Due to the prevalent theoretical status and high-performance efficiency in computer networks, face recognition, and heterogeneous data analysis, network distance has attracted significant attention and become an emerging technology in recent years. It is noted that the original difference-based network distance is inadequate for describing network discrepancies by proportionally expanding/contracting the weight function. To mitigate this shortcoming, this paper proposes ratio-based distortion and network distance, and defines a proportional strong/weak isomorphism which is compatible with the new setting. Several results with theoretical underpinning are deduced by means of mathematical analysis. Additionally, this paper conducts a similarity analysis experiment on a randomly generated network structure dataset using the proposed network distance formula. The analysis of the experimental results, including the corresponding hierarchical clustering diagrams and heatmaps, indicates that the proposed network distance has practical value for applications. The code and data of this paper are completely public at https://github.com/AizhEngHN/Ratio-based-distortion-and-network-distance.

A graph G is a fractional (a,b,m)-deleted graph if deleting any m edges from G, the resulting subgraph still admits a fractional [a,b]-factor. As an exclusive graph-based parameter, isolated toughness and its variant are utilized to measure the vulnerability of networks which are modelled by graph topology structures. Early studies have shown the inherent connection between isolated toughness and the existence of fractional factors in specific settings. This paper provides a theoretical perspective on the sufficient conditions for fractional (a,b,m)-deleted graphs with respect to isolated toughness and its variant. The sharpness of the given isolated toughness bounds is explained by counterexamples.

This paper extracts entirely new complex logarithmic prototypes to the nonlinear Black-Scholes equation used to investigate the price in economy. Newly submitted to the literature, (1/G ')-expansion method is used, successfully. Complex, logarithmic and hyperbolic non-traveling wave solutions are extracted to the considered model. Examined is the solution's response for various conformable operator values. To verify the sensitivity of the numerical outcomes, tables with the stability and absolute error analysis findings are also provided. Von Neumann Stability analysis looks at the circumstances in which the model's numerical findings are stable. Critical features, such as stability analysis and error evaluation, are presented, providing comprehensive information on the reliability and accuracy of the method. Errors that arise during the approximation process and the impact of parameter values are displayed through tables and graphs when employing the numerical technique. It offers a thorough comprehension of the financial phenomenon that the model under examination represents.

A graph G is a fractional (k, m)-deleted graph if removing any m edges from G, the resulting subgraph still admits a fractional k-factor. Let k ≥ 2 and m ≥ 1 be integers. Denote ⌊2m k⌋^∗=⌊2m k⌋ if 2m k is not an integer, and ⌊2m k⌋^∗=⌊2m k⌋ - 1 if 2m k is an integer. In this paper, we prove that G is a fractional (k, m)-deleted graph if δ(G) ≥ k + m and isolated toughness meets I( G ) > {3 - 1 m, if k = 2 and m ≥ 3,k + ⌊2m k⌋^*m + 1 - ⌊2m k⌋^*, otherwise. . Furthermore, we show that the isolated toughness bound is tight.

Multi-objective optimization problems (MOPs) aim to obtain a set of Pareto-optimal solutions, and as the number of objectives increases, the quantity of these optimal solutions grows exponentially. However, a plethora of optimal solutions can impose significant decision stress on decision-makers. Subset selection, as the extension of a model, can extract a representative set of solutions, thereby alleviating the decision-makers' choice pressure. In addition, extending a model undoubtedly incurs additional time costs. To cope with the foregoing issues, a fast subset selection method named ranking-based subset selection (RBSS) is proposed in this paper. It can efficiently select a small number of optimal solutions within an unbounded external archive and can be directly applied to any multi-objective evolutionary algorithm. This allows it to maintain good distribution and diversity with very little time investment. We employed a ranking-based approach to map the objective space to a ranking space (an integer space) defined by us and then selected the corresponding subset in the ranking space. The well-behaved mathematical properties of the ranking space and the advantages of using integer calculations accelerated the subset selection process. Experimental results indicate that compared to several state-of-the-art subset selection methods, RBSS is capable of selecting a set of representative and diverse solutions across different types of MOPs, while consuming significantly less time. Specifically, for problems where the Pareto front is a two-dimensional manifold and a one-dimensional manifold, the time consumption of RBSS is approximately only 0.028% to 27.5% and 4.6e-4% to 0.15% of that required by other algorithms, respectively.

The goal of this work is to look at how a nonlinear model describes hematopoiesis and its complexities utilizing commonly used techniques with historical and material links. Based on time delay, the Mackey–Glass model is explored in two instances. To offer a range, the relevance of the parameter impacting stability (bifurcation) is recorded. The power spectrum of the considered model is collected in order to analyze the periodic behavior of a solution in a differential equation. The complex nature of the system is relayed on a parameter which is illustrated in the bifurcation plot. Due to the fact that the considered model is associated with blood‐related diseases, the effect coefficients are effectively captured. The corresponding parameters‐based consequences of the generalized model in different order are deduced. The parametric charts for both examples reveal intriguing results. The current work enables investigations into complex real‐world problems as well as forecasts of essential techniques.

