Over-smoothing is a persistent challenge in Graph Neural Networks (GNNs), where node embeddings become indistinguishable as network depth increases, fundamentally limiting their effectiveness on tasks requiring fine-grained distinctions. This issue arises from the reliance on diffusion-based propagation mechanisms, which suppress high-frequency information essential for preserving feature diversity. To mitigate this, we propose a wave-driven GNN framework that redefines feature propagation through the wave equation. Unlike diffusion, the wave equation incorporates second-order dynamics, balancing smoothing with oscillatory behavior to retain high-frequency components while ensuring effective information flow. To enhance the stability and convergence of wave equation discretization on graphs, an energy-based mechanism inspired by kinetic and potential energy dynamics is introduced, balancing temporal evolution and structural alignment to stabilize propagation. Extensive experiments on benchmark datasets, including Cora, Citeseer, and PubMed, as well as real-world graphs, demonstrate that the proposed framework achieves state-of-the-art performance, effectively mitigating over-smoothing and enabling deeper, more expressive architectures. The code is available at https://github.com/rene0329/EWGNN/.

The security of Graph Neural Networks (GNNs) is crucial for ensuring the reliability and protection of the systems they are integrated within real-world applications. However, current approaches lack the ability to prevent GNNs from learning high-risk information, including edges, nodes, convolutions, etc. In this paper, we propose a secure GNN learning framework called Reinforcement Learning-based Secure Training Algorithm . We first introduce a model conversion technique that transforms the training process of GNNs into a verifiable Markov Decision Process model. To maintain the security of model we employ Deep Q-Learning algorithm to prevent high-risk information messages. Additionally, to verify whether the strategy derived from Deep Q-Learning algorithm meets safety requirements, we design a model transformation algorithm that converts MDPs into probabilistic verification models, thereby ensuring our method’s security through formal verification tools. The effectiveness and feasibility of our proposed method are demonstrated by achieving a 6.4% improvement in average accuracy on open-source datasets under adversarial attack graphs.

Polynomial-based learnable spectral graph neural networks (GNNs) utilize polynomial to approximate graph convolutions and have achieved impressive performance on graphs. Nevertheless, there are three progressive problems to be solved. Some models use polynomials with better approximation for approximating filters, yet perform worse on real-world graphs. Carefully crafted graph learning methods, sophisticated polynomial approximations, and refined coefficient constraints leaded to overfitting, which diminishes the generalization of the models. How to design a model that retains the ability of polynomial-based spectral GNNs to approximate filters while it possesses higher generalization and performance? In this paper, we propose a spectral GNN with triple filter ensemble (TFE-GNN), which extracts homophily and heterophily from graphs with different levels of homophily adaptively while utilizing the initial features. Specifically, the first and second ensembles are combinations of a set of base low-pass and high-pass filters, respectively, after which the third ensemble combines them with two learnable coefficients and yield a graph convolution (TFE-Conv). Theoretical analysis shows that the approximation ability of TFE-GNN is consistent with that of ChebNet under certain conditions, namely it can learn arbitrary filters. TFE-GNN can be viewed as a reasonable combination of two unfolded and integrated excellent spectral GNNs, which motivates it to perform well. Experiments show that TFE-GNN achieves high generalization and new state-of-the-art performance on various real-world datasets.

Unbounded Petri nets (UPNs) can describe and analyze discrete event systems with infinite states (DESIS). Due to the infinite state space and the combination explosion problem, the reachability analysis of UPNs is an NP-Hard problem. The existing reachability analysis methods cannot achieve an accurate result at reasonable costs (computational time and space) due to the finite reachability tree with$\omega$-numbers. Based on the idea of approximating infinite space with finite states, given some limited reachable markings of a UPN, we propose a method that can quantitatively solve the UPN’s reachability problem with machine learning. Firstly, we define the probabilistic reachability of markings and transform the UPN’s reachability problem into the prediction problem of markings. The proposed method based on positive and unlabeled learning (PUL) and bagging trains a classifier to predict the probabilistic reachability of unknown markings. Finally, to predict the markings outside the positive sample set and unlabeled sample set, an iterative strategy is designed to update the classifier. Based on seven general UPNs, the results of the experiments show that the proposed method has a good performance in the accuracy and time consumption for the UPN’s reachability problem.Note to Practitioners—In discrete event systems, the reachability problem mainly studies reachable states of the system and the relationship between states, which is the basis of the system’s states, behaviors, attributes and performance analysis. For discrete event systems with infinite states, it is hard to analyze the reachable relationship between states within a finite time due to the infinite state space and the combination explosion problem. The main motivation of the paper is to propose a method that can predict the reachable relationship between the states with a probability value within a finite time. By machine learning algorithms, the method learns the feature information of the known reachable states. The reachability of unknown states in the infinite state space can be predicted approximately. The proposed approximation method can be applied to analyze the reachability properties of general discrete event systems with infinite states, such as checking whether a fault occurs in operating systems, whether a message is delivered in communication and so on.

