Graph Theory Concepts Research Articles

Bayesian inference of frequency-specific functional connectivity in MEG imaging using a spectral graph model

Abstract Understanding the relationship between structural connectivity (SC) and functional connectivity (FC) of the human brain is an important goal of neuroscience. Highly detailed mathematical models of neural masses exist that can simulate the interactions between functional activity and structural wiring. These models are often complex and require intensive computation. Most importantly, they do not provide a direct or intuitive interpretation of this structure–function relationship. In this study, we employ the emerging concepts of spectral graph theory to obtain this mapping in terms of graph harmonics, which are eigenvectors of the structural graph’s Laplacian matrix. In order to imbue these harmonics with biophysical underpinnings, we leverage recent advances in parsimonious spectral graph modeling (SGM) of brain activity. Here, we show that such a model can indeed be cast in terms of graph harmonics, and can provide a closed-form prediction of FC in an arbitrary frequency band. The model requires only three global, spatially invariant parameters, yet is capable of generating rich FC patterns in different frequency bands. Only a few harmonics are sufficient to reproduce realistic FC patterns. We applied the method to predict FC obtained from pairwise magnitude coherence of source-reconstructed resting-state magnetoencephalography (MEG) recordings of 36 healthy subjects. To enable efficient model inference, we adopted a deep neural network-based Bayesian procedure called simulation-based inference. Using this tool, we were able to speedily infer not only the single most likely model parameters, but also their full posterior distributions. We also implemented several other benchmark methods relating SC to FC, including graph diffusion and coupled neural mass models. The present method was shown to give the best performance overall. Notably, we discovered that a single biophysical parameterization is capable of fitting FCs from all relevant frequency bands simultaneously, an aspect that did not receive adequate attention in prior computational studies.

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.

Metric Dimension of Second Order Complete Graph with Honeycomb Network Cross Operation Product

Metric dimension is a concept in graph theory that has been developed in terms of the concept and its application. Let G be a connected graph and S be a vertex subset on connected graph G. The set S is called a resolving set for G if every vertex on graph G has a distinct representation of one to each other of S. A resolving set containing a minimum cardinality is called basis. The metric dimension on graph G is cardinality of basis on graph G, notated with dim (G). In this case, the cross-product graph will be used for the research. The aim of this research is to determine the metric dimension of the second order complete graph (K2) with honeycomb networks (HC(n)) cross-operation product. Utilizing mathematical induction, we generated dim(K2×HC(n)) = 3.

Computational insights into zinc silicate MOF structures: topological modeling, structural characterization and chemical predictions

Metal-organic frameworks (MOFs) play a pivotal role in modern material science, offering unique properties such as flexibility, substantial pore space, distinctive structure, and large surface area. Recently, zinc-based MOFs have attracted significant attention, particularly in the biomedical arena, owing to their versatile applications in drug delivery, biosensing, and cancer imaging. However, there remains a crucial need to explore and understand the structural properties of zinc silicate-based MOFs to fully exploit their potential in various applications. The objective of this study is to address this need by employing topological modeling techniques to characterize zinc silicate networks. Utilizing connection number concept of chemical graph theory and novel AL molecular descriptors, we aim to investigate the structural intricacies of these MOFs. More precisely, zinc silicate-based MOF networks are topologically modeled via novel AL topological indices, and derived mathematical closed form formulae for them. By comparing experimental and calculated values and constructing linear regression models, the predictive capabilities of the proposed descriptors are evaluated. Specifically, the performance of derived topological indices against the physico-chemical properties of octane isomers is assessed, which provide valuable insights into their predictive potential. The findings of this study demonstrated the potential of novel AL indices in predicting a wide range of important physico-chemical properties, further enhancing their practicality in materials science and beyond.

