Graph Isomorphism Research Articles

ISOMORPH: an efficient application on GPU for detecting graph isomorphism

PurposeThe authors will review the main concepts of graphs, present the implemented algorithm, as well as explain the different techniques applied to the graph, to achieve an efficient execution of the algorithm, both in terms of the use of multiple cores that the authors have available today, and the use of massive data parallelism through the parallelization of the algorithm, bringing the graph closer to the execution through CUDA on GPUs.Design/methodology/approachIn this work, the authors approach the graphs isomorphism problem, approaching this problem from a point of view very little worked during all this time, the application of parallelism and the high-performance computing (HPC) techniques to the detection of isomorphism between graphs.FindingsResults obtained give compelling reasons to ensure that more in-depth studies on the HPC techniques should be applied in these fields, since gains of up to 722x speedup are achieved in the most favorable scenarios, maintaining an average performance speedup of 454x.Originality/valueThe paper is new and original.

Set-independence graphs of vector spaces and partial quasigroups

As a generalization of independence graphs of vector spaces and groups, we introduce the notions of set-independence graphs of vector spaces and partial quasigroups. The former are characterized for finite-dimensional vector spaces over finite fields. Further, we prove that every finite simple graph is isomorphic to either the independence graph of a partial quasigroup or an induced subgraph of the latter. We also prove that isomorphic partial quasigroups give rise to isomorphic set-independence graphs. As an illustrative example, all finite graphs of order $n\leq 5$ are identified with the independence graph of a partial quasigroup of the same order.

Reconstruction from one labelled card and more

The deck D of a graph H is its multiset of one-vertex deleted subgraphs. Ulam's Reconstruction Conjecture claims that every graph having more than two vertices is reconstructible from D. We seek minimal graph parameters that need to accompany one card G to enable the unique reconstruction of H. A card needs to be feasible. We consider specific graph invariants associated with the 0–1 eigenspaces of H, which need to supplement a feasible card, for the unique reconstruction of H. Non–isomorphic graphs sharing a common card turn out to have linearly independent one-dimensional eigenspaces, whereas non–isomorphic graphs, sharing a common eigenspace, have no labelled card in common. Moreover, we determine certain families of graphs for which reconstruction is possible from any card.

GINCM-DTA: A graph isomorphic network with protein contact map representation for potential use against COVID-19 and Omicron subvariants BQ.1, BQ.1.1, XBB.1.5, XBB.1.16

The COVID-19 pandemic’s profound impact on global health and the economy underscores the need for new computational strategies to rapidly identify potential antiviral drugs against emerging and re-emerging infectious diseases. AI-based drug repurposing methods have the potential to significantly shorten the drug discovery process, rendering them a crucial and efficacious approach for screening anti-SARS-CoV-2 compounds. In this paper, we proposed GINCM-DTA, an AI-based graph isomorphism network with protein contact map representation for drug–target affinity (DTA) prediction, and presented the COVID-DTA dataset for SARS-CoV-2-specific drug repurposing. Pre-training GINCM-DTA on large-scale Davis and KIBA datasets achieved state-of-the-art performance, while fine-tuning on the COVID-DTA dataset using transfer learning resulted in a mean square error of 0.026 and concordance index of 0.967. Utilizing the fine-tuned GINCM-DTA model, we predicted DTAs for drugs against SARS-CoV-2 main targets, focusing on key conserved proteins and unmutated spike proteins for virtual screening. We integrated GINCM-DTA predictions with molecular docking simulations to identify five candidates. In vitro assays revealed Hydralazine (5.737 nM) as a potential inhibitor of ACE2 and TMPRSS2, effectively blocking SARS-CoV-2 infection. Moreover, the results highlight the potential of Hydralazine as a broad-spectrum drug candidate, demonstrating its efficacy in obstructing the binding between ACE2 and spike proteins of multiple Omicron variants, including the currently dominant strains BQ.1, BQ.1.1, XBB.1.5, and XBB.1.16. Our method offers a novel computational strategy for COVID-19 antiviral screening and prepares for potential future epidemics incited by emerging viruses.

