Graph Isomorphism Research Articles

Multimodal Representation Learning via Graph Isomorphism Network for Toxicity Multitask Learning.

Toxicity is paramount for comprehending compound properties, particularly in the early stages of drug design. Due to the diversity and complexity of toxic effects, it became a challenge to compute compound toxicity tasks. To address this issue, we propose a multimodal representation learning model, termed multimodal graph isomorphism network (MMGIN), to address this challenge for compound toxicity multitask learning. Based on fingerprints and molecular graphs of compounds, our MMGIN model incorporates a multimodal representation learning model to acquire a comprehensive compound representation. This model adopts a two-channel structure to independently learn fingerprint representation and molecular graph representation. Subsequently, two feedforward neural networks utilize the learned multimodal compound representation to perform multitask learning, encompassing compound toxicity classification and multiple compound category classification simultaneously. To test the effectiveness of our model, we constructed a novel data set, termed the compound toxicity multitask learning (CTMTL) data set, derived from the TOXRIC data set. We compare our MMGIN model with other representative machine learning and deep learning models on the CTMTL and Tox21 data sets. The experimental results demonstrate significant advancements achieved by our MMGIN model. Furthermore, the ablation study underscores the effectiveness of the introduced fingerprints, molecular graphs, the multimodal representation learning model, and the multitask learning model, showcasing the model's superior predictive capability and robustness.

MVD-HG: multigranularity smart contract vulnerability detection method based on heterogeneous graphs

Smart contracts have significant losses due to various types of vulnerabilities. However, traditional vulnerability detection methods rely extensively on expert rules, resulting in low detection accuracy and poor adaptability to novel attacks. To address these problems, in this paper, deep learning methods are combined with smart contract vulnerability code detection approaches. syntax trees (ASTs), which are special isomorphic graph structures, are an important bridge between source code and graph neural networks. By learning the AST, the model can understand the semantics of the source code. Moreover, graph neural networks have an increasing ability to address complex heterogeneous graphs. Therefore, control flow graphs are fused with data flow graphs on the basis of the ASTs to build heterogeneous graphs with richer code semantics. Furthermore, multigranularity analysis of the vulnerability detection results is performed, including coarse-grained contract-level vulnerability detection and fine-grained line-level vulnerability detection. Through this multigranularity detection approach, vulnerabilities in contracts can be identified and analysed more comprehensively, providing a richer perspective and more solutions for vulnerability detection. The experimental results show that the proposed multigranularity vulnerability detection method based on heterogeneous graphs (MVD-HG) improves both the accuracy and range of the detected vulnerability types in contract-level vulnerability detection tasks; moreover, in the line-level vulnerability detection task, the MVD-HG model achieves significant results and addresses the shortcomings of existing methods. In addition, based on code generation methods used in related fields, a data enhancement method based on the source code is developed, which effectively expands the experimental dataset to address the reduced credibility of the results due to insufficient amounts of data.

State of the Art and Potentialities of Graph-level Learning

Graphs have a superior ability to represent relational data, such as chemical compounds, proteins, and social networks. Hence, graph-level learning, which takes a set of graphs as input, has been applied to many tasks, including comparison, regression, classification, and more. Traditional approaches to learning a set of graphs heavily rely on hand-crafted features, such as substructures. While these methods benefit from good interpretability, they often suffer from computational bottlenecks, as they cannot skirt the graph isomorphism problem. Conversely, deep learning has helped graph-level learning adapt to the growing scale of graphs by extracting features automatically and encoding graphs into low-dimensional representations. As a result, these deep graph learning methods have been responsible for many successes. Yet, no comprehensive survey reviews graph-level learning starting with traditional learning and moving through to the deep learning approaches. This article fills this gap and frames the representative algorithms into a systematic taxonomy covering traditional learning, graph-level deep neural networks, graph-level graph neural networks, and graph pooling. In addition, the evolution and interaction between methods from these four branches within their developments are examined to provide an in-depth analysis. This is followed by a brief review of the benchmark datasets, evaluation metrics, and common downstream applications. Finally, the survey concludes with an in-depth discussion of 12 current and future directions in this booming field.

Exploiting Neighbor Effect: Conv-Agnostic GNN Framework for Graphs With Heterophily.

