Graph Theory Research Articles

A charge-preserving method for solving graph neural diffusion networks

The aim of this paper is to give a systematic mathematical interpretation of the diffusion problem on which Graph Neural Networks (GNNs) models are based. The starting point of our approach is a dissipative functional leading to dynamical equations which allows us to study the symmetries of the model. We provide a short review of graph theory and its relation with network σ-models adapted to our analysis. We discuss the conserved charges and provide a charge-preserving numerical method for solving the dynamical equations. In any dynamical system and also in GRAph Neural Diffusion (GRAND), knowing the charge values and their conservation along the evolution flow could provide a way to understand how GNNs and other networks work with their learning capabilities.

Shortest path counting in complex networks based on powers of the adjacency matrix.

Complex networks describe a broad range of systems in nature and society. As a fundamental concept of graph theory, the path connecting nodes and edges plays a crucial role in network science, where the computation of shortest path lengths and numbers has garnered substantial focus. It is well known that powers of the adjacency matrix can calculate the number of walks, specifying their corresponding lengths. However, developing methodologies to quantify both the number and length of shortest paths through the adjacency matrix remains a challenge. Here, we extend powers of the adjacency matrix from walks to shortest paths. We address the all-pairs shortest path count problem and propose a fast algorithm based on powers of the adjacency matrix that counts both the number and the length of all shortest paths. Numerous experiments on synthetic and real-world networks demonstrate that our algorithm is significantly faster than the classical algorithms across various network types and sizes. Moreover, we verified that the time complexity of our proposed algorithm significantly surpasses that of the current state-of-the-art algorithms. The superior property of the algorithm allows for rapid calculation of all shortest paths within large-scale networks, offering significant potential applications in traffic flow optimization and social network analysis.

Solving Max-cut Problem with a mixed penalization method for Semidefinite Programming

This paper is an attempt to exploit the opportunity of Semi Definite Programming (SDP), which is an area of convex and conic optimization. Indeed, Numerous NP-hard problems can be solveby using this approach. Hence, we intend to investigate the strength of SDP to model and provide tight relaxations of combinatorial and quadratic problems in order to present a new polynomial time algorithm for solving this robust model. This algorithm was firstly use to solve the nonlinear programs, the reason for which we search to extend it to the SDP programs. Actually, this algorithm designs the combination of two penalization methods. The first one is a primal-dual interior point (PDIM) method while the second one is a primal-dual exterior point (PDEM) method. Unlike the first method, which converges globally, the second one, also called the primal dual nonlinear rescaling method, has local super linear/quadratic convergence. Therefore, it seems appropriate to use a mixed algorithm based on the interior-exterior point method (IEPM). In fact, this resolution starts from the interior method, and at a certain level of execution, it proceeds to exterior method. Hence, a convergence evaluation function is use to know the level of permutation. Through evaluation, it has been approve that our approach is use to solve some instances of max-cut problem. This problem is a central graph theory model that occurs in many real problems and it is one of many NP-hard problems, which has attracted many researchers over the years. Then, we have used a semi definite programming solver SDPA (Semi Definite Programming Algorithm) that is modify to include the exterior point method subroutine. From the computational performance, we conclude that as the problem size increases, interior-exterior point algorithm gets relatively faster. The numerical results obtained are promising.

On the hyperspaces of meager and regular continua

Given a metric continuum X, we consider the collection of all regular subcontinua of X and the collection of all meager subcontinua of X, these hyperspaces are denoted by D(X) and M(X), respectively. It is known that D(X) is compact if and only if D(X) is finite.In this way, we find some conditions related about the cardinality of D(X) and we reduce the fact to count the elements of D(X) to a Graph Theory problem, as an application of this, we prove in particular that | D ( X ) | ∉ { 2,3,4,5,8 , 9 } h t ) for any continuum X. Also, we prove that D(X) is never homeomorphic to ℕ . On the other hand, given a point p ∈ X , we consider the meager composant and the filament composant of p in X, denoted by MXp and FcsX(p), respectively, and we study some relations between MXp and FcsX(p) such as the equality of them as a subset of X. Also, we construct examples showing that the collection Fcs ( X ) = { FcsX ( p ) : p ∈ X } can be homeomorphic to: any finite discrete space, the harmonic sequence, the closure of the harmonic sequence and the Cantor set. Finally, we study the contractibility of M(X); we prove the arc of pseudo-arcs, which is a no contractible continuum, satisfies that its hyperspace of meager subcontinua is contractible, given a solution to an open problem. Also, we rise open problems.