ANALYSIS OF RESOLVING EFFICIENT DOMINATING SET AND ITS APPLICATION SCHEME IN SOLVING ETLE PROBLEMS

This research focuses on the analysis of Resolving Efficient Dominating Set (REDS) and its application in solving Electronic Traffic Law Enforcement (ETLE) problems using the Spatial Temporal Graph Neural Network (STGNN). Resolving Efficient Dominating Set (REDS) is a concept in graph theory that studies a set of points in a graph that efficiently monitors other points. It involves ensuring that each point v ∈ V (G) - D is dominated by exactly one point in D, with no adjacent points in D, and the representation of point v ∈ V (G) concerning D is not the same, which is termed as a resolving efficient dominating set. In the context of Electronic Traffic Law Enforcement (ETLE), the analysis of REDS has a significant impact. The theorem resulting from the analysis of REDS enables the determination of the number of traffic violation sensors required. Furthermore, by taking simulation data from road points, violation forecasting can be performed. The accurate predictions from this forecasting can assist authorities in anticipating and addressing traffic violation issues more effectively.

EDA-Graph: Graph Signal Processing of Electrodermal Activity for Emotional States Detection.

The continuous detection of emotional states has many applications in mental health, marketing, human-computer interaction, and assistive robotics. Electrodermal activity (EDA), a signal modulated by sympathetic nervous system activity, provides continuous insight into emotional states. However, EDA possesses intricate nonstationary and nonlinear characteristics, making the extraction of emotion-relevant information challenging. We propose a novel graph signal processing (GSP) approach to model EDA signals as graphical networks, termed EDA-graph. The GSP leverages graph theory concepts to capture complex relationships in time-series data. To test the usefulness of EDA-graphs to detect emotions, we processed EDA recordings from the CASE emotion dataset using GSP by quantizing and linking values based on the Euclidean distance between the nearest neighbors. From these EDA-graphs, we computed the features of graph analysis, including total load centrality (TLC), total harmonic centrality (THC), number of cliques (GNC), diameter, and graph radius, and compared those features with features obtained using traditional EDA processing techniques. EDA-graph features encompassing TLC, THC, GNC, diameter, and radius demonstrated significant differences (p < 0.05) between five emotional states (Neutral, Amused, Bored, Relaxed, and Scared). Using machine learning models for classifying emotional states evaluated using leave-one-subject-out cross-validation, we achieved a five-class F1 score of up to 0.68.

Extremal graph of super line graph operation via generalized Randić index

Graph operations enable the modification and transformation of graphs, enhancing data representation as well as effectiveness in fields such as computer science, mathematics, and data analysis. Super line graph operations are a concept of advanced graph theory that is used in advanced mathematics and computer science. They are used to solve specific mathematical problems, especially in graph theory and Combinatorics. Topological numbers are numerical values connected with graph structures that are used to analyze structural characteristics and solve problems in fields such as chemistry and mathematics. They are important for simplifying complex data and enabling quantitative analysis. The general Randić number of some super line graph operations of number two with a three-diameter is investigated in this study. The pendant vertices are inserted without disturbing their diameter. Seven graphs of order five, each with a diameter of three, can be used to create twenty-three generalized super line graphs.

Optimizing Graph Theory Algorithms for Social Network Analysis

Social network analysis (SNA) leverages graph theory to understand and visualize the complex relationships and structures within social networks. This research paper explores the optimization of graph theory algorithms tailored for SNA, focusing on efficiency improvements in handling large-scale networks. The study reviews key graph theory concepts, identifies common challenges in SNA, and evaluates various optimization techniques. Practical applications and case studies are presented to demonstrate the impact of these optimizations in real-world scenarios.