Due to the homophily assumption in graph convolution networks (GCNs), a common consensus in the graph node classification task is that graph neural networks (GNNs) perform well on homophilic graphs but may fail on heterophilic graphs with many interclass edges. However, the previous interclass edges' perspective and related homo-ratio metrics cannot well explain the GNNs' performance under some heterophilic datasets, which implies that not all the interclass edges are harmful to GNNs. In this work, we propose a new metric based on the von Neumann entropy to reexamine the heterophily problem of GNNs and investigate the feature aggregation of interclass edges from an entire neighbor identifiable perspective. Moreover, we propose a simple yet effective Conv-Agnostic GNN framework (CAGNNs) to enhance the performance of most GNNs on the heterophily datasets by learning the neighbor effect for each node. Specifically, we first decouple the feature of each node into the discriminative feature for downstream tasks and the aggregation feature for graph convolution (GC). Then, we propose a shared mixer module to adaptively evaluate the neighbor effect of each node to incorporate the neighbor information. The proposed framework can be regarded as a plug-in component and is compatible with most GNNs. The experimental results over nine well-known benchmark datasets indicate that our framework can significantly improve performance, especially for the heterophily graphs. The average performance gain is 9.81%, 25.81%, and 20.61% compared with graph isomorphism network (GIN), graph attention network (GAT), and GCN, respectively. Extensive ablation studies and robustness analysis further verify the effectiveness, robustness, and interpretability of our framework. Code is available at https://github.com/JC-202/CAGNN.

Trust enhanced POI recommendation with collaborative learning

With the rapid development and popularization of smart mobile devices, users tend to share their visited points-of-interest (POIs) on the network with attached location information, which forms a location-based social network (LBSN). LBSNs contain a wealth of valuable information, including the geographical coordinates of POIs and the social connections among users. Nowadays, lots of trust-enhanced approaches have fused the trust relationships of users together with other auxiliary information to provide more accurate recommendations. However, in the traditional trust-aware approaches, the embedding processes of the information on different graphs with different properties (e.g., user-user graph is an isomorphic graph, user-POI graph is a heterogeneous graph) are independent of each other and different embedding information is directly fused together without guidance, which limits their performance. More effective information fusion strategies are needed to improve the performance of trust-enhanced recommendation. To this end, we propose a Trust Enhanced POI recommendation approach with Collaborative Learning (TECL) to merge geographic information and social influence. Our proposed model integrates two modules, a GAT-based graph autoencoder as trust relationships embedding module and a multi-layer deep neural network as a user-POI graph learning module. By applying collaborative learning strategy, these two modules can interact with each other. The trust embedding module can guide the selection of user’s potential features, and in turn the user-POI graph learning module enhances the embedding process of trust relationships. Different information is fused through the two-way interaction of information, instead of travelling in one direction. Extensive experiments are conducted using real-world datasets, and results illustrate that our suggested approach outperforms state-of-the-art methods.

Two-Matchings with Respect to the General Sum-Connectivity Index of Trees

A vertex-degree-based topological index φ associates a real number to a graph G which is invariant under graph isomorphism. It is defined in terms of the degrees of the vertices of G and plays an important role in chemical graph theory, especially in QSPR/QSAR investigations. A subset of k edges in G with no common vertices is called a k-matching of G, and the number of such subsets is denoted by mG,k. Recently, this number was naturally extended to weighted graphs, where the weight function is induced by the topological index φ. This number was denoted by mkG,φ and called the k-matchings of G with respect to the topological index φ. It turns out that m1G,φ=φG, and so for k≥2, the k-matching numbers mkG,φ can be viewed as kth order topological indices which involve both the topological index φ and the k-matching numbers. In this work, we solve the extremal value problem for the number of 2-matchings with respect to general sum-connectivity indices SCα, over the set Tn of trees with n vertices, when α is a real number in the interval −1,0.

DGCL: dual-graph neural networks contrastive learning for molecular property prediction.

In this paper, we propose DGCL, a dual-graph neural networks (GNNs)-based contrastive learning (CL) integrated with mixed molecular fingerprints (MFPs) for molecular property prediction. The DGCL-MFP method contains two stages. In the first pretraining stage, we utilize two different GNNs as encoders to construct CL, rather than using the method of generating enhanced graphs as before. Precisely, DGCL aggregates and enhances features of the same molecule by the Graph Isomorphism Network and the Graph Attention Network, with representations extracted from the same molecule serving as positive samples, and others marked as negative ones. In the downstream tasks training stage, features extracted from the two above pretrained graph networks and the meticulously selected MFPs are concated together to predict molecular properties. Our experiments show that DGCL enhances the performance of existing GNNs by achieving or surpassing the state-of-the-art self-supervised learning models on multiple benchmark datasets. Specifically, DGCL increases the average performance of classification tasks by 3.73$\%$ and improves the performance of regression task Lipo by 0.126. Through ablation studies, we validate the impact of network fusion strategies and MFPs on model performance. In addition, DGCL's predictive performance is further enhanced by weighting different molecular features based on the Extended Connectivity Fingerprint. The code and datasets of DGCL will be made publicly available.