The liveness of Petri nets (PNs) means that every event can occur in any state, establishing a close relationship with the deadlock-free property of existing systems. Due to the problem of state space explosion and the infinite state space of unbounded PNs (UPNs), the time complexity and space complexity of the liveness decision are difficult to give accurate measures; at least, they are both NP-hard. Except for some particular subclasses of UPNs, there has not been an accurate method to decide the liveness of generalized UPNs. Thus, a liveness decision method from a machine learning perspective is proposed to predict probability values about UPNs’ liveness within a finite time. The method aims to learn the feature information on UPNs by deep neural networks and establish the mapping relationship between the UPNs and the liveness. First, the concept of approximating infinite space with finite states is applied to generate reachability graphs at different moments, following the firing rules of a UPN. Then, the graph convolutional network (GCN)-based reachability graph feature representation module and the gated recurrent unit (GRU)-based UPN feature representation module are designed to map the reachability graphs at different moments into the low-dimensional feature space. And the feature vector that can characterize the UPN’s liveness is obtained to decide the liveness probabilistically. Finally, three datasets, including 50000 samples, are constructed. Based on these datasets and some case studies, the experimental results validate the method’s ability to make liveness decisions for UPNs, demonstrating its strong performance in terms of effectiveness and generalization.

Graph neural networks, which generalize deep learning to graph-structured data, have achieved significant improvements in numerous graph-related tasks. Petri nets (PNs), on the other hand, are mainly used for the modeling and analysis of various event-driven systems from the perspective of prior knowledge, mechanisms, and tasks. Compared with graph data, net data can simulate the dynamic behavioral features of systems and are more suitable for representing real-world problems. However, the problem of large-scale data analysis has been puzzling the PN field for decades, and thus, limited its universal applicability. In this article, a framework of net learning (NL) is proposed. NL contains the advantages of PN modeling and analysis with the advantages of graph learning computation. Then, two kinds of NL algorithms are designed for performance analysis of stochastic PNs, and more specifically, the hidden feature information of the PN is obtained by mapping net information to the low-dimensional feature space. Experiments demonstrate the effectiveness of the proposed model and algorithms on the performance analysis of stochastic PNs.

The existing Stochastic Petri Net (SPN) analysis methods are based on a series of steps, including generating the reachable graph, solving the state equation, etc. Unfortunately, these methods cannot perform performance analysis when the state equation has no unique solution. The end-to-end deep learning methods can build a mapping relationship from SPN to performance indicators, which avoids solving the state equation. However, there is a lack of benchmark datasets for SPN learning and training. This paper proposes an automatic generation method of SPN datasets, including SPN random generation, data labeling, data enhancement, and filtering. To relieve the local aggregation problem of random-based organization, a grid-based data organization method is proposed to ensure the diversity of the datasets. In the experimental section, the generated datasets are trained and tested on three types of neural networks. The results show that the generated benchmark datasets are useful, and the increase of dataset size will significantly improve the learning performance.

Summary Information systems have been widely used to realize business processes. The business processes can be described by formal or semi‐formal models. In practice, there are a number of realistic processes involving loop semantics. As an insufficient design or improvement of the information system, event logs generated from these systems may not be consistent with the existing models. By comparing an existing process model with the event logs, we can detect inconsistencies called deviations and verify and extend the business process model. In this paper, some abnormal activities in business processes are formally defined based on Petri nets. An efficient approach to detect deviations between the process model with loop structures and event logs is proposed. A clinical process in a healthcare information system is used as a case study to illustrate our work. Experimental results show the effectiveness and efficiency of the presented approach.