Enhancing algorithmic reasoning and critical thinking through game- based learning: A graph theory approach

In recent years, the educational landscape has witnessed a transformative shift with the emergence of game-based learning (GBL) as a dynamic and effective strategy. This innovative educational approach engages students in a captivating manner, fostering the acquisition of essential skills vital for their intellectual growth. Among these skills are problem-solving, critical thinking, and algorithmic reasoning, which play pivotal roles in preparing students for the challenges of the modern world. This research article takes a pioneering step by delving into the integration of graph theory concepts into educational games. This approach is conceived as a novel method to amplify cognitive skills, offering a unique blend of theoretical knowledge and practical application. Graph theory, renowned for its abstraction and representation of relationships between entities, brings a fresh dimension to the educational gaming experience, promising enhanced learning outcomes for students. The core objective of this study is to meticulously explore the impact of graph-based educational games on students' overall learning outcomes. Furthermore, the research endeavors to assess how these immersive games contribute to the development and application of algorithmic thinking in real-world scenarios. By seamlessly integrating graph theory into the fabric of educational games, this study aims to uncover the potential synergies between engaging gameplay and the cultivation of critical cognitive skills.

Discernible interindividual patterns of global efficiency decline during theoretical brain surgery

The concept of functional localization within the brain and the associated risk of resecting these areas during removal of infiltrating tumors, such as diffuse gliomas, are well established in neurosurgery. Global efficiency (GE) is a graph theory concept that can be used to simulate connectome disruption following tumor resection. Structural connectivity graphs were created from diffusion tractography obtained from the brains of 80 healthy adults. These graphs were then used to simulate parcellation resection in every gross anatomical region of the cerebrum by identifying every possible combination of adjacent nodes in a graph and then measuring the drop in GE following nodal deletion. Progressive removal of brain parcellations led to patterns of GE decline that were reasonably predictable but had inter-subject differences. Additionally, as expected, there were deletion of some nodes that were worse than others. However, in each lobe examined in every subject, some deletion combinations were worse for GE than removing a greater number of nodes in a different region of the brain. Among certain patients, patterns of common nodes which exhibited worst GE upon removal were identified as “connectotypes”. Given some evidence in the literature linking GE to certain aspects of neuro-cognitive abilities, investigating these connectotypes could potentially mitigate the impact of brain surgery on cognition.

Power law of path multiplicity in complex networks.

Complex networks describe a wide range of systems in nature and society. As a fundamental concept of graph theory, the path connecting nodes and edges plays a vital role in network science. Rather than focusing on the path length or path centrality, here we draw attention to the path multiplicity related to decision-making efficiency, which is defined as the number of shortest paths between node pairs and thus characterizes the routing choice diversity. Notably, through extensive empirical investigations from this new perspective, we surprisingly observe a "hesitant-world" feature along with the "small-world" feature and find a universal power-law of the path multiplicity, meaning that a small number of node pairs possess high path multiplicity. We demonstrate that the power-law of path multiplicity is much stronger than the power-law of node degree, which is known as the scale-free property. Then, we show that these phenomena cannot be captured by existing classical network models. Furthermore, we explore the relationship between the path multiplicity and existing typical network metrics, such as average shortest path length, clustering coefficient, assortativity coefficient, and node centralities. We demonstrate that the path multiplicity is a distinctive network metric. These results expand our knowledge of network structure and provide a novel viewpoint for network design and optimization with significant potential applications in biological, social, and man-made networks.

A general fractional breast cancer model: Model graph energy, Caputo-Fabrizio derivative existence and uniqueness plus numerical simulation

In the study of epidemiology, mathematical modelling is crucial because it improves understanding of the underlying mechanisms that cause illnesses to spread and offers the possibility of developing preventative interventions. In this paper, we have discussed a fractional mathematical model. A set of fractional differential equations is used to build the model. We have applied a Caputo Fabrizio (CF) to describe the fractional breast cancer model. The model consists of five subpopulations that make up a population. They are disease-free (D), cardiotoxic (E), stages 1 and 2 (A), 3 (B), and 4 (C). We have used a digraph chart to present the breast cancer model. We have used some graph theory concepts to compute the energy of the breast cancer model corresponding digraph. We have demonstrated the existence, uniqueness, and positivity of the model solutions. The fixed point hypothesis demonstrates both the existence of a solution to the suggested fractional breast cancer model as well as the fact that this solution is one of a kind. Numerical simulations are a useful tool for explaining the suggested notion. We have presented a numerical method for simulating the problem. We have used an algorithm modelled after the Adams-Bashforth-Moulton method (ABMM) to solve the model. Numerical simulations of the model in various fractional orders illustrate the theoretical conclusions. We have used the Matlab application to calculate every solution. The results demonstrate the efficacy of the proposed fractional model from an epidemiological standpoint.