Portable Network Resolving Huge-Graph Isomorphism Problem

Abstract The graph isomorphism, as a key task in graph data analysis, is of great significance for the understanding, feature extraction, and pattern recognition of graph data. The best performance of traditional methods is the quasi-polynomial time complexity, which is infeasible for huge graphs. This paper aims to propose a lightweight graph network to resolve the problem of isomorphism in huge graphs. We propose a partitioning algorithm with linear time complexity based on isomorphic necessary condition to handle network graphs that cannot be fully computed by a single computer. We use anti-weight convolution analysis and skip connections to obtain a more stable representation with fewer layers and parameters. We implement simulations on personal computer (PC) with graphs consisting of hundred millions of edges, demonstrating the identification performance ($>98\%$) and time efficiency.

Eccentric connectivity index in complementary prisms

The topological index is just one of several very useful tools that graph theory has made available to chemists. Topological indices are invariants of real numbers under graph isomorphisms. Several topological indices have been defined. Some of them are used to model chemical, pharmaceutical and other properties of molecules. The eccentric connectivity index (eci) is also a topological index. The eci of [Formula: see text], denoted by [Formula: see text], is defined as [Formula: see text], where [Formula: see text] represents the degree of a vertex [Formula: see text] and [Formula: see text] is its eccentricity. In this paper, exact formula for the eci of complementary prisms is derived.

Partial Imaginary Transition State (ITS) Graphs: A Formal Framework for Research and Analysis of Atom-to-Atom Maps of Unbalanced Chemical Reactions and Their Completions

Atom-to-atom maps (AAMs) are bijections that establish the correspondence of reactant and product atoms across chemical reactions. They capture crucial features of the reaction mechanism and thus play a central role in modeling chemistry at the level of graph transformations. AAMs are equivalent to so-called “imaginary transition state” (ITS) graphs, making it possible to reduce tasks such as the computational comparison of AAMs to testing graph isomorphisms. In many application scenarios, nonetheless, only partial information is available, i.e., only partial maps or, equivalently, only subgraphs of the ITS graphs, are known. Here, we investigate whether and how, and to what extent, such partial chemical data can be completed and compared. The focus of this contribution is entirely on the development of a solid mathematical foundation for the analysis of partial AAMs and their associated partial ITS graphs.

GIAE-DTI: Predicting Drug-Target Interactions Based on Heterogeneous Network and GIN-based Graph Autoencoder.

Accurate prediction of drug-target interactions (DTIs) is essential for advancing drug discovery and repurposing. However, the sparsity of DTI data limits the effectiveness of existing computational methods, which primarily focus on sparse DTI networks and have poor performance in aggregating information from neighboring nodes and representing isolated nodes within the network. In this study, we propose a novel deep learning framework, named GIAE-DTI, which considers cross-modal similarity of drugs and targets and constructs a heterogeneous network for DTI prediction. Firstly, the model calculates the cross-modal similarity of drugs and proteins from the relationships among drugs, proteins, diseases, and side effects, and performs similarity integration by taking the average. Then, a drug-target heterogeneous network is constructed, including drug-drug interactions, protein-protein interactions, and drug-target interactions processed by weighted K nearest known neighbors. In the heterogeneous network, a graph autoencoder based on a graph isomorphism network is employed for feature extraction, while a dual decoder is utilized to achieve better self-supervised learning, resulting in latent feature representations for drugs and targets. Finally, a deep neural network is employed to predict DTIs. The experimental results indicate that on the benchmark dataset, GIAE-DTI achieves AUC and AUPR scores of 0.9533 and 0.9619, respectively, in DTI prediction, outperforming the current state-of-the-art methods. Additionally, case studies on four 5-hydroxytryptamine receptor-related targets and five drugs related to mental diseases show the great potential of the proposed method in practical applications.