Dynamics of online debates: insights from textual network analysis

AbstractTextual data analysis is critical for monitoring changing themes over time. To overcome challenges posed by data richness, graph theory emerges as a tool for investigating word-topic associations. We present an approach to clustering co-occurrence word networks that prioritises network similarity quantification over time. Addressing theoretical and network geometrical constraints, a statistical framework for manifold data analysis facilitates the grouping of semantic networks, partitioning the observed time frame into periods, and identifying dominant topics in each period via tensor decomposition. The analysis of Brexit-related tweets demonstrates the efficacy of modern methods for identifying social media patterns on public discourse.

Rényi–Sobolev Inequalities and Connections to Spectral Graph Theory

Rényi–Sobolev Inequalities and Connections to Spectral Graph Theory

Mathematical Nonlinear Graph Theory Topology Layer Model for Photoelectric Tracking System

The performance of the photoelectric tracking system mainly depends on the tracking accuracy. In order to achieve the purpose of high precision tracking, controlling the power dragging device, the main component of the photoelectric tracking system, is the main means to achieve this purpose. In order to improve the tracking speed and accuracy of photoelectric tracking systems, the author proposed a mathematical nonlinear graph theory topology layer model for photoelectric tracking systems. The topology layer model and the motion node servo mechanism model of the photoelectric tracking system are studied, and the two-stage disturbance sources that affect the tracking stability are analyzed. The results show that the interference estimation error range designed by the author reaches 0.003 through comparison, it can be seen that the estimation error designed by the author is significantly smaller than the ESO estimation error, and the buffeting is small. Through comparison, it can be concluded that the estimation error of the algorithm and the disturbance observer designed by the author are significantly smaller than the estimation error of ESO, and the buffeting is small, and they have strong compensation ability for the disturbance of different frequencies. The interference observer can quickly and accurately estimate the interference and prove its stability. The effectiveness of the proposed observer and disturbance observer is fully demonstrated. The finite time integrated site selection mode disturbance observer (F-ISMDOB) is designed to quickly estimate and compensate for equivalent interference, and effectively improves the anti-interference performance of the system structure layer.

Text Encryption with Graph Theory Based Key Generation

This work presents a new encryption technique that ensures safe communication through two distinct phases: utilizing monoalphabetic substitution ciphers that rely on agreed-upon character-swapping arrangements between authorized individuals and applying an alphabetical encryption table. In addition, the research utilizes the Kurskal technique to calculate the minimum spanning tree, which improves security by creating intricate encrypted text using a shared key based on ideas drawn from graph theory.

Web search results processing in an information support system for decision-making processes

Objectives. The problem of the web search results processing in an information support system for decision[1]making is solved in order to create and correct a meaningful description of a problem situation. An approach to solving this problem is proposed based on the use of thematic text corpora (collections of texts on a specific topic) as knowledge about the subject area, as well as a knowledge representation model based on verbal associations. When solving problems of the web search results processing in a decision-making information support system, five main goals are pursued: the formation of an extended description of a problem situation, the synthesis of a search prescription, an Internet search for information about decisions made, the synthesis of a retelling of the information found, and an assessment of the quality of the found analogues of the decisions made.Methods. Methods of set theory, graph theory and mathematical linguistics are used.Results. A mathematical model has been developed for the web search results processing in an information support system for decision making. The concepts of verbal association of words and texts, as well as pragmatically complete lexical structure, are formalized. The proven properties of such structures provide algorithmization of information processes in the model under consideration.Conclusion. The approach to modeling is based on the formalization of the concepts of the informativeness of words, sentences, texts and the informativeness of verbal associations between them. As an implementation of the model proposed in the article, algorithms have been developed for creating a dictionary of pragmatically complete lexical structures, creating structural-lexical templates for sentences, texts and subject areas, synthesizing a brief retelling of the information found, and assessing the quality of the found analogues of the decisions made.

