Isomorphism Testing Research Articles

Combinatorial refinement on circulant graphs

The combinatorial refinement techniques have proven to be an efficientapproach to isomorphism testing for particular classes of graphs.If the number of refinement rounds is small, this putsthe corresponding isomorphism problem in a low-complexity class.We investigate the round complexity of the two-dimensional Weisfeiler--Leman algorithmon circulant graphs, i.e., on Cayley graphs of the cyclic group Zn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb{Z}_n$$\\end{document}, and prove that the number of rounds until stabilization is bounded by O(d(n)logn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{O}(d(n)\\log n)$$\\end{document},where d(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d(n)$$\\end{document} is the number of divisors of n. As a particular consequence, isomorphism can be tested in NC for connected circulant graphs of order pℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p^\\ell$$\\end{document} with p an odd prime, ℓ>3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell>3$$\\end{document} and vertex degree Δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta$$\\end{document} smaller than p.We also show that the color refinement method (also known as the one-dimensional Weisfeiler--Leman algorithm)computes a canonical labeling for every non-trivial circulant graph with a prime number of verticesafter individualization of two appropriately chosen vertices.Thus, the canonical labeling problem for this class of graphs has at most the samecomplexity as color refinement, which results in a time bound of O(Δnlogn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{O}(\\Delta \\, n\\log n)$$\\end{document}.Moreover, this provides a first example wherea sophisticated approach to isomorphism testing put forward by Tinhofer has a real practical meaning.

Exponentially Many Graphs Are Determined By Their Spectrum

Abstract As a discrete analogue of Kac’s celebrated question on ‘hearing the shape of a drum’ and towards a practical graph isomorphism test, it is of interest to understand which graphs are determined up to isomorphism by their spectrum (of their adjacency matrix). A striking conjecture in this area, due to van Dam and Haemers, is that ‘almost all graphs are determined by their spectrum’, meaning that the fraction of unlabelled n-vertex graphs which are determined by their spectrum converges to 1 as $n\to\infty$. In this paper, we make a step towards this conjecture, showing that there are exponentially many n-vertex graphs which are determined by their spectrum. This improves on previous bounds (of shape $\it{e}^{c\sqrt{n}}$). We also propose a number of further directions of research.

Structural Synthesis of Planar Kinematic Chains for Non-Redundant Parallel Manipulators

Abstract This work focuses on the synthesis of non-redundant planar kinematic chains applied to parallel manipulators, where the mobility of the kinematic chains is constrained by the planar workspace, allowing for kinematic chains with up to three degrees-of-freedom. The synthesis process consists of two stages. First, a graph generator enumerates non-isomorphic biconnected graphs representing planar kinematic chains with up to three degrees-of-freedom and seven loops. Subsequently, graphs associated to chains with rigid subchains are removed using a degeneracy identification method based on matroid theory. Concise results are presented for chains with up to five loops. For chains with six and seven loops, results are categorized based on link partitions for a direct comparison with previous results. These findings align with prior research, with minor variations in the reported chain numbers. These variations can be attributed to several factors, including graph generation, isomorphism testing, fractionation, and subchain identification. These factors are comprehensively discussed and examined.

Hypergraph Isomorphism Computation.

The isomorphism problem, crucial in network analysis, involves analyzing both low-order and high-order structural information. Graph isomorphism algorithms focus on structural equivalence to simplify solver space, aiding applications like protein design, chemical pathways, and community detection. However, they fall short in capturing complex high-order relationships, unlike hypergraph isomorphism methods. Traditional hypergraph methods face challenges like high memory use and inaccurate identification, leading to poor performance. To overcome these, we introduce a hypergraph Weisfeiler-Lehman (WL) test algorithm, extending the WL test from graphs to hypergraphs, and develop a hypergraph WL kernel framework with two variants: the Hypergraph WL Subtree Kernel and Hypergraph WL Hyperedge Kernel. The Hypergraph WL Subtree Kernel counts different types of rooted subtrees and generates the final feature vector for a given hypergraph by comparing the number of different types of rooted subtrees. The Subtree Kernel identifies different rooted subtrees, while the Hyperedge Kernel focuses on hyperedges' vertex labels, enhancing feature vector generation. In order to fulfill our research objectives, a comprehensive set of experiments was meticulously designed, including seven graph classification datasets and 12 hypergraph classification datasets. Results on graph classification datasets indicate that the Hypergraph WL Subtree Kernel can achieve the same performance compared with the classical Graph Weisfeiler-Lehman Subtree Kernel. Results on hypergraph classification datasets show significant improvements compared to other typical kernel-based methods, which demonstrates the effectiveness of the proposed methods. In our evaluation, our proposed methods outperform the second-best method in terms of runtime, running over 80 times faster when handling complex hypergraph structures. This significant speed advantage highlights the great potential of our methods in real-world applications.