Planar #CSP Equality Corresponds to Quantum Isomorphism – A Holant Viewpoint

Recently, Mančinska and Roberson proved that two graphs G and \(G^{\prime }\) are quantum isomorphic if and only if they admit the same number of homomorphisms from all planar graphs. We extend this result to planar #CSP with any pair of sets \(\mathcal {F}\) and \(\mathcal {F}^{\prime }\) of real-valued, arbitrary-arity constraint functions. Graph homomorphism is the special case, in which each of \(\mathcal {F}\) and \(\mathcal {F}^{\prime }\) contains a single symmetric 0-1-valued binary constraint function. Our treatment uses the framework of planar Holant problems. To prove that quantum isomorphic constraint function sets give the same value on any planar #CSP instance, we apply a novel form of holographic transformation of Valiant, using the quantum permutation matrix \(\mathcal {U}\) defining the quantum isomorphism. Due to the noncommutativity of \(\mathcal {U}\) ’s entries, it turns out that this form of holographic transformation is only applicable to planar Holant. To prove the converse, we introduce the quantum automorphism group \(Qut(\mathcal {F})\) of a set \(\mathcal {F}\) of constraint functions/tensors and characterize the intertwiners of \(Qut(\mathcal {F})\) as the signature matrices of planar \(\text{Holant}{\mathcal {F}\,|\,\mathcal {EQ})\) quantum gadgets. Then, we define a new notion of (projective) connectivity for constraint functions and reduce their arity while preserving their inclusion in the original intertwiner space. Finally, to address the challenges posed by generalizing from 0-1 valued to real-valued constraint functions, we adapt a technique of Lovász in the classical setting for isomorphisms of real-weighted graphs to the setting of quantum isomorphisms.

Logical equivalences, homomorphism indistinguishability, and forbidden minors

Two graphs G and H are homomorphism indistinguishable over a graph class F if for all graphs F∈F the number of homomorphisms from F to G is equal to the number of homomorphisms from F to H. Many graph isomorphism relaxations such as (quantum) isomorphism, spectral, and logical equivalences can be characterised as homomorphism indistinguishability relations over certain graph classes.Abstracting from the wealth of such instances, we show in this paper that equivalences w.r.t. any self-complementarity logic admitting a characterisation as homomorphism indistinguishability relation can be characterised by homomorphism indistinguishability over a minor-closed graph class. Self-complementarity is a mild property satisfied by most well-studied logics.Furthermore, we classify all graph classes which are in a sense finite and satisfy the maximality condition of being homomorphism distinguishing closed, i.e. adding any graph to the class strictly refines its homomorphism indistinguishability relation. Thereby, we answer various questions raised by Roberson (2022) on properties of the homomorphism distinguishing closure.

Lasserre Hierarchy for Graph Isomorphism and Homomorphism Indistinguishability

We show that feasibility of the $t^\text{th}$ level of the Lasserre semidefinite programming hierarchy for graph isomorphism can be expressed as a homomorphism indistinguishability relation. In other words, we define a class $\mathcal{L}_t$ of graphs such that graphs $G$ and $H$ are not distinguished by the $t^\text{th}$ level of the Lasserre hierarchy if and only if they admit the same number of homomorphisms from any graph in $\mathcal{L}_t$. By analysing the treewidth of graphs in $\mathcal{L}_t$, we prove that the $3t^\text{th}$ level of Sherali--Adams linear programming hierarchy is as strong as the $t^\text{th}$ level of Lasserre. Moreover, we show that this is best possible in the sense that $3t$ cannot be lowered to $3t-1$ for any $t$. The same result holds for the Lasserre hierarchy with non-negativity constraints, which we similarly characterise in terms of homomorphism indistinguishability over a family $\mathcal{L}_t^+$ of graphs. Additionally, we give characterisations of level-$t$ Lasserre with non-negativity constraints in terms of logical equivalence and via a graph colouring algorithm akin to the Weisfeiler--Leman algorithm. This provides a polynomial time algorithm for determining if two given graphs are distinguished by the $t^\text{th}$ level of the Lasserre hierarchy with non-negativity constraints.

Dynamic graph topology generating mechanism: Framework for feature-level multimodal information fusion applied to lower-limb activity recognition

Lower-limb activity recognition (LLAR), which is driven by wearable sensors, has recently become a popular research topic in human assistive devices. Although multimodal information fusion is crucial for improving the accuracy of LLAR, traditional feature-level fusion strategies cannot directly model the relationship between multimodal features according to the lower-limb activity patterns and they realize fusion based on the relationship. To address these limitations, a dynamic graph topology generation (DGTG) mechanism that uses a graph topology to model the relationship between multimodal features mined by a convolutional neural network (CNN) is proposed in this study. A weighted graph isomorphism network (WGIN) that can realize multimodal feature fusion based on the relationship (i.e., graph topology) is also proposed, resulting in a novel feature-level multimodal information fusion framework called CNN-DGTG-WGIN. This proposed method achieved the best average accuracies of 96.3 and 99.7% for two publicly available datasets. This method integrates the determination of the graph topology into an end-to-end learning process without requiring any prior knowledge or original graph structures. Further, CNN-DGTG-WGIN effectively incorporates both linear and nonlinear correlations between features during the DGTG process. Analysis results revealed that the graph topology (multimodal feature relationship) learned by CNN-DGTG-WGIN was closely related to lower-limb activity patterns, encoding relevant information about these patterns, which is the essence of its superior performance.

