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# The Logical Expressiveness of Graph Neural Networks

ICLR, (2020)

EI

Abstract

The ability of graph neural networks (GNNs) for distinguishing nodes in graphs has been recently characterized in terms of the Weisfeiler-Lehman (WL) test for checking graph isomorphism. This characterization, however, does not settle the issue of which Boolean node classifiers (i.e., functions classifying nodes in graphs as true or fals...More

Introduction

- Graph neural networks (GNNs) (Merkwirth & Lengauer, 2005; Scarselli et al, 2009) are a class of neural network architectures that has recently become popular for a wide range of applications dealing with structured data, e.g., molecule classification, knowledge graph completion, and Web page ranking (Battaglia et al, 2018; Gilmer et al, 2017; Kipf & Welling, 2017; Schlichtkrull et al, 2018).
- One such FOC2 classifier is γ(x) in Equation (4), but there are infinitely many and even simpler FOC2 formulas that cannot be captured by AC-GNNs. Intuitively, the main problem is that an ACGNN has only a fixed number L of layers and the information of local aggregations cannot travel further than at distance L of every node along edges in the graph.

Highlights

- Graph neural networks (GNNs) (Merkwirth & Lengauer, 2005; Scarselli et al, 2009) are a class of neural network architectures that has recently become popular for a wide range of applications dealing with structured data, e.g., molecule classification, knowledge graph completion, and Web page ranking (Battaglia et al, 2018; Gilmer et al, 2017; Kipf & Welling, 2017; Schlichtkrull et al, 2018)
- We prove that each FOC2 formula can be captured by an ACR-Graph neural networks
- We show that on synthetic graph data conforming to FOC2 formulas, ACGNNs struggle to fit the training data while ACR-Graph neural networks can generalize even to graphs of sizes not seen during training
- We perform two sets of experiments: experiments to show that ACR-Graph neural networks can learn a very simple FOC2 node classifier that AC-Graph neural networks cannot learn, and experiments involving complex FOC2 classifiers that need more intermediate readouts to be learned
- We report results on a real benchmark (PPI) where we did not observe an improvement of ACR-Graph neural networks over AC-Graph neural networks
- Separating AC-Graph neural networks and ACR-Graph neural networks We consider a very simple FOC2 formula defined by α(x) := Red(x) ∧ ∃y Blue(y), which is satisfied by every red node in a graph provided that the graph contains at least one blue node

Results

- To see how a readout function could help in capturing non-local properties, consider again the logical classifier γ(x) in Equation (4), that assigns true to every red node v as long as there is another node not connected with v having two blue neighbors.
- The construction is similar to that of Proposition 4.1 and uses simple, homogeneous ACR-GNNs— that is, the readout function is just the sum of all the local node feature vectors.
- The authors leave as a challenging open problem whether FOC2 classifiers are exactly the logical classifiers captured by ACR-GNNs. 5.3 COMPARING THE NUMBER OF READOUT LAYERS
- The proof of Theorem 5.1 constructs GNNs whose number of layers depends on the formula being captured—that is, readout functions are used unboundedly many times in ACR-GNNs for capturing different FOC2 classifiers.
- It is based on a refinement of the GIN architecture proposed by Xu et al (2019) to obtain as much information as possible about the local neighborhood in graphs, followed by a readout and combine functions that use this information to deal with non-local constructs in formulas.
- The authors report results on a real benchmark (PPI) where the authors did not observe an improvement of ACR-GNNs over AC-GNNs. Separating AC-GNNs and ACR-GNNs The authors consider a very simple FOC2 formula defined by α(x) := Red(x) ∧ ∃y Blue(y), which is satisfied by every red node in a graph provided that the graph contains at least one blue node.

Conclusion

- This combined with the fact that random graphs that are more dense make the maximum distances between nodes shorter, may explain the boost in performance for AC-GNNs. Complex FOC2 properties In the second experiment the authors consider classifiers αi(x) constructed as α0(x) := Blue(x), αi+1(x) := ∃[N,M]y αi(y) ∧ ¬E(x, y) , (6)
- The authors' work is close in spirit to that of Xu et al (2019) and Morris et al (2019) establishing the correspondence between the WL test and GNNs. In contrast to the work, they focus on graph classification and do not consider the relationship with logical classifiers.

- Table1: Results on synthetic data for nodes labeled by classifier α(x) := Red(x) ∧ ∃y Blue(y)
- Table2: Results on E-R synthetic data for nodes labeled by classifiers αi(x) in Equation (6)
- Table3: Synthetic data for the experiment with classifier α(x) := Red(x) ∧ ∃y Blue(y)
- Table4: Detailed results for Erdos-Renyi synthetic graphs with different connectivities
- Table5: Synthetic data for the experiment with classifier αi(x) in Equation (6)
- Table6: Performance of AC-GNN and ACR-GNN in the PPI benchmark

Funding

- This work was partly funded by the Millennium Institute for Foundational Research on Data2

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