Graph Neural Network-based framework for predicting top Betweenness Centrality Nodes across Unseen Graphs

Measuring node importance in complex networks has great theoretical and practical significance for network stability and robustness. A variety of network centrality criteria have been presented to address this problem, but each of them focuses only on certain aspects and results in loss of information. Therefore, this paper proposes a relatively comprehensive and effective method to evaluate node importance in complex networks using a multicriteria decision‐making method. This method not only takes into account degree centrality, closeness centrality, and betweenness centrality, but also uses an entropy weighting method to calculate the weight of each criterion, which can overcome the influence of the subjective factor. To illustrate the effectiveness and feasibility of the proposed method, four experiments were conducted to rank node importance on four real networks. The experimental results showed that the proposed method can rank node importance more comprehensively and accurately than a single centrality criterion.

In recent years, the identification of the essential nodes in complex networks has attracted significant attention because of their theoretical and practical significance in many applications, such as preventing and controlling epidemic diseases and discovering essential proteins. Several importance measures have been proposed from diverse perspectives to identify crucial nodes more accurately. In this paper, we propose a novel importance metric called node propagation entropy, which uses a combination of the clustering coefficients of nodes and the influence of the first- and second-order neighbor numbers on node importance to identify essential nodes from an entropy perspective while considering the local and global information of the network. Furthermore, the susceptible–infected–removed and susceptible–infected–removed–susceptible epidemic models along with the Kendall coefficient are used to reveal the relevant correlations among the various importance measures. The results of experiments conducted on several real networks from different domains show that the proposed metric is more accurate and stable in identifying significant nodes than many existing techniques, including degree centrality, betweenness centrality, closeness centrality, eigenvector centrality, and H-index.

Accurate measurement of important nodes in complex networks has great practical and theoretical significance. Mining important nodes should not only consider the core nodes, but also take into account the locations of the nodes in the network. Despite much research on assessing important nodes, the importance of nodes in the structural holes is still easily ignored. Therefore, a local measuring method is proposed, which evaluates the nodes importance by the total constraints caused by the lack of primary structural holes and secondary structural holes around the nodes. This method simultaneously considers both the centrality and the bridging property of the nodes' first-order and second-order neighbours. Further to prove the accuracy of the method TCM, we carry out deliberate attack simulations through selective deletion in a certain proportion of network nodes. Then, we calculate the decreased ratio of network efficiency in the before-and-after attacks. Experiment results show that the average effect of the TCM in four real networks are improved by 50.64% and 14.92% compared to the clustering coefficient index and the k-shell decomposition method, respectively. Obviously, the TCM is more accurate to mine important nodes than other two methods, and it is suitable for quantitative analysis in large-scale networks.

Identifying influential nodes in complex networks: Effective distance gravity model

Considering the high computational complexity of Closeness Centrality and the structural characteristics of community in complex networks, The BridgeRank (BR) algorithm to evaluate node importance of complex networks based on community division is proposed. The complex network is divided into several communities. The node with the largest local betweenness centrality in each community is extracted as the keynode of the community. The importance ranking of nodes is obtained by adopting the BR algorithm to caculate the sum of the shortest paths from each node to these key nodes in the network. The BR algorithm is achieved based on the Kernighan-Lin, Girvan-Newman, Newman fast and Louvain algorithm respectively. The experimental shows that BR algorithm based on Louvain community division has the highest accuracy in estimating node importance. On the basis that Community Size BridgeRank (CS-BR) and the Local Betweenness Centrality BridgeRank (LBC-BR) are proposed based on the size of the community and the weight of the local betweenness centrality of key nodes respectively. The experiments results on four detasets show that compared with some classic algorithms, the important nodes evaluated by the three proposed methods have stronger spread ability in SIR model, Moreover,the improved CS-BR and LBC-BR algorithm are better than BR.

Importance evaluation of power network nodes based on community division and characteristics of coupled network

Node influence ranking in complex networks: A local structure entropy approach

Group centralization of network indices

Graph neural networks (GNNs) learn representations from network data with naturally distributed architectures, rendering them well-suited candidates for decentralized learning. Oftentimes, this decentralized graph support changes with time due to link failures or topology variations. These changes create a mismatch between the graphs on which GNNs were trained and the ones on which they are tested. Online learning can be used to retrain GNNs at testing time, overcoming this issue. However, most online algorithms are centralized and work on convex problems (which GNNs rarely lead to). This paper proposes the Wide and Deep GNN (WD-GNN), a novel architecture that can be easily updated with distributed online learning mechanisms. The WD-GNN comprises two components: the wide part is a bank of linear graph filters and the deep part is a GNN. At training time, the joint architecture learns a nonlinear representation from data. At testing time, the deep part (nonlinear) is left unchanged, while the wide part is retrained online, leading to a convex problem. We derive convergence guarantees for this online retraining procedure and further propose a decentralized alternative. Experiments on the robot swarm control for flocking corroborate theory and show potential of the proposed architecture for distributed online learning.