Using knowledge graphs and contrastive learning for detecting APT Malware on Endpoint systems

Advanced persistent threat (APT) attacking campaigns have been a common method for cyber-attackers to attack and exploit end-user computers (workstations) in recent years. In this study, to enhance the effectiveness of the APT malware detection, a combination of deep graph networks and contrastive learning is proposed. The idea is that several deep graph networks such as Graph Convolution Networks (GCN), Graph Isomorphism Networks (GIN), are combined with some popular contrastive learning models like N-pair Loss, Contrastive Loss, and Triplet Loss, in order to optimize the process of APT malware detection and classification in endpoint workstations. The proposed approach consists of three main phases as follows. First, the behaviors of APT malware are collected and represented as graphs. Second, GIN and GCN networks are used to extract feature vectors from the graphs of APT malware. Finally, different contrastive learning models, i.e. N-pair Loss, Contrastive Loss, and Triplet Loss are applied to determine which feature vectors belong to APT malware, and which ones belong to normal files. This combination of deep graph networks and contrastive learning algorithm is a novel approach, that not only enhances the ability to accurately detect APT malware but also reduces false alarms for normal behaviors. The experimental results demonstrate that the proposed model, whose effectiveness ranges from 88% to 94% across all performance metrics, is not only scientifically effective but also practically significant. Additionally, the results show that the combination of GIN and N-pair Loss performs better than other combined models. This provides a base malware detection system with flexible parameter selection and mathematical model choices for optimal real-world applications.

Graph isomorphism U-Net

Graph embedding learning is a fundamental task when dealing with diverse datasets. While encoder–decoder architectures, such as U-Nets, have shown great success in image pixel-wise prediction tasks, applying similar methods to graph data poses challenges due to the lack of natural pooling and up-sampling operations for graphs. Recent methods leverage learnable parameters to extract structural information from neural networks and extend pooling and unpooling to graphs using node features and graph structural information. This paper proposes a novel model called GIUNet (Graph Isomorphism U-Net) for the graph classification task. The proposed Graph U-Net structure is based on graph isomorphism convolution while using a comprehensive pqPooling layer. The pqPooling layer in our approach effectively combines node features and graph structure information during the graph down-sampling stage. To incorporate graph structure information, we utilize both the spectral representation and node centrality measurements. Node centrality measurements capture various structural aspects of nodes in the graph, while the spectral representation helps us focus on the informative low-frequency components of the graph structure. Through ablation studies, we have demonstrated that leveraging the GIUNet model leads to significant improvements compared to state-of-the-art methods across multiple benchmark datasets.

Algorithms for Efficiently Computing Structural Anonymity in Complex Networks

This article proposes methods for efficiently computing the anonymity of entities in networks. We do so by partitioning nodes into equivalence classes where a node is k -anonymous if it is equivalent to k -1 other nodes. This assessment of anonymity is crucial when one wants to share data and must ensure the anonymity of entities represented is compliant with privacy laws. Additionally, in such an assessment, it is necessary to account for a realistic amount of information in the hands of a possible attacker that attempts to de-anonymize entities in the network. However, measures introduced in earlier work often assume a fixed amount of attacker knowledge. Therefore, in this work, we use a new parameterized measure for anonymity called d - k -anonymity. This measure can be used to model the scenario where an attacker has perfect knowledge of a node’s surroundings up to a given distance d . This poses nontrivial computational challenges, as naive approaches would employ large numbers of possibly computationally expensive graph isomorphism checks. This article proposes novel algorithms that severely reduce this computational burden. In particular, we present an iterative approach, assisted by techniques for preprocessing nodes that are trivially automorphic and heuristics that exploit graph invariants. We evaluate our algorithms on three well-known graph models and a wide range of empirical network datasets. Results show that our approaches significantly speed up the computation by multiple orders of magnitude, which allows one to compute d - k -anonymity for a range of meaningful values of d on large empirical networks with tens of thousands of nodes and over a million edges.

