Can graph neural networks count substructures?

NeurIPS 2020, 2020.

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weisfeiler lehmanmean squared errorgraph isomorphism testingFolklore Weisfeiler-LehmanInvariant Graph NetworksMore(13+)
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We propose a theoretical framework to study the expressive power of classes of graph neural networks based on their ability to count substructures

Abstract:

The ability to detect and count certain substructures in graphs is important for solving many tasks on graph-structured data, especially in the contexts of computational chemistry and biology as well as social network analysis. Inspired by this, we propose to study the expressive power of graph neural networks (GNNs) via their ability t...More

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Introduction
Highlights
  • In recent years, graph neural networks (GNNs) have achieved empirical success on processing data from various fields such as social networks, quantum chemistry, particle physics, knowledge graphs and combinatorial optimization (Scarselli et al, 2008; Bruna et al, 2013; Duvenaud et al, 2015; Kipf and Welling, 2016; Defferrard et al, 2016; Bronstein et al, 2017; Dai et al, 2017; Nowak et al, 2017; Ying et al, 2018; Zhou et al, 2018; Choma et al, 2018; Zhang and Chen, 2018; You et al, 2018a,b, 2019; Yao et al, 2019; Ding et al, 2019; Stokes et al, 2020)
  • Instead of performing iterative equivariant aggregations of information as is done in Message Passing Neural Networks (MPNNs) and Invariant Graph Networks (IGNs), we propose a type of locally powerful models based on the observation that substructures present themselves in local neighborhoods known as egonets
  • GIN, 2-Invariant Graph Networks (2-IGNs) and spectral GNN (sGNN) produce much smaller test error than the variance of the ground truth counts for the 3-star tasks, consistent with their theoretical power to perform containment-count of stars
  • We propose a theoretical framework to study the expressive power of classes of GNNs based on their ability to count substructures
  • We prove that neither MPNNs nor 2-IGNs can matching-count any connected structure with 3 or more nodes; k-Invariant Graph Functions (k-IGNs) and k-WL can containment-count and matching-count any pattern of size k
  • We build the foundation for using substructure counting as an intuitive and relevant measure of the expressive power of GNNs, and our concrete results for existing GNNs motivate the search for more powerful designs of GNNs
Methods
  • The authors verify the theoretical results on two graph-level regression tasks: matching-counting triangles and containment-counting 3-stars, with both patterns unattributed, as illustrated in Figure 3.
  • By. Theorem 2 and Corollary 1, MPNNs and 2-IGNs can perform matching-count of triangles.
  • Note that since a triangle is a clique, its matching-count and containment-count are equal.
  • The authors generate the ground-truth counts of triangles in each graph with an counting algorithm proposed by Shervashidze et al (2009).
Results
  • The results on the two tasks are shown in Table 1, measured by the MSE on the test set divided by the variance of the ground truth counts of the pattern computed over all graphs in the dataset.
  • The almost-negligible errors of LRP on all the tasks supports the theory that depth-1 LRP is powerful enough for counting triangles and 3-stars, both of which are patterns with radius 1.
  • GIN, 2-IGN and sGNN produce much smaller test error than the variance of the ground truth counts for the 3-star tasks, consistent with their theoretical power to perform containment-count of stars.
Conclusion
  • The authors propose a theoretical framework to study the expressive power of classes of GNNs based on their ability to count substructures.
  • The authors provide an upper bound on the size of “path-shaped” substructures that finite iterations of k-WL can matching-count.
  • To establish these results, the authors prove an equivalence between approximating graph functions and discriminating graphs.
  • The authors build the foundation for using substructure counting as an intuitive and relevant measure of the expressive power of GNNs, and the concrete results for existing GNNs motivate the search for more powerful designs of GNNs
Summary
  • Introduction:

