Mixed Graph Research Articles

Edge‐arc‐disjoint paths in semicomplete mixed graphs

AbstractThe so‐called weak‐2‐linkage problem asks for a given digraph and distinct vertices of whether has arc‐disjoint paths so that is an ‐path for . This problem is NP‐complete for general digraphs but the first author showed that the problem is polynomially solvable and that all exceptions can be characterized when is a semicomplete digraph, that is, a digraph with no pair of nonadjacent vertices. In this paper we extend these results to paths which are both edge‐disjoint and arc‐disjoint in semicomplete mixed graphs, that is, a mixed graph in which every pair of distinct vertices has either an arc, an edge, or both an arc and an edge between them. We give a complete characterization of the negative instances and explain how this gives rise to a polynomial algorithm for the problem.

Wiener index of inverse fuzzy mixed graphs with application in education system

Wiener index of inverse fuzzy mixed graphs with application in education system

Mixed orthogonality graphs for continuous-time stationary processes

In this paper, we introduce different concepts of Granger causality and contemporaneous correlation for multivariate stationary continuous-time processes to model different dependencies between the component processes. Several equivalent characterisations are given for the different definitions, in particular by orthogonal projections. We then define two mixed graphs based on different definitions of Granger causality and contemporaneous correlation, the (mixed) orthogonality graph and the local (mixed) orthogonality graph. In these graphs, the components of the process are represented by vertices, directed edges between the vertices visualise Granger causal influences and undirected edges visualise contemporaneous correlation between the component processes. Further, we introduce various notions of Markov properties in analogy to Eichler (2012), which relate paths in the graphs to different dependence structures of subprocesses, and we derive sufficient criteria for the (local) orthogonality graph to satisfy them. Finally, as an example, for the popular multivariate continuous-time AR (MCAR) processes, we explicitly characterise the edges in the (local) orthogonality graph by the model parameters.

A Graph-Informed Modeling Framework Empowering Gene Pathway Discovery.

This study introduces a novel graph-informed modeling framework for improving the statistical analysis of gene expression data, particularly in the context of identifying differentially expressed gene pathways and gene expression-assisted disease classification in a high-dimensional data setting. By integrating gene regulatory network information into hypothesis testing for the difference between mean vectors and linear discriminant analysis, we aim to effectively capture and utilize previously validated external gene interaction information. Our method leverages a block-coordinate descent approach which enables us to incorporate mixed graph information into linear structural equation modeling, accommodating directed/undirected edges and potential cycles in gene regulatory networks. Extensive simulations under various data scenarios have demonstrated the effectiveness of our approach with improved power for gene pathway tests and disease classification over existing methods. An application to a lung cancer dataset from the Cancer Genome Atlas Program (TCGA) further exemplifies the potential of our graph-informed approach in empowering the detection of differentially expressed gene pathways and gene expression-based classification of different lung cancer stages. Our findings underscore the potential utility of incorporating gene regulatory network information in gene pathway analysis, setting the stage for future advancements in gene pathway discovery, disease diagnosis, and treatment strategies.

Structural and Spectral Properties of Chordal Ring, Multi-Ring, and Mixed Graphs

The chordal ring (CR) graphs are a well-known family of graphs used to model some interconnection networks for computer systems in which all nodes are in a cycle. Generalizing the CR graphs, in this paper, we introduce the families of chordal multi-ring (CMR), chordal ring mixed (CRM), and chordal multi-ring mixed (CMRM) graphs. In the case of mixed graphs, we can have edges (without direction) and arcs (with direction). The chordal ring and chordal ring mixed graphs are bipartite and 3-regular. They consist of a number r (for r≥1) of (undirected or directed) cycles with some edges (the chords) joining them. In particular, for CMR, when r=1, that is, with only one undirected cycle, we obtain the known families of chordal ring graphs. Here, we used plane tessellations to represent our chordal multi-ring graphs. This allowed us to obtain their maximum number of vertices for every given diameter. Additionally, we computationally obtained their minimum diameter for any value of the number of vertices. Moreover, when seen as a lift graph (also called voltage graph) of a base graph on Abelian groups, we obtained closed formulas for the spectrum, that is, the eigenvalue multi-set of its adjacency matrix.