An efficient approach of graph isomorphism identification using loop theory and hopfield neural networks

An efficient approach of graph isomorphism identification using loop theory and hopfield neural networks

Geometric learning for computational mechanics, Part III: Physics-constrained response surface of geometrically nonlinear shells

This paper presents a graph-manifold iterative algorithm to predict the configurations of geometrically exact shells subjected to external loading. The finite element solutions are first stored in a weighted graph where each graph node stores the nodal displacement and nodal director. This collection of solutions is embedded onto a low-dimensional latent space through a graph isomorphism encoder. This graph embedding step reduces the dimensionality of the nonlinear data and makes it easier for the response surface to be constructed. The decoder, in return, converts an element in the latent space back to a weighted graph that represents a finite element solution. As such, the deformed configuration of the shell can be obtained by decoding the predictions in the latent space without running extra finite element simulations. For engineering applications where the shell is often subjected to concentrated loads or a local portion of the shell structure is of particular interest, we use the solutions stored in a graph to reconstruct a smooth manifold where the balance laws are enforced to control the curvature of the shell. The resultant computer algorithm enjoys both the speed of the nonlinear dimensional reduced solver and the fidelity of the solutions at locations where it matters.

Distributed Optimization for Graph Matching.

Graph matching, or the determination of the vertex correspondences between a pair of graphs, is a crucial task in various problems in different science and engineering disciplines. This article aims to propose a distributed optimization approach for graph matching (GM) between two isomorphic graphs over multiagent networks. For this, we first show that for a class of asymmetric graphs, GM of two isomorphic graphs is equivalent to a convex relaxation where the set of permutation matrices is replaced by the set of pseudostochastic matrices. Then, we formulate GM as a distributed convex optimization problem with equality constraints and a set constraint, over a network of multiple agents. For arbitrary labelings of the vertices, each agent only has information about just one vertex and its neighborhood, and can exchange information with its neighbors. A projected primal-dual gradient method is developed to solve the constrained optimization problem, and globally exponential convergence of the agents' states to the optimal permutation is achieved. Finally, we illustrate the effectiveness of the algorithm through simulation examples.

A Heuristic for Direct Product Graph Decomposition

In this paper we describe a heuristic for decomposing a directed graph into factors according to the direct product (also known as Kronecker, cardinal or tensor product). Given a directed, unweighted graph $G$ with adjacency matrix $\mathbf{Adj}(G)$, our heuristic aims at identifying two graphs $G_1$ and $G_2$ such that $G = G_1 \times G_2$, where $G_1 \times G_2$ is the direct product of $G_1$ and $G_2$. For undirected, connected graphs it has been shown that graph decomposition is "at least as difficult" as graph isomorphism; therefore, polynomial-time algorithms for decomposing a general directed graph into factors are unlikely to exist. Although graph factorization is a problem that has been extensively investigated, the heuristic proposed in this paper represents - to the best of our knowledge - the first computational approach for general directed, unweighted graphs. We have implemented our algorithm using the MATLAB environment; we report on a set of experiments that show that the proposed heuristic solves reasonably-sized instances in a few seconds on general-purpose hardware. Although the proposed heuristic is not guaranteed to find a factorization, even if one exists; however, it always succeeds on all the randomly-generated instances used in the experimental evaluation.

Aα-spectrum of duplicate and corona operations in graphs

Let G be a graph of order n, A(G) its adjacency matrix and D(G) the diagonal matrix of degrees of G. In 2017, for every α in [0, 1], Nikiforov defined the matrix Aα(G) = αD(G) + (1 − α)A(G). In this paper, we investigate the Aα-spectrum of graphs obtained from the duplicate and corona operations. As an application of our results, we provide conditions for the construction of some pairs of non isomorphic Aα-cospectral graphs.