Zhou (2023) introduced the concept of path-factor critical avoidable graph and determined several parameter bounds for (P≥2,n) or (P≥3,n)-factor critical avoidable graphs. Certain instances are listed in Zhou (2023) to show that these toughness or isolated toughness bounds are sharp. However, the derivation processes are flawed. In this remark, we directly modify the derivation processes and obtain the revised bounds. The counterexamples are updated to satisfy κ(G)=n+2. Finally, richness bounds on binding number for a graph to be path-factor critical avoidable are determined.

This paper focuses on the application of an efficient technique, namely, the fractional natural decomposition method (FNDM). The numerical solutions of the model containing the water transport in unsaturated porous media, called Richards equation, are extracted. This model is used to describe the non-locality behaviors which cannot be modeled under the framework of classical calculus. To demonstrate the effectiveness and efficiency of the scheme used, two cases with time-fractional problems are considered in detail. The numerical stimulation is presented with results accessible in the literature, and corresponding consequences are captured with different values of parameters of fractional order. The attained consequences confirm that the projected algorithm is easy to implement and very effective to examine the behavior of nonlinear models. The reliable algorithm applied in this paper can be used to generate easily computable solutions for the considered problems in the form of rapidly convergent series.

A graph G is a fractional (a, b, n)-critical graph if removing any n vertices from G, the resulting subgraph still admits a fractional [a, b] factor. In this paper, we determine the exact tight isolated toughness bound for fractional (a, b, n)-critical graphs. To be specific, a graph G is fractional (a, b, n)-critical if delta (G) >= a + n and I(G) > a - 1 + n+1 /n(a,b ), where na,b >= 2 is an integer satisfies (n(a,b) - 1)a <= b <= n(a,b)a - 1. Furthermore, the sharpness of bounds is showcased by counterexamples. Our contribution improves a result from [W. Gao, W. Wang, and Y. Chen, Tight isolated toughness bound for fractional (k, n) critical graphs, Discrete Appl. Math. 322 (2022), 194-202] which established the tight isolated toughness bound for fractional (k, n) critical graphs.

The motivation of this work is to analyse the nonlinear models and their complex nature with generalized tools associated with material and history-based properties. With the help of well-known and widely used numerical scheme, we study the stimulating behaviours of the financial system in this work. The impact of parameters on price index, rate of interest, investment demand, influence changes and investment cost with respect to saving amount, and the elasticity of commercial markets demand are discussed. The consequences of generalizing the model within the arbitrary order are derived. The existence of the solution for the considered system is presented. This study helps beginner researchers to investigate complex real-world problems and predict the corresponding consequences.

The compressible Murnaghan’s materials composed in the hyperelastic rod describing the far-field equation as the KdV equation for the weak nonliear waves traversing inside the rod is studied with the F expansion method and solved to get the exact elliptic function solutions. The necessary condition for the each Jacobi solutions are reported. The 2D and 3D graphicals are simulated for certain periodic waves. The necessary condition of selection of material and geometric constants for distraction free wave travelling are suggested.

The main aim of the graph model is to implement the structured representation of data, and the hypergraph can be regarded as a promotion of graph which is used in a wider range of data representation scenarios. Bipolar fuzzy sets are employed to describe the uncertainty of the positive and negative of objectives, and fuzzy graphs are modelled to structure the description of uncertain data. In this work, for bipolar Pythagorean fuzzy set (BPFS) and bipolar intuitionistic fuzzy set (BIFS), the corresponding definitions of bipolar Pythagorean fuzzy hypergraph (BPFH) and bipolar intuitionistic fuzzy hypergraph (BIFH) are given, and they are described from the perspective of set theory. The characteristics under the bipolar hypergraph framework of this representation are discussed.

The stability of the network is an important factor considered during the network designing. Network engineers need to determine the structure of the network in advance and confirm that this structure has a certain degree of security. From the perspective of graph topology, the vulnerability of the corresponding network structure can be judged by calculating graph parameters. As an index to measure the stability of the network, the binding number is closely connected to the existence of the fractional factor. This article determines the binding number bounds for fractional (k, m)-covered graphs, and shows that this condition is the best. Then, we correct the counterexample which implies the sharpness of isolated toughness bound for the existence of fractional k-factor. Finally, the sun toughness conditions for a graph to be (P_≥ 2,k) - and (P_≥ 3,k) -factor critical covered are given, and we show that these sun toughness bounds are tight.