Mixed Graph Colouring as Scheduling a Partially Ordered Set of Interruptible Multi-Processor Tasks with Integer Due Dates

We investigate relationships between scheduling problems with the bottleneck objective functions (minimising makespan or maximal lateness) and problems of optimal colourings of the mixed graphs. The investigated scheduling problems have integer durations of the multi-processor tasks (operations), integer release dates and integer due dates of the given jobs. In the studied scheduling problems, it is required to find an optimal schedule for processing the partially ordered operations, given that operation interruptions are allowed and indicated subsets of the unit-time operations must be processed simultaneously. First, we show that the input data for any considered scheduling problem can be completely determined by the corresponding mixed graph. Second, we prove that solvable scheduling problems can be reduced to problems of finding optimal colourings of corresponding mixed graphs. Third, finding an optimal colouring of the mixed graph is equivalent to the considered scheduling problem determined by the same mixed graph. Finally, due to the proven equivalence of the considered optimisation problems, most of the results that were proven for the optimal colourings of mixed graphs generate similar results for considered scheduling problems, and vice versa.

Graph reinforcement learning for flexible job shop scheduling under industrial demand response: A production and energy nexus perspective

Amidst the global energy crisis, the industrial sector is facing unparalleled energy conservation challenges as a primary energy consumer. Industrial demand response (IDR) is a promising strategy to enhance the energy efficiency of manufacturing enterprises. This paper introduces a graph reinforcement learning (GRL)-based method for flexible job shop scheduling under IDR from a production and energy nexus perspective. The scheduling problem is initially formulated as a Markov Decision Process (MDP). On this basis, the scheduling state is represented by a unique heterogeneous disjunctive graph incorporating IDR features, and the reward is constructed through a tailored generalized electricity consumption index (GECI). Moreover, a mixed graph neural network scheduler is designed to tackle this MDP, which integrates the heterogeneous attention mechanism and adaptive greedy sampling strategy. Furthermore, a proximal policy optimization algorithm is employed for training to obtain optimal scheduling schemes. Empirical case studies indicate that the proposed method can reduce GECI by up to 14.44% and 2.22%, respectively, outperforming existing well-known dispatching rules and another scheduling method based on GRL.

The network of sports: using network analysis to understand the relationship between sports and socio-physiological factors in contemporary China

This study examines sports and physical activities among Chinese aged 18–65, using network analysis on a significant random sample. It categorizes sports into 11 groups based on public selection, with a community detection algorithm aiding classification. Variables like age, gender, and education were integrated, revealing how life stages, gender disparities, and social class influence activity participation. The mixed graph model shows both positive and negative correlations among activities, highlighting the role of sports as both a social integrator and divider, reflective of broader societal norms and inequalities. The exponential random graph model further illustrates a complex network of demographic-driven participation patterns. The purpose of this investigation is twofold: to advance methodological approaches in the study of sports-related social networks and to explore the broader implications such networks may have on individual and collective behaviors within this field.

Graph-enhanced deep material network: multiscale materials modeling with microstructural informatics

AbstractThis study addresses the fundamental challenge of extending the deep material network (DMN) to accommodate multiple microstructures. DMN has gained significant attention due to its ability to be used for fast and accurate nonlinear multiscale modeling while being only trained on linear elastic data. Due to its limitation to a single microstructure, various works sought to generalize it based on the macroscopic description of microstructures. In this work, we utilize a mechanistic machine learning approach grounded instead in microstructural informatics, which can potentially be used for any family of microstructures. This is achieved by learning from the graph representation of microstructures through graph neural networks. Such an approach is a first in works related to DMN. We propose a mixed graph neural network (GNN)-DMN model that can single-handedly treat multiple microstructures and derive their DMN representations. Two examples are designed to demonstrate the validity and reliability of the approach, even when it comes to the prediction of nonlinear responses for microstructures unseen during training. Furthermore, the model trained on microstructures with complex topology accurately makes inferences on microstructures created under different and simpler assumptions. Our work opens the door for the possibility of unifying the multiscale modeling of many families of microstructures under a single model, as well as new possibilities in material design.

