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line graph neural networks for link weight prediction

PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS 2025年 661卷

作者： Liang, Jinbi Pu, Cunlai Shu, Xiangbo Xia, Yongxiang Xia, Chengyi Nanjing Univ Sci & Technol Sch Comp Sci & Engn Nanjing 210094 Jiangsu Peoples R China Hangzhou Dianzi Univ Sch Commun Engn Hangzhou 310018 Zhejiang Peoples R China Tiangong Univ Sch Artificial Intelligence Tianjin 300387 Peoples R China

In real-world networks, predicting the weight (strength) of links is as crucial as predicting the existence of the links themselves. Previous studies have primarily used shallow graph features for link weight prediction, limiting the prediction performance. In this paper, we propose a new link weight prediction method, namely line graph Neural Networks for Link Weight Prediction (LGLWP), which learns intrinsic graph features through deep learning. In our method, we first extract the enclosing subgraph around a target link and then employ a weighted graph labeling algorithm to label the subgraph nodes. Next, we transform the subgraph into the line graph and apply graph convolutional neural networks to learn the node embeddings in the line graph, which can represent the links in the original subgraph. Finally, the node embeddings are fed into a fully-connected neural network to predict the weight of the target link, treated as a regression problem. Our method directly learns link features, surpassing previous methods that splice node features for link weight prediction. Experimental results on six network datasets of various sizes and types demonstrate that our method outperforms state-of-the-art methods.

关键词： Complex network Link weight prediction line graph graph neural network

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Extremal Values on the Kirchhoff Index of the line graph of Unicyclic Networks

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CIRCUITS SYSTEMS AND SIGNAL PROCESSING 2025年第3期44卷 1523-1544页

作者： Sardar, Muhammad Shoaib Xu, Shou-Jun Harbin Engn Univ Coll Math Sci Harbin Peoples R China Lanzhou Univ Gansu Ctr Appl Math Sch Math & Stat Lanzhou 730000 Gansu Peoples R China

Let G be a simple connected graph with vertex set V(G) and edge set E(G). Suppose N represents a network derived from G by substituting a 1-ohm resistor for each edge of G. In that case, the resistance between u,v is an element of V(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u, v \in V(G)$$\end{document} is analogous to the resistance between two equivalent nodes in network N. The Kirchhoff index of G is the summation of the resistance distances between all pairs of vertices in G. The line graph LG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_G$$\end{document} of G is a graph whose vertices correspond to the edges of G, and any two vertices of LG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_G$$\end{document} are adjacent if and only if the corresponding edges of G are incident with the same vertex of G. A unicyclic graph is a connected graph containing exactly one cycle. In this paper, we will identify the extremal values and unicyclic graphs for the Kirchhoff index of the line graph of unicyclic graphs by utilizing techniques derived from electrical networks.

关键词： Distance Resistance distance line graph Unicyclic graphs Kirchhoff index 05C12 05C35 05C81

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LGSMOTE-IDS: line graph based Weighted-Distance SMOTE for imbalanced network traffic detection

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EXPERT SYSTEMS WITH APPLICATIONS 2025年 281卷

作者： Zhao, Guyu Li, Linwei He, Hongdou Ren, Jiadong Yanshan Univ Sch Informat Sci & Engn Qinhuangdao 066000 Hebei Peoples R China