Practical Teaching Mode Structure for Ideological and Political Theory Course Based on Fuzzy System Theory

The application of graph theory is instrumental in the construction of a practical teaching mode structure for an ideological and political theory (IPE) course rooted in fuzzy system theory. In recent years, IPE education has received more and more attention. System theory can well construct the IPE teaching mode, so we need to have a good grasp of system theory. This paper starts with the significance of system theory in the practical teaching of IPE, finds the IPE curriculum that conforms to system theory, and constructs the basic mode of practical teaching of IPE on this basis. Using the idea of a multi-level fuzzy comprehensive evaluation to quantify teachers’ teaching quality, this paper establishes an algorithm model to quantitatively measure teachers’ teaching quality fuzzy.

On the Various Energy Forms of the Join of Complete Graphs

The concept of graph energy, first introduced in 1978, has been a focal point of extensive research within the field of graph theory, leading to the publication of numerous articles. Graph energy, originally associated with the eigenvalues of the adjacency matrix of a graph, has since been extended to various other matrices. These include the maximum degree matrix, Randić matrix, sum-connectivity matrix, and the first and second Zagreb matrices, among others. In this paper, we focus on calculating the energy of several such matrices for the join graph of complete graphs, denoted as J m ( K n ) . Specifically, we compute the energies for the maximum degree matrix, Randić matrix, sum-connectivity matrix, first Zagreb matrix, second Zagreb matrix, reverse first Zagreb matrix, and reverse second Zagreb matrix for J m ( K n ) . Our results provide new insights into the structural properties of the join graph and contribute to the broader understanding of the mathematical characteristics of graph energy for different matrix representations. This work extends the scope of graph energy research by considering these alternative matrix forms, offering a deeper exploration into the algebraic and spectral properties of graph energy in the context of join graphs.

Графовые алгоритмы для расчёта распределения следов амурского тигра на территории Приморского края

The paper analyzes the distribution of the Amur tiger (Panthera tigris altaica) tracks in the Primorsky Krai municipalities according to observations obtained during the one-time winter count organized by the Pacific Institute of Geography of the Far Eastern Branch of the Russian Academy of Sciences in 2005. This task is related to the need to calculate the number of animals based on the tracks of their vital activity. Based on the initial data we form a sample to analyze a set of tracks to identify intersection areas of various tiger habitats in Primorsky Krai. The map of Primorsky Krai municipalities was presented in the form of a planar graph to identify their intersections. A recurrent algorithm for identifying tiger clusters was developed using graph theory methods. According to them, the territory of Primorsky Krai is divided into northern, central, southern and southwestern zones with intersections. The districts of Primorsky Krai included in the designated zones are listed. The total number of tracks in intersecting adjacent zones was counted to identify possible tiger migrations in Primorsky Krai. A quantitative assessment of the degree of intersection of the selected zones by the number of tiger tracks is given. The proposed calculation scheme can be used to clarify and verify the results of modeling the spatial distribution and total population size of the Amur tiger.

Using Schur Rings to Produce GRRs for Dihedral Groups

If Γ is a finite group and G a graph such that Aut(G) ≡ Γ acts regularly on V(G), then we say that G is a graphical regular representation (GRR) of Γ. The question asking which finite groups have at least one GRR was an important question in algebraic graph theory and it was completely solved as a result of work done by several researchers. However, it remains a challenge to discern whether a group known to have GRRs has GRRs with specific properties, such as being trivalent. In this paper, we shall be deriving simple conditions on the parameters of a subset of a dihedral group for easily constructing trivalent graphical regular representations (GRR) of the group. Specifically, we shall prove the following: Let n be an odd integer greater than 5 and let r, s, and t be integers less than n such that the difference of any two of them is relatively prime to n. If 3r – 2s = t (mod n), then Cay(Dn, {abr, abs, abt}) is a GRR of Dn. We will also be looking at very convenient corollaries of this result. But another main aim of this paper is to show how a simple use of Schur rings can be used to derive such results. This paper therefore also serves as a review of some basic results about Schur rings which we feel should be among the standard armory of an algebraic graph theorist.