Drug Toxicity Prediction by Machine Learning Approaches

Drug property prediction, especially toxicity, helps reduce risks in a range of real-world applications. In this paper, we aim to apply various machine-learning models for solving the drug toxicity prediction problem. Among various machine-learning approaches, we select five suitable representatives: random forest, multi-layer perceptron, logistic regression, graph convolutional neural network, and graph isomorphism network (GIN) for conducting experiments on six datasets for toxicity prediction, including Tox 21, ClinTox, ToxCast, SIDER, HIV, and BACE. We design the GIN with four hidden layers and select the Adam optimizer with the learning rate [Formula: see text] and the batch size [Formula: see text]. Furthermore, we use a batch norm layer inside each of the GIN hidden layers. Experimental results show that the designed GIN model is most efficient in distinguishing between safe and toxic drugs and outperforms the others under the supervision of ROC AUC score and recall.

Solving Panel Block Assembly Line Scheduling Problem via a Novel Deep Reinforcement Learning Approach

The panel block is a quite important “intermediate product” in the shipbuilding process. However, the assembly efficiency of the panel block assembly line is not high. Therefore, rational scheduling optimization is of great significance for improving shipbuilding efficiency. Currently, the processing sequence of the panel blocks in the panel block assembly line is mainly determined using heuristic and metaheuristic algorithms. However, these algorithms have limitations, such as small problem-solving capacity and low computational efficiency. To address these issues, this study proposes an end-to-end approach based on deep reinforcement learning to solve the scheduling problem of the ship’s panel block assembly line. First, a Markov decision model is established, and a disjunctive graph is creatively used to represent the current scheduling status of the panel block assembly line. Then, a policy function based on a graph isomorphism network is designed to extract information from the disjunctive graph’s state and train it using Proximal Policy Optimization algorithms. To validate the effectiveness of our method, tests on both real shipbuilding data and publicly available benchmark datasets are conducted. We compared our proposed end-to-end deep reinforcement learning algorithm with heuristic algorithms, metaheuristic algorithms, and the unimproved reinforcement learning algorithm. The experimental results demonstrate that our algorithm outperforms other baseline methods in terms of model performance and computation time. Moreover, our model exhibits strong generalization capabilities for larger instances.

Enhancing attack resilience of cyber-physical systems through state dependency graph models

AbstractThis paper presents a method that utilizes graph theory and state modelling algorithms to perform automatic complexity analysis of the architecture of cyber-physical systems (CPS). It describes cyber physical systems risk assessment (CPSRA), a tool to provide automatic decision support for enhancing the overall resilience of CPS architectures often used in critical infrastructures. CPRSA is built to enhance industrial risk assessment and improve the resilience of CPS architecture against malicious attacks on the cyber domain that can affect industrial processes, which is critical in a distributed cyber environment. Such attacks often compromise execution states on physical components and lead to hazards or even disasters through plant malfunction. CPSRA is tested against a real-world testbed model of a large SCADA system that is infused with real-world CVE vulnerabilities in some of its components. The tool creates an isomorphic graph of the CPS process model and uses graph algorithms and network analytics on the model to test cyber-attacks and evaluate attack resilience aspects. The tool’s output is then used to pinpoint high-complexity components in terms of influence on the overall CPS architecture and suggest mitigation points for security measure implementation while considering every potential subattack path and subliminal path on the model’s attack graph. The paper complements standardized assessment reports and contributes to automatic architecture assessment for critical infrastructure environments and can be used as the basis to model dependencies and threat propagation in larger digital twins, a need outlined in major NIST publications concerning the security of industrial systems that was previously done manually, without automatic insight into state and vulnerability influences.