The application of graph Neural Networks (GNNs) to Network Intrusion Detection Systems (NIDS) has become a prominent research focus. However, NIDS often struggles to classify minority attack samples due to the severe class imbalance in NIDS datasets, where the number of samples varies significantly across classes. Additionally, prior studies have frequently overlooked the importance of edge features in GNNs. To address these challenges, we propose LGSMOTE-IDS, a novel framework that integrates a L line G graph based Weighted-Distance SMOTE for I Intrusion D Detection S Systems. First, we define the fine-grained protocol service graph (PSG) and transform it into its corresponding protocol service line graph (L(PSG)). This transformation provides a novel perspective for describing network traffic interactions and enables the conversion of the edge classification task into a node classification task. Second, we introduce Weighted-Distance SMOTE, an oversampling algorithm specifically tailored to NIDS datasets, which employs an improved interpolation strategy to generate synthetic minority class samples. Finally, we utilize a GNN-based classifier to predict labels for all samples. We conduct experiments on three widely used datasets-NF-UNSW-NB15, NF-BoT-IoT, and NF-ToN-IoT. LGSMOTE-IDS achieves average increases of 18.11%, 45.91%, and 36.41% in weighted F1-scores for five, one, and three minority classes across the three datasets, respectively, compared to baseline method. Moreover, LGSMOTE-IDS successfully detects attack types that previous models fail to recognize. To the best of our knowledge, LGSMOTE-IDS is the first framework to integrate GNNs with an oversampling algorithm to address the class imbalance issue in NIDS datasets.

关键词： Network Intrusion Detection System Weighted-Distance SMOTE graph Neural Network line graph

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Extremal values on the Kirchhoff index of the line graph of trees

KUWAIT JOURNAL OF SCIENCE

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KUWAIT JOURNAL OF SCIENCE 2025年第1期52卷

作者： Sardar, Muhammad Shoaib Xu, Shou-Jun Pan, Xiang-Feng Lanzhou Univ Gansu Ctr Appl Math Sch Math & Stat Lanzhou 730000 Peoples R China Harbin Engn Univ Coll Math Sci Harbin Peoples R China Anhui Univ Sch Math Sci Hefei 230601 Anhui Peoples R China

The computation of resistance distance and the Kirchhoff index is a classical problem that has been extensively investigated by numerous mathematicians, physicists, and scientists. Consider a simple connected graph G with vertex set V(G) and edge set E(G). If we replace each edge of G with a resistor of 1 ohm resistance, we create an electrical network N;in that case, the distance between any two nodes between network N is called resistance distance. The Kirchhoff index is a mathematical term that quantifies the complexity of a graph;it is defined as the total resistance distance among each pair of vertices in G. The line graph L-G is constructed from G by swapping out the edges for vertices;in L-G, if two vertices share endpoints in G, then they are connected in L-G. In this study, extremal values for the Kirchhoff index of the line graph of trees are calculated. Further, we will also calculate the Kirchhoff index for the line graph of some special trees and establish the relationship between them.

关键词： Distance Resistance distance Kirchhoff index line graph Tree Series-reduced tree

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The decycling number of a line graph

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DISCRETE MATHEMATICS 2025年第2期348卷

作者： Ma, Mingyuan Ren, Han East China Normal Univ Sch Math Sci Shanghai 200241 Peoples R China

关键词： line graph Decycling number Cycle rank

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Multiplicity of Powers of Path Ideals of A line graph

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BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY 2025年第1期48卷 1-16页

作者： Shan, Jiawen Wang, Zexin Lu, Dancheng Soochow Univ Sch Math Sci Suzhou 215006 Peoples R China

Let R=K[x1,& mldr;,xn]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R=\mathbb {K}[x_1,\ldots ,x_n]$$\end{document} be the polynomial ring over a field K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {K}$$\end{document}, and let I subset of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I\subseteq R$$\end{document} be the t-path ideal of the line graph Ln\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_n$$\end{document} with n-vertices. It is shown that the set of associated prime ideals of Is\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I<^>s$$\end{document} is equal to the set of minimal prime ideals of I for all s >= 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\ge 1$$\end{document}, and we provide an explicit description of these prime ideals. Additionally, as the main contribution of this paper, we derive an explicit formula for the multiplicity of R/Is\documentclass[12p

关键词： Path ideals line graph Multiplicity Normally torsion-free

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3-Path Decompositions of the line graph of the Complete Bipartite graph

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JOURNAL OF INTERCONNECTION NETWORKS 2025年第1期25卷

作者： Gao, Limin Yang, Weihua Taiyuan Univ Technol Dept Math Taiyuan 030024 Shanxi Peoples R China

The line graph is a very popular research object in graph theory, in complex networks and also in social networks recently. Let L(K-m,K-n) be the line graph of the complete bipartite graph K-m,K-n and Pk+1 be a path of length k. In this paper, we give necessary and sufficient conditions for the existence of P4-decompositions of L(K-m,K-n).