Mapping Financial Connections: Market Integration in Emerging Economies through Graph Theory

In this study, we explore the financial and economic integration of BRICS nations (Brazil, Russia, India, China, and South Africa) and key emerging economies (Egypt, Saudi Arabia, and the UAE) using graph theory, aiming to map intersectoral connections and their impact on financial stability and market risk. The research addresses a critical gap in the literature; while political and economic linkages between nations have been widely studied, the specific connectivity between sectors within these economies remains underexplored. Our methodology utilizes eigenvector centrality and Euclidean distance to construct a comprehensive network of 106 publicly listed firms from 2013 to 2022, across sectors such as energy, telecommunications, retail, and technology. The primary hypothesis is that sectors with higher centrality scores—indicative of their interconnectedness within the broader financial network—demonstrate greater resilience to market volatility and contribute disproportionately to sectoral profitability. The analysis yielded several key insights. For instance, BHARTI AIRTEL LIMITED in telecommunications exhibited an eigenvector centrality score of 0.9615, positioning it as a critical node in maintaining sectoral stability, while AMBEV SA in the retail sector, with a centrality score of 0.9938, emerged as a pivotal player influencing both profitability and risk. Sectors led by companies with high centrality showed a 20% increase in risk-adjusted returns compared to less connected entities, supporting the hypothesis that central firms act as stabilizers in fluctuating market conditions. The findings underscore the practical implications for policymakers and investors alike. Understanding the structure of these networks allows for more informed decision making in terms of investment strategies and macroeconomic policy. By identifying the central entities within these economic systems, both policymakers and investors can target their efforts more effectively, either to support growth initiatives or to mitigate systemic risks. This study advances the discourse on emerging market integration by providing a quantitative framework to analyze intersectoral connections, offering critical insights into how sectoral dynamics in emerging economies influence global financial trends.

Machine-learning-based classification of obstructive sleep apnea using 19-channel sleep EEG data

ObjectiveThis study aimed to investigate the neurophysiological effects of obstructive sleep apnea (OSA) using multi-channel sleep electroencephalography (EEG) through machine learning methods encompassing various analysis methodologies including power spectral analysis, network analysis, and microstate analysis. MethodsTwenty participants with apnea-hypopnea index (AHI) ≥ 15 and 18 participants with AHI <15 were recruited. Overnight polysomnography was conducted concurrently with 19-channel EEG. Preprocessed EEG data underwent computation of relative spectral power. A weighted network based on graph theory was generated; and indices of strength, path length, eigenvector centrality, and clustering coefficient were calculated. Microstate analysis was conducted to derive four topographic maps. Machine learning techniques were employed to assess EEG features capable of differentiating two groups. ResultsAmong 71 features that showed significant differences between the two groups, seven exhibited good classification performance, achieving 88.3 % accuracy, 92 % sensitivity, and 84 % specificity. These features were power at C4 theta, P3 theta, P4 theta, and F8 gamma during NREM1 sleep and at Pz gamma during REM sleep from power spectral analysis; eigenvector centrality at F7 gamma during REM sleep from network analysis; and duration of microstate 4 during NREM2 sleep from microstate analysis. These seven EEG features were significantly correlated with polysomnographic parameters reflecting the severity of OSA. ConclusionsThe application of machine learning techniques and various EEG analytical methods resulted in a model that showed good performance in classifying moderate to severe OSA and highlights the potential of EEG to serve as a biomarker of functional changes in OSA.

Numerical Solutions for Kinematics of Multi-bar Mechanisms Using Graph Theory and Computer Simulations

Numerical Solutions for Kinematics of Multi-bar Mechanisms Using Graph Theory and Computer Simulations

Distributed Adaptive Formation Control for Fractional-Order Multi-Agent Systems with Actuator Failures and Switching Topologies

In this paper, a class of distributed adaptive formation control problems are investigated for second-order nonlinear fractional-order multi-agent systems with actuator failures and switching topologies. To address these challenges, two adaptive coupling gains based on agents’ position and velocity are incorporated into the control protocol. Using the Lyapunov method along with graph theory and matrix analysis, sufficient conditions for system stability are derived in the presence of actuator failures and switching topologies. The effectiveness of the proposed control protocol is demonstrated through numerical simulations, which show its capability to maintain stable formation control under these challenging conditions.

