On Weisfeiler-Leman invariance: Subgraph counts and related graph properties

Arvind, Vikraman; Fuhlbrück, Frank; Köbler, Johannes; Verbitsky, Oleg

doi:10.1016/j.jcss.2020.04.003

Cited by 47 publications

(74 citation statements)

References 25 publications

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“…Another idea to analyze disadvantage of GNNs is to inspect the ability of GNNs to count simple local structure indicators such as circles and triangles. Their finds are consistent with GIN [12], which can basically be concluded as: message passing GNNs can not count local structure features that 1-WL test can not count( [5], [13]- [15]). Some works inspect the GNNs' spectral ability and find that GNNs are nothing but low-pass filters [16].…”

Section: The Disadvantage Of Neighbor Averaging Gnnsmentioning

confidence: 52%

Distance and Hop-wise Structures Encoding Enhanced Graph Attention Networks

Huang¹,

Chen²,

Wang³

2021

Preprint

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Numerous works have proven that existing neighbor-averaging Graph Neural Networks cannot efficiently catch structure features, and many works show that injecting structure, distance, position or spatial features can significantly improve performance of GNNs, however, injecting overall structure and distance into GNNs is an intuitive but remaining untouched idea. In this work, we shed light on the direction. We first extracting hop-wise structure information and compute distance distributional information, gathering with node's intrinsic features, embedding them into same vector space and then adding them up. The derived embedding vectors are then fed into GATs(like GAT, AGDN) and then Correct and Smooth, experiments show that the DHSEGATs achieve competitive result. The code is available at https://github.com/hzg0601/DHSEGATs.

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Section: The Disadvantage Of Neighbor Averaging Gnnsmentioning

confidence: 52%

Distance and Hop-wise Structures Encoding Enhanced Graph Attention Networks

Huang¹,

Chen²,

Wang³

2021

Preprint

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“…Towards understanding the expressive power of the algorithm, in a related direction of research, it has been studied which graph properties the WL algorithm can detect, which may become particularly relevant in the graph-learning framework. In this context, Fürer [14] as well as Arvind et al [2] obtained results concerning the ability of the algorithm to detect and count certain subgraphs.…”

Section: :3mentioning

confidence: 99%

The Power of the Weisfeiler--Leman Algorithm to Decompose Graphs

Kiefer¹,

Neuen²

2022

SIAM J. Discrete Math.

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The Weisfeiler-Leman procedure is a widely-used approach for graph isomorphism testing that works by iteratively computing an isomorphism-invariant coloring of vertex tuples. Meanwhile, a fundamental tool in structural graph theory, which is often exploited in approaches to tackle the graph isomorphism problem, is the decomposition into bi-and triconnected components.We prove that the 2-dimensional Weisfeiler-Leman algorithm implicitly computes the decomposition of a graph into its triconnected components. Thus, the dimension of the algorithm needed to distinguish two given graphs is at most the dimension required to distinguish the corresponding decompositions into 3-connected components (assuming dimension at least 2).Our result implies that for k ≥ 2, the k-dimensional algorithm distinguishes k-separators, i.e., k-tuples of vertices that separate the graph, from other vertex k-tuples. As a byproduct, we also obtain insights about the connectivity of constituent graphs of association schemes.In an application of the results, we show the new upper bound of k on the Weisfeiler-Leman dimension of graphs of treewidth at most k. Using a construction by Cai, Fürer, and Immerman, we also provide a new lower bound that is asymptotically tight up to a factor of 2.

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“…Figure 6 illustrates the 4-by-4 Rook's graph and the Shrikhande graph. These are non-isomorphic graphs that are strongly regular of the same parameters, so they cannot be distinguished by 3-WL / 2-FWL (Arvind et al, 2020;Balcilar et al, 2021).…”

Section: F2 Proofs Of Increasing Encoder Expressivenessmentioning

confidence: 99%

“…To show that DS-GNN with 3-WL is strictly stronger than the other two models, we show that it can distinguish the Rook's graph and the Shrikhande graph, which the other two models cannot. Recall that 3-WL cannot distinguish the two because they are both strongly regular graphs of the same parameters (Arvind et al, 2020). Now, note that all depth-1 ego-nets of the Rook's graph consist of two disjoint triangles with a root node that is connected to all nodes.…”

Section: F2 Proofs Of Increasing Encoder Expressivenessmentioning

confidence: 99%

Equivariant Subgraph Aggregation Networks

Beatrice¹,

Frasca²,

Lim³

et al. 2021

Preprint

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Message-passing neural networks (MPNNs) are the leading architecture for deep learning on graph-structured data, in large part due to their simplicity and scalability. Unfortunately, it was shown that these architectures are limited in their expressive power. This paper proposes a novel framework called Equivariant Subgraph Aggregation Networks (ESAN) to address this issue. Our main observation is that while two graphs may not be distinguishable by an MPNN, they often contain distinguishable subgraphs. Thus, we propose to represent each graph as a set of subgraphs derived by some predefined policy, and to process it using a suitable equivariant architecture. We develop novel variants of the 1-dimensional Weisfeiler-Leman (1-WL) test for graph isomorphism, and prove lower bounds on the expressiveness of ESAN in terms of these new WL variants. We further prove that our approach increases the expressive power of both MPNNs and more expressive architectures. Moreover, we provide theoretical results that describe how design choices such as the subgraph selection policy and equivariant neural architecture affect our architecture's expressive power. To deal with the increased computational cost, we propose a subgraph sampling scheme, which can be viewed as a stochastic version of our framework. A comprehensive set of experiments on real and synthetic datasets demonstrates that our framework improves the expressive power and overall performance of popular GNN architectures.

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On Weisfeiler-Leman invariance: Subgraph counts and related graph properties

Cited by 47 publications

References 25 publications

Distance and Hop-wise Structures Encoding Enhanced Graph Attention Networks

Distance and Hop-wise Structures Encoding Enhanced Graph Attention Networks

The Power of the Weisfeiler--Leman Algorithm to Decompose Graphs

Equivariant Subgraph Aggregation Networks

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