Graph neural networks (GNNs) are naturally distributed architectures for\nlearning representations from network data. This renders them suitable\ncandidates for decentralized tasks. In these scenarios, the underlying graph\noften changes with time due to link failures or topology variations, creating a\nmismatch between the graphs on which GNNs were trained and the ones on which\nthey are tested. Online learning can be leveraged to retrain GNNs at testing\ntime to overcome this issue. However, most online algorithms are centralized\nand usually offer guarantees only on convex problems, which GNNs rarely lead\nto. This paper develops the Wide and Deep GNN (WD-GNN), a novel architecture\nthat can be updated with distributed online learning mechanisms. The WD-GNN\nconsists of two components: the wide part is a linear graph filter and the deep\npart is a nonlinear GNN. At training time, the joint wide and deep architecture\nlearns nonlinear representations from data. At testing time, the wide, linear\npart is retrained, while the deep, nonlinear one remains fixed. This often\nleads to a convex formulation. We further propose a distributed online learning\nalgorithm that can be implemented in a decentralized setting. We also show the\nstability of the WD-GNN to changes of the underlying graph and analyze the\nconvergence of the proposed online learning procedure. Experiments on movie\nrecommendation, source localization and robot swarm control corroborate\ntheoretical findings and show the potential of the WD-GNN for distributed\nonline learning.\n

Molecular machine learning (ML) has proven important for tackling various molecular problems, such as predicting molecular properties based on molecular descriptors or fingerprints. Since relatively recently, graph neural network (GNN) algorithms have been implemented for molecular ML, showing comparable or superior performance to descriptor or fingerprint-based approaches. Although various tools and packages exist to apply GNNs in molecular ML, a new GNN package, named MolGraph, was developed in this work with the motivation to create GNN model pipelines highly compatible with the TensorFlow and Keras application programming interface (API). MolGraph also implements amodule to accommodate the generation of small molecular graphs, which can be passed to a GNN algorithm to solve a molecular ML problem. To validate the GNNs, benchmarking was conducted using the datasets from MoleculeNet, as well as three chromatographic retention time datasets. The benchmarking results demonstrate that the GNNs performed in line with expectations. Additionally, the GNNs proved useful for molecular identification and improved interpretability of chromatographic retention time data. MolGraph is available at https://github.com/akensert/molgraph . Installation, tutorials and implementation details can be found at https://molgraph.readthedocs.io/en/latest/ .

Calculating centrality measures has been used to identify Cyber-physical System critical component for some time. However, accurate identification of critical elements within the interconnected system that are vulnerable to system failures and intentional attacks requires not only a reliable model but robust index to dynamic system operating environment. These measures include 1) Node Degree, 2) Node Importance, 3) Betweenness Centrality, 4) Closeness Centrality and 5) Eigenvector Centrality. System component criticalities have been evaluated based on these measures in this paper and their performances and rankings under different network topologies and operating conditions have been compared using 1) Pearson Correlation Coefficient and 2) Spearman Correlation Coefficient. This has enabled the quantification of the optimality of using different centrality measures in different network scenarios. The results suggest that Betweenness and Degree centralities are the most suitable indices for identifying critical electrical power system bus and cyber system node individually. Moreover, the degree of correlation between directed measures has suggested also the necessity of using directional model in Cyber-physical System studies.

Finding important nodes in complex networks is an important topic. However, the location information obtained by many previous studies is not sufficient and effective, and the types of attributes applied also have limitations. Based on K-shell and gravity model, this paper proposes a node importance measurement method based on multi-attribute fusion. In this method, the objective, comprehensive evaluation of multiple attributes is obtained by the entropy weight method. Experiments on real networks show that the proposed algorithm can effectively measure the importance of nodes.

Identifying important nodes in complex networks is a fundamental problem in network analysis. Although a plethora of measures has been proposed to identify important nodes in static (i.e., time-invariant) networks, there is a lack of tools in the context of temporal networks (i.e., networks whose connectivity dynamically changes over time). The aim of this paper is to propose a system-theoretic approach for identifying important nodes in temporal networks. In this direction, we first propose a generalization of the popular Katz centrality measure to the family of Markovian temporal networks using tools from the theory of Markov jump linear systems. We then show that Katz centrality in Markovian temporal networks can be efficiently computed using linear programming. Finally, we propose a convex program for optimizing the Katz centrality of a given node by tuning the weights of the temporal network in a cost-efficient manner. Numerical simulations illustrate the effectiveness of the obtained results.

Efficient algorithms for game-theoretic betweenness centrality