IMCSN: An improved neighborhood aggregation interaction strategy for multi-scale contrastive Siamese networks

Graph contrastive learning is an effective method for enhancing graph representation by maximizing the similarity of representations between analogous graphs while minimizing the similarity between dissimilar ones. A common approach to improve the representation capabilities of graphs involves data augmentation, which generates additional training samples through transformations applied to the original graphs. However, traditional data augmentation techniques, such as random deletion of nodes or edges, often compromise the structural integrity of graphs, result in loss of crucial information, and lead to representation instability. To overcome these drawbacks, this study introduces a novel data augmentation paradigm, the integration of directed random noise (IDR), which incorporates controlled random noise to optimize the augmentation objectives. IDR enhances the diversity and robustness of representations without sacrificing the structural integrity of the graphs. This approach significantly improves the performance of graph contrastive learning, avoiding the common issues of graph structure damage and information loss associated with conventional methods. To further refine the model, this paper proposes an improved multiscale contrastive Siamese network framework that employs three Siamese networks to process different views of input graphs. It utilizes cross-network and cross-view contrastive learning objectives to optimize graph representations, leveraging the complementary information between different views to maximize the consistency of representations among similar graphs. This enhancement improves the quality and generalizability of graph representations. Additionally, the introduction of a self-supervised loss function based on graph reconstruction within the loss framework capitalizes on the structural similarity between the original and reconstructed graphs to further refine graph representations. This loss function ensures that the global view retains more structural information from the original graph, thus enhancing the complementarity between the global and local views. The effectiveness and stability of this research framework are demonstrated through node classification tasks and visualizations on six real-world datasets, comparing favorably against existing methods such as MERIT and GraphVAT. The proposed model achieves accuracy improvements of 3.21 % and 3.71 % on the Cora dataset and 1.12 % and 1.42 % on the CiteSeer dataset. Additionally, it ranks first in an average accuracy comparison across six datasets, with scores of 80.6 % and 52.67 %. These results underscore the robustness and efficacy of the proposed model in self-supervised graph representation learning, offering substantial advancements over existing techniques.

Structural-functional connectomics in major depressive disorder following aiTBS treatment

Major depressive disorder (MDD) has been associated with changes in the structural (SC) and functional connectivity (FC) of the brain. This study investigated the effects of accelerated intermittent theta burst stimulation (aiTBS) on SC-FC coupling and graph theory measures, focusing on the association between baseline SC-FC coupling of the dorsolateral prefrontal cortex (dlPFC) and clinical improvement. In a randomized, sham-controlled, quadruple-blind, crossover study, aiTBS was delivered to the left dlPFC of depressed patients with MDD, and diffusion tensor imaging (DTI) and resting-state functional magnetic resonance imaging (rsfMRI) data were acquired. In 77 MDD patients, significantly increased whole-brain SC-FC coupling was observed, primarily driven by default mode network (DMN) SC-FC coupling, along with increased somatomotor network FC, and decreased FC between the DMN hubs and limbic regions after active aiTBS. Furthermore, significant increases were observed in structural global and local efficiency measures that were not specific to the stimulation condition (active/sham aiTBS). However, these changes did not significantly correlate with clinical improvement. Notably, baseline SC-FC coupling of the left dlPFC was a significant predictor of clinical improvement. Our findings highlight the potential of left dlPFC SC-FC coupling as a predictor of aiTBS treatment outcomes, as well as the effect of aiTBS in enhancing SC-FC coupling.

Navigating the complexity of hybrid materials without structural dependency: PerovGNN as a map

Hybrid organic-inorganic materials, spearheaded by perovskites, have showcased exceptional potential in optical and electrical applications. However, finding suitable compositions with desired properties is challenging due to the vast, high-dimensional search space. Traditional machine learning techniques often fall short in capturing crucial interactions within the composites of hybrid organic-inorganic materials, while others necessitate structural information that is often unavailable prior to material synthesis. To address this, we propose PerovGNN, a structure-agnostic graph representation learning method for property prediction. It transforms the chemical formula of each mixed perovskite into a weighted graph of atoms, with weights representing the relative frequency of interactions between neighboring atoms. By embedding both fractional ratios and atomic features, this graph formulation effectively retains crucial mixture information and learns a superior representation required for a specific targeted property. We applied PerovGNN to experimental datasets for mixed HOIPs bandgap prediction, achieving a lower mean absolute error (MAE) of 0.0483 eV compared to other methods. New perovskites with bandgaps ranging from 1.3 to 2.0 eV were experimentally characterized, yielding a consistent MAE of 0.043 eV. Our work presents a unified model for predicting properties of inorganic, organic, and hybrid crystals, serving as a map for researchers navigating vast design spaces.