Conditional independence collapsibility for acyclic directed mixed graph models

Collapsibility refers to the property that, when marginalizing over some variables that are not of interest from the full model, the resulting marginal model of the remaining variables is equivalent to the local model induced by the subgraph on these variables. This means that when the marginal model satisfies collapsibility, statistical inference results based on the marginal model and the local model are consistent. This has significant implications for small-sample data, modeling latent variable data, and reducing the computational complexity of statistical inference. This paper focuses on studying the conditional independence collapsibility of acyclic directed mixed graph (ADMG) models. By introducing the concept of inducing paths in ADMGs and exploring its properties, the conditional independence collapsibility of ADMGs is characterized equivalently from both graph theory and statistical perspectives.

Intelligent code search aids edge software development

The growth of multimedia applications poses new challenges to software facilities in edge computing. Developers must effectively develop edge computing software to accommodate the rapid expansion of multimedia applications. Code search has become a prevalent practice to enhance the efficiency of the construction of edge software infrastructure. Researchers have proposed lots of approaches for code search, and employed deep learning technology to extract features from program representations, such as token, AST, graphs, method name, and API. Nevertheless, two prominent issues remain: 1) there are only a few studies on the effective use of graph representation for code search (especially in Java language), and 2) there is a lack of empirical study on the contributions of different program representations. To address these issues, we conduct an empirical study to explore program representations, especially program graphs. To the best of our knowledge, this is the first attempt to conduct code search with mixed graphs representation for Java language, containing the control flow graph and the program dependence graph. We also present a hybrid approach to capture and fuse the features of a program with representations of Token, AST, and Mixed Graphs (TAMG). The results of our experiment show that our approach possesses the best ability (R@1 with 37% and R@10 with 67.1%). Our graph representation exhibits a positive effect, and the token and AST also have a significant contribution to the code search. Our findings can aid developers in efficiently searching for the desired code while constructing the software infrastructure for edge computing, which is crucial for the rapid expansion of multimedia applications.

Inverse of \(\alpha\)-Hermitian Adjacency Matrix of a Unicyclic Bipartite Graph

Let \(X\) be bipartite mixed graph and for a unit complex number \(\alpha\), \(H_\alpha\) be its \(\alpha\)-hermitian adjacency matrix. If \(X\) has a unique perfect matching, then \(H_\alpha\) has a hermitian inverse \(H_\alpha^{-1}\). In this paper we give a full description of the entries of \(H_\alpha^{-1}\) in terms of the paths between the vertices. Furthermore, for \(\alpha\) equals the primitive third root of unity \(\gamma\) and for a unicyclic bipartite graph \(X\) with unique perfect matching, we characterize when \(H_\gamma^{-1}\) is \(\pm 1\) diagonally similar to \(\gamma\)-hermitian adjacency matrix of a mixed graph. Through our work, we have provided a new construction for the \(\pm 1\) diagonal matrix.

The Index Weighted Hermitian Adjacency Matrices for Mixed Graphs

Based on the Hermitian adjacency matrices of second kind introduced by Mohar [1] and weighted adjacency matrices introduced in [2], we define a kind of index weighted Hermitian adjacency matrices of mixed graphs. In this paper we characterize the structure of mixed graphs which are cospectral to their underlying graphs, then we determine a upper bound on the spectral radius of mixed graphs with maximum degree Δ , and characterize the corresponding extremal graphs.

Graph Invariant Learning with Subgraph Co-mixup for Out-of-Distribution Generalization

Graph neural networks (GNNs) have been demonstrated to perform well in graph representation learning, but always lacking in generalization capability when tackling out-of-distribution (OOD) data. Graph invariant learning methods, backed by the invariance principle among defined multiple environments, have shown effectiveness in dealing with this issue. However, existing methods heavily rely on well-predefined or accurately generated environment partitions, which are hard to be obtained in practice, leading to sub-optimal OOD generalization performances. In this paper, we propose a novel graph invariant learning method based on invariant and variant patterns co-mixup strategy, which is capable of jointly generating mixed multiple environments and capturing invariant patterns from the mixed graph data. Specifically, we first adopt a subgraph extractor to identify invariant subgraphs. Subsequently, we design one novel co-mixup strategy, i.e., jointly conducting environment mixup and invariant mixup. For the environment mixup, we mix the variant environment-related subgraphs so as to generate sufficiently diverse multiple environments, which is important to guarantee the quality of the graph invariant learning. For the invariant mixup, we mix the invariant subgraphs, further encouraging to capture invariant patterns behind graphs while getting rid of spurious correlations for OOD generalization. We demonstrate that the proposed environment mixup and invariant mixup can mutually promote each other. Extensive experiments on both synthetic and real-world datasets demonstrate that our method significantly outperforms state-of-the-art under various distribution shifts.