关键词： Path decomposition complete bipartite graph line graph Cartesian product

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line graph or Scatter Plot? Automatic Selection of Methods for Visualizing Trends in Time Series

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IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER graphICS 2018年第2期24卷 1141-1154页

作者： Wang, Yunhai Han, Fubo Zhu, Lifeng Deussen, Oliver Chen, Baoquan Shandong Univ Jinan 250100 Shandong Peoples R China Univ Konstanz SIAT Shenzhen D-78457 Constance Germany Southeast Univ Nanjing 210096 Jiangsu Peoples R China

line graphs are usually considered to be the best choice for visualizing time series data, whereas sometimes also scatter plots are used for showing main trends. So far there are no guidelines that indicate which of these visualization methods better display trends in time series for a given canvas. Assuming that the main information in a time series is its overall trend, we propose an algorithm that automatically picks the visualization method that reveals this trend best. This is achieved by measuring the visual consistency between the trend curve represented by a LOESS fit and the trend described by a scatter plot or a line graph. To measure the consistency between our algorithm and user choices, we performed an empirical study with a series of controlled experiments that show a large correspondence. In a factor analysis we furthermore demonstrate that various visual and data factors have effects on the preference for a certain type of visualization.

关键词： line graph scatter plot time series trend

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line graph of combinations of generalized Bethe trees: Eigenvalues and energy

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lineAR ALGEBRA AND ITS APPLICATIONS 2011年第10期435卷 2402-2419页

作者： Rojo, Oscar Jimenez, Raul D. Univ Catolica Norte Dept Math Antofagasta Chile

We characterize the eigenvalues and energy of the line graph L(G) whenever G is (i) a generalized Bethe tree, (ii) a Bethe tree, (iii) a tree defined by generalized Bethe trees attached to a path, (iv) a tree defined by generalized Bethe trees having a common root, (v) a graph defined by copies of a generalized Bethe tree attached to a cycle and (vi) a graph defined by copies of a star attached to a cycle;in this case, explicit formulas for the eigenvalues and energy of L(G) are derived. (C) 2010 Elsevier Inc. All rights reserved.

关键词： line graph Laplacian matrix Signless Laplacian matrix Adjacency matrix Generalized Bethe tree

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line graph eigenvalues and line energy of caterpillars

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lineAR ALGEBRA AND ITS APPLICATIONS 2011年第8期435卷 2077-2086页

作者： Rojo, Oscar Univ Catolica Norte Dept Math Antofagasta Chile

The energy of a graph G is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. A caterpillar is a tree in which the removal of all pendant vertices makes it a path. Let d >= 3 and n >= 2(d - 1). Let p = vertical bar p(1), p(2), ... , p(d-1)vertical bar with p(1) >= 1.p(2) >= 1, ... , P(d-1) >= 1 such that p(1) + p(2) + ... + p(d-1) = n - d + 1. Let C(p) be the caterpillar obtained from the stars S(p1+1), S(p2+1) , ... , S(pd+i) +1 and the path P(d-1) by identifying the root ofS(pi+1) with the i-vertex of P(d-1). The line graph of C(p), denoted by 12,(C(p)), becomes a sequence of cliques K(p1+1), K(p2+2), ... , Kp(d-2+2), K(pd-1+1) in this order, such that two consecutive cliques have in common exactly one vertex. In this paper, we characterize the eigenvalues and the energy of L(C(p)). Explicit formulas are given for the eigenvalues and the energy of L(C(a)) where a = vertical bar a, a, ... , a vertical bar. Finally, a lower bound and an upper bound for the energy of C(C(p)) are derived. (C) 2011 Elsevier Inc. All rights reserved.

关键词： line graph line energy Laplacian matrix Signless Laplacian matrix Adjacency matrix Generalized Bethe tree Caterpillar

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