In recent years, the study on topological indices related to network security has attracted attention from mathematics and computer science fields. Although several gratifying theoretical results exist, definitions and conclusions under most mathematical frameworks need to be expanded. The focus topic of this paper is to use a graph model to represent the network structure, and the membership functions (MFs) of vertices and edges describe the uncertain nature of stations and channels. The connectivity index is used to describe the robustness and the stability of the fuzzy graph of the corresponding network. In this paper, the definition of the connectivity index and the derived concept are given in a bipolar intuitionistic fuzzy graph (BIFG) setting, and the characteristics of connectivity indices are given from the theoretical point of view. These results have potential applications in the field of computer security, and we analyzed topics for future research.

连通偶[2,2s]-因子H 是G的连通生成子图,且每个顶点在 H 中的度属于{2,4,…,2s}.韧度变种用于衡量网络的易受攻击程度,是网络拓扑设计的重要指标.给出图存在连通偶[2,2s]-因子的韧度变种条件,该结论在网络数据传输中有潜在的应用.

Isolated toughness of graph G is formulated by minimizing the ratio |S|/i(G−S) over all S⊆V(G) with i(G−S)≥2, where i(G−S) is the number of isolated vertices after removing vertex subset S from G. The previous works reveal that there exist explicit correlations between isolated toughness and fractional critical graphs (a.k.a. the graph admits a fractional factor after deleting given number of vertices). However, among the existing isolated toughness bounds, the term with respect to n (the number of removed vertices) is always at least n. In this paper, the exactly sharp isolated toughness bound for fractional (k,n)-critical graph is determined which reveals that the coefficient of term with regard to n can be reduced to 1/2. It is well acknowledged that some tight toughness related bounds can reach to the extreme value, while others cannot. We give an explanation why these tight bounds in various settings possess such pattern differences.

The aim of this work is to present a design of Morlet wavelet neural network (MWNN) for solving a novel prevention category (P) in the HIV system, known as HIPV mathematical model. The numerical performance of the novel HIPV mathematical model will be observed by exploiting the MWNN that works through the optimization procedures of global/local via “genetic algorithm (GA)” and local search “interior-point algorithm (IPA)”, i.e. MWNN-GA-IPA. An error function using the differential HIPV mathematical model and its initial conditions is presented and optimized by the MWNN-GA-IPA. The obtained results have been compared with the Adams method to check the competence of the MWNN-GA-IPA. For the reliability and stability of the scheme, the performance using different statistical operators has been performed based on the multiple independent trials to solve the novel HIPV mathematical model.

In this paper, some new exact traveling and oscillatory wave solutions to the Kudryashov-Sinelshchikov non-linear PDE are investigated by using Bernoulli sub-equation function method. Profiles of obtained solutions are plotted.

In this paper, we analyzed and found the solution for a suitable nonlinear fractional dynamical system that describes coronavirus (2019-nCoV) using a novel computational method. A compartmental model with four compartments, namely, susceptible, infected, reported and unreported, was adopted and modified to a new model incorporating fractional operators. In particular, by using a modified predictor–corrector method, we captured the nature of the obtained solution for different arbitrary orders. We investigated the influence of the fractional operator to present and discuss some interesting properties of the novel coronavirus infection.

Clothing attribute prediction is a fundamental image classification task in the field of computer vision. Motivated by the human recognition system, we investigate the task relationship and spatial importance where people usually utilize these useful clues to assist in recognizing clothing attributes. In this paper, we propose a novel Holistic Relation Network (HRNet) for clothing attribute recognition, considering the fusion of multiple relations, including spatial and spatial relation via a spatial relation module, spatial and task relation via a task attention module, and task and task relation via a graph context reasoning module. Specifically, we first use the backbone network to extract features from the input image, two types of attention models followed will further learn the features, then the graph context reasoning module will be used to further enhance the features, and finally, a classifier exploited to classify the clothing attributes with the learned representation information. Without using manual image feature filtering methods, this paper aims to achieve clothing attribute recognition by deeply exploring the relationships among different clothing attribute recognition tasks. In this paper, we use double-branches of the attention model to model the relevance of spatial context information and learn more discriminative feature representations from multitask features for clothing attribute prediction. Derived from the prior knowledge learned from the two above-mentioned attention models, we further propose a graph-relation model constructing relationships among different clothing attribute tasks by integrating the spatial association relationships among multitask. The proposed HRNet only uses image-level annotation but it owns a good ability for obtaining distinguishing feature representations. We obtain state-of-the-art performance, which is demonstrated by extensive experiments on three mainstream benchmarks, for example, woman clothing data set, man clothing data set, and shop-domain clothing data set.