Graphs Identified by Logics with Counting

Kiefer, Sandra; Schweitzer, Pascal; Selman, Erkal

doi:10.1007/978-3-662-48057-1_25

Cited by 27 publications

(24 citation statements)

References 26 publications

(52 reference statements)

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“…The inclusion Amenable ⊂ Compact, therefore, implies that all amenable graphs are refinable as well. The last result is independently obtained in [17] by a different argument. In the particular case of trees, this fact was observed long ago by several authors; see a survey in [25].…”

Section: Our Resultsmentioning

confidence: 65%

Graph Isomorphism, Color Refinement, and Compactness

et al. 2016

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Abstract. Color refinement is a classical technique used to show that two given graphs G and H are non-isomorphic; it is very efficient, although it does not succeed on all graphs. We call a graph G amenable to color refinement if the color-refinement procedure succeeds in distinguishing G from any nonisomorphic graph H. Tinhofer (1991) explored a linear programming approach to Graph Isomorphism and defined the notion of compact graphs: A graph is compact if its fractional automorphisms polytope is integral. Tinhofer noted that isomorphism testing for compact graphs can be done quite efficiently by linear programming. However, the problem of characterizing and recognizing compact graphs in polynomial time remains an open question. Our results are summarized below: -We determine the exact range of applicability of color refinement by showing that amenable graphs are recognizable in time O((n + m) log n), where n and m denote the number of vertices and the number of edges in the input graph.-We show that all amenable graphs are compact. Thus, the applicability range for Tinhofer's linear programming approach to isomorphism testing is at least as large as for the combinatorial approach based on color refinement. -Exploring the relationship between color refinement and compactness further, we study related combinatorial and algebraic graph properties introduced by Tinhofer and Godsil. We show that the corresponding classes of graphs form a hierarchy and we prove that recognizing each of these graph classes is P-hard. In particular, this gives a first complexity lower bound for recognizing compact graphs.

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Section: Our Resultsmentioning

confidence: 65%

Graph Isomorphism, Color Refinement, and Compactness

et al. 2016

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“…Our experimental evaluation confirms our theoretical findings as well as confirms that higher-order information is useful in the task of graph classification and regression.completing the equivalence. Since the power of the 1-WL has been completely characterized, see, e.g., (Arvind et al 2015;Kiefer, Schweitzer, and Selman 2015), we can transfer these results to the case of GNNs, showing that both approaches have the same shortcomings.Going further, we leverage these theoretical relationships to propose a generalization of GNNs, called k-GNNs, which are neural architectures based on the k-dimensional WL algorithm (k-WL), which are strictly more powerful than GNNs. The key insight in these higher-dimensional variants is that they perform message passing directly between subgraph structures, rather than individual nodes.…”

mentioning

confidence: 65%

“…completing the equivalence. Since the power of the 1-WL has been completely characterized, see, e.g., (Arvind et al 2015;Kiefer, Schweitzer, and Selman 2015), we can transfer these results to the case of GNNs, showing that both approaches have the same shortcomings.…”

mentioning

confidence: 65%

Weisfeiler and Leman Go Neural: Higher-Order Graph Neural Networks

Morris

Ritzert

Fey

et al. 2019

AAAI

925

888

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In recent years, graph neural networks (GNNs) have emerged as a powerful neural architecture to learn vector representations of nodes and graphs in a supervised, end-to-end fashion. Up to now, GNNs have only been evaluated empirically-showing promising results. The following work investigates GNNs from a theoretical point of view and relates them to the 1-dimensional Weisfeiler-Leman graph isomorphism heuristic (1-WL). We show that GNNs have the same expressiveness as the 1-WL in terms of distinguishing non-isomorphic (sub-)graphs. Hence, both algorithms also have the same shortcomings. Based on this, we propose a generalization of GNNs, so-called k-dimensional GNNs (k-GNNs), which can take higher-order graph structures at multiple scales into account. These higher-order structures play an essential role in the characterization of social networks and molecule graphs. Our experimental evaluation confirms our theoretical findings as well as confirms that higher-order information is useful in the task of graph classification and regression.completing the equivalence. Since the power of the 1-WL has been completely characterized, see, e.g., (Arvind et al. 2015;Kiefer, Schweitzer, and Selman 2015), we can transfer these results to the case of GNNs, showing that both approaches have the same shortcomings.Going further, we leverage these theoretical relationships to propose a generalization of GNNs, called k-GNNs, which are neural architectures based on the k-dimensional WL algorithm (k-WL), which are strictly more powerful than GNNs. The key insight in these higher-dimensional variants is that they perform message passing directly between subgraph structures, rather than individual nodes. This higher-order form of message passing can capture structural information that is not visible at the node-level.Graph kernels based on the k-WL have been proposed in the past (Morris, Kersting, and Mutzel 2017). However, a key advantage of implementing higher-order message passing in GNNs-which we demonstrate here-is that we can design hierarchical variants of k-GNNs, which combine graph representations learned at different granularities in an end-to-end trainable framework. Concretely, in the presented hierarchical approach the initial messages in a k-GNN are based on the output of lower-dimensional k -GNN (with k < k), which allows the model to effectively capture graph structures of varying granularity. Many real-world graphs inherit a hierarchical structure-e.g., in a social network we must model both the ego-networks around individual nodes, as well as the coarse-grained relationships between entire communities, see, e.g., (Newman 2003)-and our experimental results demonstrate that these hierarchical k-GNNs are able to consistently outperform traditional GNNs on a variety of graph classification and regression tasks. Across twelve graph regression tasks from the QM9 benchmark, we find that our hierarchical model reduces the mean absolute error by 54.45% on average. For graph classification, we find that our hierarchical models...

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“…While it is possible to describe precisely the graphs of WL-dimension 1 [25,1] (i.e., graphs definable with a 2-variable sentence in first order logic with counting), it appears difficult to make such statements for higher dimensions. However, for various graph classes for which the isomorphism problem is known to be polynomial time solvable, one can give upper bounds on the dimension.…”

Section: Theorem 1 the Weisfeiler-leman Dimension Of The Class Of Plmentioning

confidence: 99%

The Weisfeiler-Leman dimension of planar graphs is at most 3

Kiefer¹,

Ponomarenko²,

Schweitzer³

2017

2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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We prove that the Weisfeiler-Leman (WL) dimension of the class of all finite planar graphs is at most 3. In particular, every finite planar graph is definable in first-order logic with counting using at most 4 variables. The previously best known upper bounds for the dimension and number of variables were 14 and 15, respectively.First we show that, for dimension 3 and higher, the WL-algorithm correctly tests isomorphism of graphs in a minor-closed class whenever it determines the orbits of the automorphism group of any arc-colored 3-connected graph belonging to this class.Then we prove that, apart from several exceptional graphs (which have WLdimension at most 2), the individualization of two correctly chosen vertices of a colored 3-connected planar graph followed by the 1-dimensional WL-algorithm produces the discrete vertex partition. This implies that the 3-dimensional WLalgorithm determines the orbits of a colored 3-connected planar graph.As a byproduct of the proof, we get a classification of the 3-connected planar graphs with fixing number 3. This is an extended version of the paper with the same title published in the

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Graphs Identified by Logics with Counting

Cited by 27 publications

References 26 publications

Graph Isomorphism, Color Refinement, and Compactness

Graph Isomorphism, Color Refinement, and Compactness

Weisfeiler and Leman Go Neural: Higher-Order Graph Neural Networks

The Weisfeiler-Leman dimension of planar graphs is at most 3

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