3 Lawrence Carin 2 1 Infinia ML, Inc. 2 Department of ECE, Duke University 3 Colledge of Computing, Georgia Tech June 13, 2019
Problem Statement and Proposed Method Given two graphs, we aim to achieve ◮ Graph matching: Finding a correspondence between their nodes. ◮ Node embedding: Embedding their nodes in the same space. Unify them in our Gromov-Wasserstein Learning (GWL) framework. � i , j , i ′ , j ′ L ( c s ij , c t i ′ j ′ ) T ii ′ T jj ′ = min T ∈ Π( µ s , µ t ) � L ( C s , C t , T ) , T � . d GW ( G s , G t ) := min T ∈ Π( µ s , µ t ) Relational matching between graphs Cost = | d(A, D) - d(1, 2) | 5 1 4 B A 8 7 2 3 D 6 C E 1 2 3 4 5 6 7 8 A B C D Optimal E transport
Gromov-Wasserstein Learning � min T ∈ Π( µ s , µ t ) � L ( C s ( X s ) , C t ( X t ) , T ) , T � min + α � K ( X s , X t ) , T � + β k = s , t R ( K ( X k , X k ) , C k ) . X s , X t � �� � � �� � � �� � Gromov-Wasserstein discrepancy Wasserstein discrepancy prior information Relational matching between graphs 5 1 Cost = | d(A, D) - d(1, 2) | 4 Update optimal transport based B A 8 on embeddings 7 and graph 2 3 topology D 6 C E 1 2 3 4 5 6 7 8 A B C D Optimal E transport Absolute matching Update embeddings between graphs based on optimal Cost = K(A, 1) transport and graph topology Embedding space of nodes
Experimental Results GWD GWL-C (OT) GWL-C (Embedding) GW Discrepancy 90 100 0.0008 0.00200 80 0.0007 95 0.00175 0.0006 70 90 Node Correctness (%) Node Correctness (%) 0.00150 GW discrepancy 0.0005 GW discrepancy 60 85 0.00125 0.0004 50 80 0.0003 0.00100 40 75 0.0002 0.00075 30 0.0001 70 0.00050 20 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 The percentage of noisy nodes and edges The percentage of noisy nodes and edges Table 1. Communication network matching results. Call → Email (Sparse) Call → Email (Dense) Method Node Correctness (%) Node Correctness (%) GAA 34.22 0.53 LRSA 38.20 2.93 TAME 37.39 2.67 GRAAL 39.67 0.48 MI-GRAAL 35.53 0.64 MAGNA++ 7.88 0.09 HugAlign 36.21 3.86 NETAL 36.87 1.77 GWD 23.16 ± 0.46 1.77 ± 0.22 GWL-R 39.64 ± 0.57 3.80 ± 0.23 GWL-C 40.45 ± 0.53 4.23 ± 0.27
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