Dynamic Multi-Task Graph Isomorphism Network for Classification of Alzheimer’s Disease

Alzheimer’s disease (AD) is a progressive, irreversible neurodegenerative disorder that requires early diagnosis for timely treatment. Functional magnetic resonance imaging (fMRI) is a non-invasive neuroimaging technique for detecting brain activity. To improve the accuracy of Alzheimer’s disease diagnosis, we propose a new network architecture called Dynamic Multi-Task Graph Isomorphism Network (DMT-GIN). This approach uses fMRI images transformed into brain network structures to classify Alzheimer’s disease more effectively. In the DMT-GIN architecture, we integrate an attention mechanism with the Graph Isomorphism Network (GIN) to capture node features and topological structure information. To further enhance AD classification performance, we incorporate auxiliary tasks of gender and age classification prediction alongside the primary AD classification task in the network. This is achieved through sharing network parameters and adaptive weight adjustments for simultaneous task optimization. Additionally, we introduce a method called GradNorm for dynamically balancing gradient updates between tasks. Evaluation results demonstrate that the DMT-GIN model outperforms existing baseline methods on the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database, leading in various metrics with a prediction accuracy of 90.44%. This indicates that our DMT-GIN model effectively captures brain network features, providing a powerful auxiliary means for the early diagnosis of Alzheimer’s disease.

Graph isomorphism: physical resources, optimization models, and algebraic characterizations

In the (G, H)-isomorphism game, a verifier interacts with two non-communicating players (called provers), by privately sending each of them a random vertex from either G or H. The goal of the players is to convince the verifier that the graphs G and H are isomorphic. In recent work along with Atserias et al. (J Comb Theory Ser B 136:89–328, 2019) we showed that a verifier can be convinced that two non-isomorphic graphs are isomorphic, if the provers are allowed to share quantum resources. In this paper we model classical and quantum graph isomorphism by linear constraints over certain complicated convex cones, which we then relax to a pair of tractable convex models (semidefinite programs). Our main result is a complete algebraic characterization of the corresponding equivalence relations on graphs in terms of appropriate matrix algebras. Our techniques are an interesting mix of algebra, combinatorics, optimization, and quantum information.

A conjecture related to the nilpotency of groups with isomorphic non-commuting graphs

In this work we discuss whether the non-commuting graph of a finite group can determine its nilpotency. More precisely, Abdollahi, Akbari and Maimani conjectured that if G and H are finite groups with isomorphic non-commuting graphs and G is nilpotent, then H must be nilpotent as well (Conjecture 2). We characterize the structure of such an H when G is a finite AC-group, that is, a finite group in which all centralizers of non-central elements are abelian. As an application, we prove Conjecture 2 for finite AC-groups whenever |Z(G)|≥|Z(H)|.

A Comparison of Four Notions of Isomorphism-Based Security for Graphs

A graph is a powerful abstraction for representing information. We address the problem of publishing a secure version of a graph that does not leak information to an adversary who may possess prior information about portions of the graph, and may have unbounded computational power. In this context, we revisit four notions of security, all of which are based on variants of graph isomorphism, that have been proposed in two different application contexts in the literature. We compare the four notions to one another,first from from the standpoint of strength, i.e., whether meeting one notion implies meeting another, and then from the standpoint of computational hardness, i.e., what the exact computational complexity is for the problem of checking whether a graph meets a notion. For the latter, we identify that for two of the notions we consider, the problem is <b>NP</b>-complete, and for the two others, it is <b>ISO</b>-complete, where <b>ISO</b> is the class of problems induced by graph isomorphism. We observe that strength is not necessarily correlated to computational hardness. In summary, our work makes contributions at the foundations of an important notion of security for graphs.

Order Reconfiguration under Width Constraints

In this work, we consider the following order reconfiguration problem: Given a graph $G$ together with linear orders $\omega$ and $\omega'$ of the vertices of $G$, can one transform $\omega$ into $\omega'$ by a sequence of swaps of adjacent elements in such a way that, at each time step, the resulting linear order has cutwidth (pathwidth) at most $k$? We show that this problem always has an affirmative answer when the input linear orders $\omega$ and $\omega'$ have cutwidth (pathwidth) of at most $k/2$. This result also holds in a weighted setting. Using this result, we establish a connection between two apparently unrelated problems: the reachability problem for two-letter string rewriting systems and the graph isomorphism problem for graphs of bounded cutwidth. This opens an avenue for the study of the famous graph isomorphism problem using techniques from term rewriting theory.