Topological awareness towards collision-free multi-axis curved layer additive manufacturing

We present a topology-aware computational framework that enables global collision-free and support-structure-free material extrusion for curved layer multi-axis printing of a range of high-genus and simply connected models. The nature of multi-axis additive manufacturing facilitates continuous variation of build direction, which paves the way for numerous approaches of geometry-based curved layer design. However, due to unawareness of global topological features, prevention of collision between the printer nozzle and the workpiece can either remain uncertain or demand for exhaustive algorithmic approaches, depending on the topological and geometrical complexity of a given mesh model. In this paper, we present a topological analysis framework that draws inspiration from the concepts of Reeb graph and Morse theory, to distinguish between collision-prone and collision-free regions of mesh models. We provide an exact definition for global collision-inducing features in regions of risk and optimise the nozzle orientation vector field to avoid global collisions whilst adhering to the overhang angle constraints. A new shape analysis method is then proposed to find the model orientation which maximises the manufacturability of bifurcating simply connected models. To validate our algorithmic framework, several high-genus and bifurcating simply connected models are printed using a robotic multi-axis printer. The experimental results demonstrate the feasibility and effectiveness of our framework for minimising global collisions in multi-axis curved layer additive manufacturing for a range of mesh models.

Tuple regularity and 𝑘-ultrahomogeneity for finite groups

Abstract For k , ℓ ∈ N k,\ell\in\mathbb{N} , we introduce the concepts of 𝑘-ultrahomogeneity and ℓ-tuple regularity for finite groups. Inspired by analogous concepts in graph theory, these form a natural generalization of homogeneity, which was studied by Cherlin and Felgner as well as Li, and automorphism transitivity, which was investigated by Zhang. Additionally, these groups have an interesting algorithmic interpretation. We classify the 𝑘-ultrahomogeneous and ℓ-tuple regular finite groups for k , ℓ ≥ 2 k,\ell\geq 2 . In particular, we show that every 2-tuple regular finite group is ultrahomogeneous.

2-distance Zero Forcing Sets in Graphs

In this paper, we introduce new concept in graph theory called 2-distance zero forcing. We give some properties of this new parameter and investigate its connections with other parameters such as zero forcing and hop domination. We show that 2-distance zero forcing and hop domination (respectively, zero forcing parameter) are incomparable. Moreover, we characterize 2-distance zero forcing sets in some special graphs, and finally derive the exact values or bounds of the parameter using these results.

Implementation of Shortest Path Algorithms

An important area of study that models the relationships between items is called graph theory. The shortest path between two objects is one of the most important concepts in graph theory. Many algorithms, such as Dijkstra's Algorithm, Prim’s Algorithm, Floyd Warshall Algorithm have been created for this purpose.

Study of the Agricultural Applications of the PageRank Algorithm

As a traditional agricultural country, agriculture is the foundation of China's economic and social development. Therefore, promoting the development of agriculture is a necessary way to realize common prosperity. Applying the PageRank algorithm to agriculture can not only liberate the labor force, but also improve the crop yield to achieve economic optimization and promote social development and progress. By exploring the advantages and disadvantages of the PageRank algorithm, based on the concept of graph theory, different weights are assigned to the links according to the attribute characteristics of the links themselves. Then according to the data validation, experimental analysis, and the results of weight size, match and screen the corresponding web pages, and according to the matching degree and importance of the sequential order, and then collect data. Then the relevant knowledge of graph theory is used to illustrate the crop growth environment and growth factors, and the growth status of the target crops is observed in real-time to optimize the growth conditions at any time. Thus, the optimal growing environment is determined, and the yield of crops is improved to promote economic development.