Turán problems for mixed graphs

We investigate natural Turán problems for mixed graphs, generalizations of graphs where edges can be either directed or undirected. We study a natural Turán density coefficient that measures how large a fraction of directed edges an F-free mixed graph can have; we establish an analogue of the Erdős-Stone-Simonovits theorem and give a variational characterization of the Turán density coefficient of any mixed graph (along with an associated extremal F-free family).This characterization enables us to highlight an important divergence between classical extremal numbers and the Turán density coefficient. We show that Turán density coefficients can be irrational, but are always algebraic; for every positive integer k, we construct a family of mixed graphs whose Turán density coefficient has algebraic degree k.

Decompositions of the λ-Fold Complete Mixed Graph into Mixed 6-Stars

Graph and digraph decompositions are a fundamental part of design theory. Probably the best known decompositions are related to decomposing the complete graph into 3-cycles (which correspond to Steiner triple systems), and decomposing the complete digraph into orientations of a 3-cycle (the two possible orientations of a 3-cycle correspond to directed triple systems and Mendelsohn triple systems). Decompositions of the λ-fold complete graph and the λ-fold complete digraph have been explored, giving generalizations of decompositions of complete simple graphs and digraphs. Decompositions of the complete mixed graph (which contains an edge and two distinct arcs between every two vertices) have also been explored in recent years. Since the complete mixed graph has twice as many arcs as edges, an isomorphic decomposition of a complete mixed graph into copies of a sub-mixed graph must involve a sub-mixed graph with twice as many arcs as edges. A partial orientation of a 6-star with two edges and four arcs is an example of such a mixed graph; there are five such mixed stars. In this paper, we give necessary and sufficient conditions for a decomposition of the λ-fold complete mixed graph into each of these five mixed stars for all λ>1.

More on minors of Hermitian (quasi-)Laplacian matrix of the second kind for mixed graphs

More on minors of Hermitian (quasi-)Laplacian matrix of the second kind for mixed graphs

Almost Moore and the largest mixed graphs of diameters two and three

Almost Moore mixed graphs appear in the context of the degree/diameter problem as a class of extremal mixed graphs, in the sense that their order is one unit less than the Moore bound for such graphs. The problem of their existence has been considered just for diameter 2. In this paper, we give a complete characterization of these extremal mixed graphs for diameters 2 and 3. We also derive some optimal constructions for other diameters.

Mixed Graph Neural Network-Based Fake News Detection for Sustainable Vehicular Social Networks

The rapid development of the Internet of Vehicles has substantially boosted the prevalence of vehicular social networks (VSN). However, content security has gradually been a latent threat to the stable operation of VSN. The VSN is a time-varying environment and mixed with various real or fake contents, which brings great challenges to the sustainability of VSN. To establish a sustainable VSN, it is of practical value to possess a strong ability for fake content detection. Related works can be divided into the global semantics-based approaches and the local semantics-based approaches, though both with limitations. Leveraging these two different approaches, this paper proposes a fake content detection model based on the mixed graph neural networks (GNN) for sustainable VSN. It takes GNN as the bottom architecture and integrates both convolution neural networks and recurrent neural networks to capture two aspects of semantics. Such a mixed detection framework is expected to possess a better detection effect. A number of experiments were conducted on two social network datasets for evaluation, and the results indicated that the detection effect can be improved by about 5%-15% compared with baseline methods.

On Mixed Cages

Mixed graphs have both directed and undirected edges. A mixed cage is a regular mixed graph of given girth with minimum possible order. In this paper mixed cages are studied. Upper bounds are obtained by general construction methods and computer searches.