Intervals of posets of a zero-divisor graph

Abstract This article is concerned with bounded partially ordered sets P such that for every p ∈ P ∖ {1} there exists q ∈ P ∖ {0} such that 0 is the only lower bound of {p, q}. The posets P such that P ≅ Q if and only if P and Q have isomorphic zero-divisor graphs are completely characterized. In the case of finite posets, this result is generalized by proving that posets with isomorphic zero-divisor graphs form an interval under the partial order given by P ≲ Q if and only if there exists a bijective poset-homomorphism P → Q. In particular, the singleton intervals correspond to the posets that are completely determined by their zero-divisor graphs. These results are obtained by exploring universal and couniversal orderings with respect to posets that have isomorphic zero-divisor graphs.

A Note on Wiener and Hyper-Wiener Indices of Abid-Waheed Graph

A Hosoya polynomial is a polynomial connected to a molecular graph, which is a graph representation of a chemical compound with atoms as vertices and chemical bonds as edges. A graph invariant is the Hosoya polynomial; it is a graph attribute that does not change under graph isomorphism. It provides information about the number of unique non-empty subgraphs in a given graph. A molecular graph's size and branching complexity are determined by a topological metric known as the Wiener index. The Wiener index of each pair of vertices in a molecular network is the sum of those distances. The topological index, one of the various classes of graph invariants, is a real number related to a connected graph's structure .The goal of this article is to compute the Hosoya polynomial of some class of Abid-Waheed graph. Further, this research focused on a C++ algorithm to calculate the wiener index of and . The Wiener index (W∗I) and Hyper-Wiener index (H∗W∗I) are calculated using Hosoya polynomial (H∗-polynomial) of some family of Abid-Waheed graphs and . Illustrations and applications are given to enhance the research work.

Exhaustive Enumeration of Spatial Prime Structures

Prime structures are link chains with 0 DoF (degrees of freedom), not including subchains with 0 or fewer DoF, which are expected to be used in systematic kinematic and dynamic analyses of link mechanisms. This paper describes the exhaustive enumeration of spatial prime structures with three–five links. There will be more types of spatial prime structures than planar prime structures due to the variety in the DoF of kinematic pairs and the existence of prime structures with idle DoF. In the enumeration, the graphs which represent the connection of links and pairs are used. The vertices of the graphs represent the links of structures, and the edges represent the kinematic pairs. To consider the pair DoF, weights are set on edges. First, the numbers of pairs are calculated using Grübler’s equation. Second, the graphs corresponding to structures are enumerated, without considering pair DoF, and the isomorphic graphs and other inappropriate graphs are eliminated. Then, the combinations of pair DoF arrangement are enumerated, and the isomorphic graphs and graphs which correspond to non-prime structures are eliminated. Finally, prime structures with idle DoF are considered. As a result, 3, 13, and 97 kinds of spatial prime structures for 3, 4, and 5 links, respectively, are obtained.

Planar Stick Indices of Some Knotted Graphs

Two isomorphic graphs can have inequivalent spatial embeddings in 3-space. In this way, an isomorphism class of graphs contains many spatial graph types. A common way to measure the complexity of a spatial graph type is to count the minimum number of straight sticks needed for its construction in 3-space. In this paper, we give estimates of this quantity by enumerating stick diagrams in a plane. In particular, we compute the planar stick indices of knotted graphs with low crossing numbers. We also show that if a bouquet graph or a theta-curve has the property that its proper subgraphs are all trivial, then the planar stick index must be at least seven.

Enhancing Major Depressive Disorder Diagnosis With Dynamic-Static Fusion Graph Neural Networks.

Major Depressive Disorder (MDD) is a debilitating, complex mental condition with unclear mechanisms hindering diagnostic progress. Research links MDD to abnormal brain connectivity using functional magnetic resonance imaging (fMRI). Yet, existing fMRI-based MDD models suffer from limitations, including neglecting dynamic network traits, lacking interpretability, and struggling with small datasets. We present DSFGNN, a novel graph neural network framework addressing these issues for improved MDD diagnosis. DSFGNN employs a graph isomorphism encoder to model static and dynamic brain networks, achieving effective fusion of temporal and spatial information through a spatiotemporal attention mechanism, thereby enhancing interpretability. Furthermore, we incorporate a causal disentangling module and orthogonal regularization module to augment the model's expressiveness. We evaluate DSFGNN on the Rest-meta-MDD dataset, yielding superior results compared to the best baseline. Besides, extensive ablation studies and interpretability analysis confirm DSFGNN's effectiveness and potential for biomarker discovery.