Learning symmetry-aware atom mapping in chemical reactions through deep graph matching

Accurate atom mapping, which establishes correspondences between atoms in reactants and products, is a crucial step in analyzing chemical reactions. In this paper, we present a novel end-to-end approach that formulates the atom mapping problem as a deep graph matching task. Our proposed model, AMNet (Atom Matching Network), utilizes molecular graph representations and employs various atom and bond features using graph neural networks to capture the intricate structural characteristics of molecules, ensuring precise atom correspondence predictions. Notably, AMNet incorporates the consideration of molecule symmetry, enhancing accuracy while simultaneously reducing computational complexity. The integration of the Weisfeiler-Lehman isomorphism test for symmetry identification refines the model’s predictions. Furthermore, our model maps the entire atom set in a chemical reaction, offering a comprehensive approach beyond focusing solely on the main molecules in reactions. We evaluated AMNet’s performance on a subset of USPTO reaction datasets, addressing various tasks, including assessing the impact of molecular symmetry identification, understanding the influence of feature selection on AMNet performance, and comparing its performance with the state-of-the-art method. The result reveals an average accuracy of 97.3% on mapped atoms, with 99.7% of reactions correctly mapped when the correct mapped atom is within the top 10 predicted atoms.Scientific contributionThe paper introduces a novel end-to-end deep graph matching model for atom mapping, utilizing molecular graph representations to capture structural characteristics effectively. It enhances accuracy by integrating symmetry detection through the Weisfeiler-Lehman test, reducing the number of possible mappings and improving efficiency. Unlike previous methods, it maps the entire reaction, not just main components, providing a comprehensive view. Additionally, by integrating efficient graph matching techniques, it reduces computational complexity, making atom mapping more feasible.

Count-free Weisfeiler–Leman and group isomorphism

We investigate the power of counting in Group Isomorphism. We first leverage the count-free variant of the Weisfeiler–Leman Version I algorithm for groups [J. Brachter and P. Schweitzer, On the Weisfeiler–Leman dimension of finite groups, in 35th Annual ACM/IEEE Symp. Logic in Computer Science, eds. H. Hermanns, L. Zhang, N. Kobayashi and D. Miller, Saarbrucken, Germany, July 8–11, 2020 (ACM, 2020), pp. 287–300, doi:10.1145/3373718.3394786] in tandem with bounded non-determinism and limited counting to improve the parallel complexity of isomorphism testing for several families of groups. These families include: • Direct products of non-Abelian simple groups. • Coprime extensions, where the normal Hall subgroup is Abelian and the complement is an [Formula: see text]-generated solvable group with solvability class [Formula: see text]. This notably includes instances where the complement is an [Formula: see text]-generated nilpotent group. This problem was previously known to be in [Formula: see text] [Y. Qiao, J. M. N. Sarma and B. Tang, On isomorphism testing of groups with normal Hall subgroups, in Proc. 28th Symp. Theoretical Aspects of Computer Science, Dagstuhl Castle, Leibniz Center for Informatics, 2011), pp. 567–578, doi:10.4230/LIPIcs. STACS.2011.567], and the complexity was recently improved to [Formula: see text] [J. A. Grochow and M. Levet, On the parallel complexity of group isomorphism via Weisfeiler–Leman, in 24th Int. Symp. Fundamentals of Computation Theory, eds. H. Fernau and K. Jansen, Lecture Notes in Computer Science, Vol. 14292, September 18–21, 2023, Trier, Germany (Springer, 2023), pp. 234–247]. • Graphical groups of class 2 and exponent [Formula: see text] [A. H. Mekler, Stability of nilpotent groups of class 2 and prime exponent, J. Symb. Logic 46(4) (1981) 781–788] arising from the CFI and twisted CFI graphs [J.-Y. Cai, M. Fürer and N. Immerman, An optimal lower bound on the number of variables for graph identification, Combinatorica 12(4) (1992) 389–410], respectively. In particular, our work improves upon previous results of Brachter and Schweitzer [On the Weisfeiler–Leman dimension of finite groups, in 35th Annual ACM/IEEE Symp. Logic in Computer Science, eds. H. Hermanns, L. Zhang, N. Kobayashi and D. Miller, Saarbrucken, Germany, July 8–11, 2020 (ACM, 2020), pp. 287–300, doi:10.1145/3373718.3394786]. Notably, each of these families was previously known to be identified by the counting variant of the more powerful Weisfeiler–Leman Version II algorithm. We finally show that the q-ary count-free pebble game is unable to even distinguish Abelian groups. This extends the result of Grochow and Levet (ibid), who established the result in the case of [Formula: see text]. The general theme is that some counting appears necessary to place [Formula: see text] into [Formula: see text].

