Degree Matrix Research Articles

Soft-label recover based label-specific features learning

Presently, multi-label classification algorithms are mainly based on positive and negative logical labels, which have achieved good results. However, logical labeling inevitably leads to the label misclassification problem. In addition, missing labels are common in multi-label datasets. Recovering missing labels and constructing soft labels that reflect the mapping relationship between instances and labels is a difficult task. Most existing algorithms can only solve one of these problems. Based on this, this paper proposes a soft-label recover based label-specific features learning (SLR-LSF) to solve the above problems simultaneously. Firstly, the information entropy is used to calculate the confidence matrix between labels, and the membership degree of soft labels is obtained by combining the label density information. Secondly, the membership degree and confidence matrix are combined to construct soft labels, and this process not only solves the problem of missing labels but also obtains soft labels with richer semantic information. Finally, in the process of learning specific label features for soft labels. The local smoothness of the labels learned through stream regularization is complemented by the global label correlation, thus improving the classification performance of the algorithm. To demonstrate the effectiveness of the proposed algorithm, we conduct comprehensive experiments on several datasets.

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.

Spectral Bipartition via Gap Cut on DNA Sequences

Deoxyribonucleic Acid (DNA) and graph partitioning are two distinct fields of study which can be linked in the structure of biological networks. Graph partitioning has been extensively studied but not its application in the biological field. This research explored on the application of spectral graph partitioning in DNA splicing, aiming to simulate the cleavage of DNA by performing spectral bipartition on DNA sequences in a DNA splicing system. This research incorporates Fiedler theory and algebraic graph theory, which are commonly utilized in network analysis and the analysis of graph connectivity. Some DNA sequences of even length are selected and expressed in graphical representations. The adjacency matrix, Laplacian matrix, and degree matrix are computed from the graphs, as well as the Fiedler value and Fiedler vector associated with the graphs. Gap cut is used as a method of spectral bipartition which produces two partitions of DNA sequence of unequal lengths. The generalizations of gap cut on DNA sequences of even length are provided as lemmas and theorem.

Identification of Multi-Microgrid Clusters Using Terminal Spectral Clustering Algorithm

In response to the increasing impact of extreme weather on power distribution networks (PDNs), prioritizing resilience is imperative. This study introduces an innovative k-means spectral clustering algorithm to define the boundaries of microgrids (MGs) within a multi-microgrid (MMG) system. The aim is to improve reliability by clustering PDNs into resilient MGs. The power systems are modeled with nodes representing buses, and connections are represented as edges. The analysis involves computing the adjacency matrix, degree matrix, Laplacian matrix, and applying k-means clustering to group buses based on terminal point features. Silhouette coefficients (SC) are calculated to assess the quality of the clustering. The proposed method is tested on three IEEE distribution systems: IEEE 33, 69, and 118 bus systems. Findings reveal distinct clusters within each system with SC values above 0.68, particularly emphasizing the significance of terminal points as the basis for assisting power engineers in decision-making for predetermined grid partitioning.

The Complexity of Octopus Graph, Friendship Graph, and Snail Graph

Graphs are basic structures that represent objects with nodes and relationships between objects with edges. Trees are one of the parts studied in graph theory along with finding the number of spanning trees of a graph such as octopus graph, friendship graph, and snail graph. The complexity of an octopus graph is strongly dependent on the number and length of tentacles, the complexity of a friendship graph is dependent on the number of triangle cycles, and the complexity of a snail graph is dependent on the number of edges and vertices located in the shell-like part of the snail. To calculate the number of spanning trees (τ(G)) of a graph, various calculations can be used, such as the extension of Kirchhoff's formula. The extension of Kirchhoff's formula uses the determinant of the adjacency matrix and degree matrix of the graph complement of a graph. Therefore, this research applies the extension of Kirchhoff's formula to obtain the complexity of octopus graph, friendship graph, and snail graph. From the analysis, it is obtained that for any n≥2, the number of spanning trees of octopus graph and friendship graph are τ(On )=1/5 √5 [((3+√5)/2)^n-((3-√5)/2)^n ] and τ(Fn )=3^n and the number of spanning trees of snail graph is τ(SIn )=2^(n+2)+3n∙2^(n-1) for n≥1.

Merging preferences into the best solution seeking for many-objective optimization problems

Finding out a best solution of the multi-objective optimization problem (MOP) or many-objective optimization problem (MaOP) in the user’s view, is a significant and practical problem. The multi-objective evolutionary algorithms (MOEAs) are applied to solve the MaOPs. Considering many conflicting objectives, they would generate numerous nondominated solutions. Merging the preferences from the user to objectives is an effective way to measure the nondominated solutions and find out the best one (a posterior approach). But the preference relations may be contradictory and highly sparse. In this paper, we propose a process of finding out a best solution with preference relations among attributes (BSPA). In which, we emphasize to test the contradiction and ease the high sparsity of preference relations, then an available preference degree matrix is obtained and merged into the selection of best solution. Based on the proposed process, three methods of NSGA-II-BSPA, P-NSGA-II and P-GA are proposed to solve the best solution of an MOP or MaOP. The experiments are conducted, and the metrics of contradiction probability, feasibility and availability are proposed to measure the BSPA and the vigorous results are obtained. At last, three case studies are adopted to verify the proposed methods’ effectiveness and efficiency. Meanwhile, the hypothesis tests are conducted and the results indicate that the obtained best solutions of proposed methods are satisfiable.