PlasmoData.jl — A Julia framework for modeling and analyzing complex data as graphs

Datasets encountered in scientific and engineering applications appear in complex formats (e.g., images, multivariate time series, molecules, video, text strings, networks). Graph theory provides a unifying framework to model such datasets and enables the use of powerful tools that can help analyze, visualize, and extract value from data. In this work, we present PlasmoData.jl, an open-source, Julia framework that uses concepts of graph theory to facilitate the modeling and analysis of complex datasets. The core of our framework is a general data modeling abstraction, which we call a DataGraph. We show how the abstraction and software implementation can be used to represent diverse data objects as graphs and to enable the use of tools from topology, graph theory, and machine learning (e.g., graph neural networks) to conduct a variety of tasks. We illustrate the versatility of the framework by using real datasets: (i) an image classification problem using topological data analysis to extract features from the graph model to train machine learning models; (ii) a disease outbreak problem where we model multivariate time series as graphs to detect abnormal events; and (iii) a technology pathway analysis problem where we highlight how we can use graphs to navigate connectivity. Our discussion also highlights how PlasmoData.jl leverages native Julia capabilities to enable compact syntax, scalable computations, and interfaces with diverse packages. Overall, we show that the DataGraph abstraction and PlasmoData.jl Julia package are able to model data within graphs and enable useful analysis.

A Network-Group Target State and Network Topology Estimation Method Based on Signals Containing Delay, Doppler and Address

Network-group targets are a set of objectives that adhere to a shared communication protocol, perform common tasks, and exhibit relatively coordinated movements. Typically, network-group targets emit radar and communication signals. However, they often employ a silent mode to evade our perception. Despite this, they continue to exchange data through their communication networks. By intercepting the communication signals of these targets, this paper proposes a method for estimating the state and network topology of network-group targets based on random finite set (RFS) theory. This method is based on the modeling of network-group targets using a labeled multi-Bernoulli (LMB) RFS. In state estimation, the usual method involves extracting the localization parameters from the signals in the first step and use these parameters for target tracking in the second step. This study focused on estimating the kinematic states of network-group targets using communication signals containing delay and Doppler information received by multiple mobile sensor platforms. The proposed method considers the coherency between frequency-hopping pulses in the signals, resulting in an improved estimation performance. In addition, considering that network-group targets require network access for information exchange, graph theory concepts were utilized to estimate the network topology of network-group targets by address measurement. The simulation results validated the effectiveness of the proposed method and demonstrated its superior performance.

Graph Theory and Algorithms for Social Network Analysis

Graph theory and algorithms play a crucial role in the analysis of social networks, which are complex systems composed of individuals or entities connected by various types of relationships. This paper provides an overview of graph theory concepts and algorithms commonly used in social network analysis. It discusses how these tools can be applied to understand the structure and dynamics of social networks, identify key actors or communities, and analyze information flow and influence propagation. The paper also highlights some of the challenges and future directions in the field of social network analysis using graph theory. The study of graph theory and algorithms for social network analysis is of significant importance due to its wide-ranging applications in various fields. Understanding the structure and dynamics of social networks can provide valuable insights into human behavior, information diffusion, organizational dynamics, and the spread of diseases or ideas. By applying graph theory concepts such as centrality measures, community detection algorithms, and link prediction techniques, researchers can uncover hidden patterns, identify influential nodes or groups, and predict future interactions in social networks. This knowledge can be used to design more effective strategies for marketing, public health interventions, social media management, and other areas where understanding and influencing human behavior are critical. Versatile with distinctive options. MOORA is a new optimization approach that has been proposed. This objective method signifies a matrix of potential replies, but it also proposes better policies based on the rates employed. Well established. For comparison, the reference point method is another multi-objective optimization technique. After several competitions, this proved to be the best pick among the methods. From the result network models is in first position whereas Graph density is in 5th position