Complete Neural Networks for Complete Euclidean Graphs

Neural networks for point clouds, which respect their natural invariance to permutation and rigid motion, have enjoyed recent success in modeling geometric phenomena, from molecular dynamics to recommender systems. Yet, to date, no architecture with polynomial complexity is known to be complete, that is, able to distinguish between any pair of non-isomorphic point clouds. We fill this theoretical gap by showing that point clouds can be completely determined, up to permutation and rigid motion, by applying the 3-WL graph isomorphism test to the point cloud's centralized Gram matrix. Moreover, we formulate an Euclidean variant of the 2-WL test and show that it is also sufficient to achieve completeness. We then show how our complete Euclidean WL tests can be simulated by an Euclidean graph neural network of moderate size and demonstrate their separation capability on highly symmetrical point clouds.

Isomorphism Testing for Graphs Excluding Small Topological Subgraphs

We give an isomorphism test that runs in time \(n^{\operatorname{polylog}(h)} \) on all n -vertex graphs excluding some h -vertex graph as a topological subgraph. Previous results state that isomorphism for such graphs can be tested in time \(n^{\operatorname{polylog}(n)} \) (Babai, STOC 2016) and n f ( h ) for some function f (Grohe and Marx, SIAM J. Comp., 2015). Our result also unifies and extends previous isomorphism tests for graphs of maximum degree d running in time \(n^{\operatorname{polylog}(d)} \) (SIAM J. Comp., 2023) and for graphs of Hadwiger number h running in time \(n^{\operatorname{polylog}(h)} \) (SIAM J. Comp., 2023).

SEGCN: a subgraph encoding based graph convolutional network model for social bot detection

Message passing neural networks such as graph convolutional networks (GCN) can jointly consider various types of features for social bot detection. However, the expressive power of GCN is upper-bounded by the 1st-order Weisfeiler–Leman isomorphism test, which limits the detection performance for the social bots. In this paper, we propose a subgraph encoding based GCN model, SEGCN, with stronger expressive power for social bot detection. Each node representation of this model is computed as the encoding of a surrounding induced subgraph rather than encoding of immediate neighbors only. Extensive experimental results on two publicly available datasets, Twibot-20 and Twibot-22, showed that the proposed model improves the accuracy of the state-of-the-art social bot detection models by around 2.4%, 3.1%, respectively.

Isomorphism Testing Parameterized by Genus and Beyond

Isomorphism Testing Parameterized by Genus and Beyond

How to Find the Equivalence Classes in a Set of Linear Codes in Practice?

An algorithm for equivalence of linear codes over finite fields is presented. Its main advantage is that it can extract exactly one representative from each equivalence class among a large number of linear codes. It can also be used as a test for isomorphism of binary matrices. The algorithm is implemented in the program LCequivalence, which is designed to obtain the inequivalent codes in a set of linear codes over a finite field with q<64 elements. This program is a module of the free software package QextNewEdition for constructing, classifying and studying linear codes.

Weisfeiler-Leman Indistinguishability of Graphons

The color refinement algorithm is mainly known as a heuristic method for graph isomorphism testing. It has surprising but natural characterizations in terms of, for example, homomorphism counts from trees and solutions to a system of linear equations. Grebík and Rocha (2022) have recently shown how color refinement and notions that characterize it generalize to graphons, which emerged as limit objects in the theory of dense graph limits. In particular, they show that these characterizations are still equivalent in the graphon case. The $k$-dimensional Weisfeiler-Leman algorithm ($k$-WL) is a more powerful variant of color refinement that colors $k$-tuples instead of single vertices, where the terms $1$-WL and color refinement are often used interchangeably since they compute equivalent colorings. We show how to adapt the result of Grebík and Rocha to $k$-WL or, in other words, how $k$-WL and its characterizations generalize to graphons. In particular, we obtain characterizations in terms of homomorphism densities from multigraphs of bounded treewidth and linear equations. We give a simple example that parallel edges make a difference in the more general case of graphons, which means that, there, the equivalence between $1$-WL and color refinement does not hold anymore. We also show how this equivalence can be recovered by defining a variant of $k$-WL that corresponds to homomorphism densities from simple graphs of bounded treewidth.