    Graph neural networks (GNNs) have achieved empirical success on processing data from various fields such as social networks, quantum chemistry, particle physics, knowledge graphs and combinatorial optimization (Scarselli et al, 2008; Bruna et al, 2013; Duvenaud et al, 2015; Kipf and Welling, 2016; Defferrard et al, 2016; Bronstein et al, 2017; Dai et al, 2017; Nowak et al, 2017; Ying et al, 2018; Zhou et al, 2018; Choma et al, 2018; Zhang and Chen, 2018; You et al, 2018a,b, 2019; Yao et al, 2019; Ding et al, 2019; Stokes et al, 2020).
  • From the viewpoint of graph isomorphism testing, existing GNNs are in some sense already not far from being maximally powerful, which could make the pursuit of more powerful GNNs appear unnecessary
  • Objectives:

    The general case can be proved in the same way but with more subscripts.
  • (In particular, for the counterexamples, (69) can be shown to hold for each of the d0 feature dimensions.) Define a set S = {(1, 2), (2, 1), (1 + m, 2 + m), (2 + m, 1 + m), (1, 2 + m), (2 + m, 1), (1 + m, 2), (2, 1 + m)}, which represents the “special” edges that capture the difference between G[1] and G[2].
  • Methods:

    The authors verify the theoretical results on two graph-level regression tasks: matching-counting triangles and containment-counting 3-stars, with both patterns unattributed, as illustrated in Figure 3.
  • By. Theorem 2 and Corollary 1, MPNNs and 2-IGNs can perform matching-count of triangles.
  • Note that since a triangle is a clique, its matching-count and containment-count are equal.
  • The authors generate the ground-truth counts of triangles in each graph with an counting algorithm proposed by Shervashidze et al (2009).
  • Results:

    The results on the two tasks are shown in Table 1, measured by the MSE on the test set divided by the variance of the ground truth counts of the pattern computed over all graphs in the dataset.
  • The almost-negligible errors of LRP on all the tasks supports the theory that depth-1 LRP is powerful enough for counting triangles and 3-stars, both of which are patterns with radius 1.
  • GIN, 2-IGN and sGNN produce much smaller test error than the variance of the ground truth counts for the 3-star tasks, consistent with their theoretical power to perform containment-count of stars.
  • Conclusion:

    The authors propose a theoretical framework to study the expressive power of classes of GNNs based on their ability to count substructures.
  • The authors provide an upper bound on the size of “path-shaped” substructures that finite iterations of k-WL can matching-count.
  • To establish these results, the authors prove an equivalence between approximating graph functions and discriminating graphs.
  • The authors build the foundation for using substructure counting as an intuitive and relevant measure of the expressive power of GNNs, and the concrete results for existing GNNs motivate the search for more powerful designs of GNNs
Tables
  • Table1: Performance of different GNNs on matching-counting triangles and containment-counting 3-stars on the two datasets, measured by test MSE divided by variance of the ground truth counts. Shown here are the best and the median performances of each model over five runs. Note that we select the best out of four variants for each of GCN, GIN and sGNN, and the better out of two variants for 2-IGN. Details of the GNN architectures and raw results can be found in Appendices J, K
  • Table2: Test MSE loss for all models with chosen parameters as specified in Appendix J. We run each model for five times and picked the best and the median (3rd best) results for Table 1. Note that each of GCN, GIN and sGNN has four variants while 2-IGN has two variants. The reported rows in Table 1 are bolded here
Download tables as Excel
Funding
  • This work is partially supported by the Alfred P
  • SV is partly supported by NSF DMS 1913134, EOARD FA9550-18-1-7007 and the Simons Algorithms and Geometry (A&G) Think Tank
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  • On one hand, by construction, 2-WL will not be able to distinguish G[1] from G[2]. This is intuitive if we compare the rooted subtrees in the two graphs, as there exists a bijection from V [1] to V [2] that preserves the rooted subtree structure. A rigorous proof is given at the end of this section. In addition, we note that this is also consequence of the direct proof of Corollary 4 given in Appendix I, in which we will show that the same pair of graphs cannot be distinguished by 2-IGNs. Since 2-IGNs are no less powerful than 2-WL (Maron et al., 2019b), this implies that 2-WL cannot distinguish them either.
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