Non-IID Transfer Learning on Graphs

Transfer learning refers to the transfer of knowledge or information from a relevant source domain to a target domain. However, most existing transfer learning theories and algorithms focus on IID tasks, where the source/target samples are assumed to be independent and identically distributed. Very little effort is devoted to theoretically studying the knowledge transferability on non-IID tasks, e.g., cross-network mining. To bridge the gap, in this paper, we propose rigorous generalization bounds and algorithms for cross-network transfer learning from a source graph to a target graph. The crucial idea is to characterize the cross-network knowledge transferability from the perspective of the Weisfeiler-Lehman graph isomorphism test. To this end, we propose a novel Graph Subtree Discrepancy to measure the graph distribution shift between source and target graphs. Then the generalization error bounds on cross-network transfer learning, including both cross-network node classification and link prediction tasks, can be derived in terms of the source knowledge and the Graph Subtree Discrepancy across domains. This thereby motivates us to propose a generic graph adaptive network (GRADE) to minimize the distribution shift between source and target graphs for cross-network transfer learning. Experimental results verify the effectiveness and efficiency of our GRADE framework on both cross-network node classification and cross-domain recommendation tasks.

Scalable maximal subgraph mining with backbone-preserving graph convolutions

Maximal subgraph mining is increasingly important in various domains, including bioinformatics, genomics, and chemistry, as it helps identify common characteristics among a set of graphs and enables their classification into different categories. Existing approaches for identifying maximal subgraphs typically rely on traversing a graph lattice. However, in practice, these approaches are limited to relatively small subgraphs due to the exponential growth of the search space and the NP-completeness of the underlying subgraph isomorphism test. In this work, we propose SCAMA, an approach that addresses these limitations by adopting a divide-and-conquer strategy for efficient mining of maximal subgraphs. Our approach involves initially partitioning a graph database into equivalence classes using bootstrapped backbones, which are tree-shaped frequent subgraphs. We then introduce a learning process based on a novel graph convolutional network (GCN) to extract maximal backbones for each equivalence class. A critical insight of our approach is that by estimating each maximal backbone directly in the embedding space, we can avoid the exponential traversal of the graph lattice. From the extracted maximal backbones, we construct the maximal frequent subgraphs. Furthermore, we outline how SCAMA can be extended to perform top-k largest frequent subgraph mining and how the discovered patterns facilitate graph classification. Our experimental results demonstrate the effectiveness of SCAMA in identifying almost perfectly maximal frequent subgraphs, while exhibiting approximately 10 times faster performance compared to the best baseline technique.

On QSAR-based cardiotoxicity modeling with the expressiveness-enhanced graph learning model and dual-threshold scheme.

Introduction: Given the direct association with malignant ventricular arrhythmias, cardiotoxicity is a major concern in drug design. In the past decades, computational models based on the quantitative structure-activity relationship have been proposed to screen out cardiotoxic compounds and have shown promising results. The combination of molecular fingerprint and the machine learning model shows stable performance for a wide spectrum of problems; however, not long after the advent of the graph neural network (GNN) deep learning model and its variant (e.g., graph transformer), it has become the principal way of quantitative structure-activity relationship-based modeling for its high flexibility in feature extraction and decision rule generation. Despite all these progresses, the expressiveness (the ability of a program to identify non-isomorphic graph structures) of the GNN model is bounded by the WL isomorphism test, and a suitable thresholding scheme that relates directly to the sensitivity and credibility of a model is still an open question. Methods: In this research, we further improved the expressiveness of the GNN model by introducing the substructure-aware bias by the graph subgraph transformer network model. Moreover, to propose the most appropriate thresholding scheme, a comprehensive comparison of the thresholding schemes was conducted. Results: Based on these improvements, the best model attains performance with 90.4% precision, 90.4% recall, and 90.5% F1-score with a dual-threshold scheme (active: ; non-active: ). The improved pipeline (graph subgraph transformer network model and thresholding scheme) also shows its advantages in terms of the activity cliff problem and model interpretability.

Enhancing Molecular Representations Via Graph Transformation Layers.

Molecular representation learning is an essential component of many molecule-oriented tasks, such as molecular property prediction and molecule generation. In recent years, graph neural networks (GNNs) have shown great promise in this area, representing a molecule as a graph composed of nodes and edges. There are increasing studies showing that coarse-grained or multiview molecular graphs are important for molecular representation learning. Most of their models, however, are too complex and lack flexibility in learning different granular information for different tasks. Here, we proposed a flexible and simple graph transformation layer (i.e., LineEvo), a plug-and-use module for GNNs, which enables molecular representation learning from multiple perspectives. The LineEvo layer transforms fine-grained molecular graphs into coarse-grained ones based on the line graph transformation strategy. Especially, it treats the edges as nodes and generates the new connected edges, atom features, and atom positions. By stacking LineEvo layers, GNNs can learn multilevel information, from atom-level to triple-atoms level and coarser level. Experimental results show that the LineEvo layers can improve the performance of traditional GNNs on molecular property prediction benchmarks on average by 7%. Additionally, we show that the LineEvo layers can help GNNs have more expressive power than the Weisfeiler-Lehman graph isomorphism test.

Isomorphism Testing for Graphs Excluding Small Minors

We prove that there is a graph isomorphism test running in time on -vertex graphs excluding some -vertex graph as a minor. Previously known bounds were [I. N. Ponomarenko, J. Soviet Math., 55 (1991), pp. 1621–1643] and [L. Babai, Proceedings of the 48th Annual ACM Symposium on Theory of Computing, 2016, pp. 684–697]. For the algorithm we combine recent advances in the group-theoretic graph isomorphism machinery with new graph-theoretic arguments.

Explainability in subgraphs-enhanced Graph Neural Networks

Recently, subgraphs-enhanced Graph Neural Networks (SGNNs) have been introduced to enhance the expressive power of Graph Neural Networks (GNNs), which was proved to be not higher than the 1-dimensional Weisfeiler-Leman isomorphism test. The new paradigm suggests using subgraphs extracted from the input graph to improve the model's expressiveness, but the additional complexity exacerbates an already challenging problem in GNNs: explaining their predictions. In this work, we adapt PGExplainer, one of the most recent explainers for GNNs, to SGNNs. The proposed explainer accounts for the contribution of all the different subgraphs and can produce a meaningful explanation that humans can interpret. The experiments that we performed both on real and synthetic datasets show that our framework is successful in explaining the decision process of an SGNN on graph classification tasks.

Canonisation and Definability for Graphs of Bounded Rank Width

We prove that the combinatorial Weisfeiler-Leman algorithm of dimension (3 k +4) is a complete isomorphism test for the class of all graphs of rank width at most k . Rank width is a graph invariant that, similarly to tree width, measures the width of a certain style of hierarchical decomposition of graphs; it is equivalent to clique width. It was known that isomorphism of graphs of rank width k is decidable in polynomial time (Grohe and Schweitzer, FOCS 2015), but the best previously known algorithm has a running time n f(k) for a non-elementary function f . Our result yields an isomorphism test for graphs of rank width k running in time n O(k) . Another consequence of our result is the first polynomial-time canonisation algorithm for graphs of bounded rank width. Our second main result is that fixed-point logic with counting captures polynomial time on all graph classes of bounded rank width.

Improving Graph Neural Network Expressivity via Subgraph Isomorphism Counting.

While Graph Neural Networks (GNNs) have achieved remarkable results in a variety of applications, recent studies exposed important shortcomings in their ability to capture the structure of the underlying graph. It has been shown that the expressive power of standard GNNs is bounded by the Weisfeiler-Leman (WL) graph isomorphism test, from which they inherit proven limitations such as the inability to detect and count graph substructures. On the other hand, there is significant empirical evidence, e.g. in network science and bioinformatics, that substructures are often intimately related to downstream tasks. To this end, we propose "Graph Substructure Networks" (GSN), a topologically-aware message passing scheme based on substructure encoding. We theoretically analyse the expressive power of our architecture, showing that it is strictly more expressive than the WL test, and provide sufficient conditions for universality. Importantly, we do not attempt to adhere to the WL hierarchy; this allows us to retain multiple attractive properties of standard GNNs such as locality and linear network complexity, while being able to disambiguate even hard instances of graph isomorphism. We perform an extensive experimental evaluation on graph classification and regression tasks and obtain state-of-the-art results in diverse real-world settings including molecular graphs and social networks.