A trust relationship network-based consensus model in large-scale TODIM group decision-making

PurposeIn large-scale group decision-making (LSGDM) situations, existing TODIM group decision-making methods often fail to account for the influence of social network relationships and the bounded rationality of decision-makers (DMs). To address this issue, a new TODIM-based group decision-making method is proposed that considers the current trust relationships among DMs in a large-scale trust relationship network.Design/methodology/approachThis method consists of two main stages. In the first stage, the large-scale group is partitioned into several sub-clusters based on trust relationships among DMs. The dominance degree matrix of each sub-cluster is then aggregated into the large-scale group dominance degree. In the second stage, after aggregating the large-scale group dominance degree, the consensus index is calculated to identify any inconsistent sub-clusters. Feedback adjustments are made based on trust relationships until a consensus is reached. The TODIM method is then applied to calculate the corresponding ranking results. Finally, an illustrative example is applied to show the feasibility of the proposed model.FindingsThe proposed method is practical and effective which is verified by the real case study. By taking into account the trust relationships among DMs in the core process of LSGDM, it indeed has an impact on the decision outcomes. We also specifically address this issue in Chapter Five. The proposed method fully incorporates the bounded rationality of DMs, namely their tendency to accept the opinions of trusted experts, which aligns more with their psychology. The two-stage consensus model proposed in this paper effectively addresses the limitations of traditional assessment-based methods.Originality/valueThis study establishes a two-stage consensus model based on trust relationships among DMs, which can assist DMs in better understanding trust issues in complex decision-making, enhancing the accuracy and efficiency of decisions, and providing more scientific decision support for organizations such as businesses and governments.

Direct reconstruction of the temperature field of lithium-ion battery based on mapping characteristics

Lithium-ion (Li-ion) batteries are susceptible to the working temperature so that the monitoring is essential to the battery internal temperature. The electrochemical-thermal coupling model completely reflects the battery internal behavior. However, the high complexity and multi-field coupling make it difficult to be applied to reconstruct the battery temperature field based on state observer. Based on mapping characteristics, this work proposes a temperature field direct reconstruction scheme. And the electrochemical-thermal coupling model plays a large role in this scheme. Mapping characteristic vectors of the transient temperature field are obtained from the step response model of the Li-ion battery temperature field to charge and discharge current. Furthermore, the quantitative connection of the mapping characteristic vectors between the surface points of and the internal nodes of the battery is characterized by a relevance degree matrix. And a direct reconstruction model of the internal transient temperature field is established. The internal temperature field is reconstructed directly online and in real time via exploiting the temperature measurements on battery surface. In numerical experiments, the three-dimensional temperature field of the Li-ion battery charge and discharge processes is reconstructed. The effects of charge and discharge rates, model mismatch, and measurement errors on the reconstruction results are investigated. During the charge and discharge processes, the maximum values of the transient average error of the reconstructed temperature field are about 0.4 K and 0.3 K, respectively. And the transient average error is generally below 0.8 K when the standard deviation of measurement errors is 0.5 K.

China’s urbanisation evolution and metropolitan area expansion, based on the Prolonged Artificial Nighttime-light Dataset (PANDA, 1984–2020)

ABSTRACT China’s massive urbanisation development will undoubtedly serve as a global reference. Due to the uncertainty of statistical data, nighttime light (NTL) data have emerged as alternative data for urbanisation evaluation. Here, the application of the Prolonged Artificial NTL Dataset of China (1984–2020) has been newly expanded. Total-value statistical, complex NTL index (CNLI), NTL concentration degree, rank-size rule, and Markov transfer matrix were used to systematically mine information about urban change from multiple perspectives. China’s urbanisation exhibited rapid growth and outwards expansion. The total percentage increase in NTL brightness and NTL area was 409.38% and 302.58%, respectively. The Yangtze River Delta urban agglomeration of East China had the fastest urbanisation (CNLI Trend = 0.0057/a, p < 0.01). Low NTL areas transitioned to medium NTL areas, and high and extremely-high NTL areas diffused to medium NTL areas. The cities in Eastern China and Southern China typically exhibited extremely-high type NTL areas, whereas other cities primarily exhibited medium NTL areas. The gaps between city sizes decreased over time (q Trend = – 0.0200/a, p < 0.01). Lowest-ranked cities exhibited the highest stability (>95%) in city size type transition. The spatiotemporal changes in NTL obtained were of great significance for monitoring urban expansion patterns, making government decisions, and quantifying China’s sustainable development.

Simulation-Based Resilience Evaluation for Urban Rail Transit Transfer Stations

Disturbances often occur in transfer stations; however, little is known about the weaknesses of transfer stations and their ability to cope with passenger flows. Therefore, this paper introduces resilience into the study of transfer stations to enhance their emergency response processes and improve the sustainability of URT networks. It establishes a two-level fuzzy evaluation model, using the G1 weighting method, to assess resilience across various scenarios (daily operation, heavy passenger flow, and emergencies) and identify weaknesses; then, corresponding enhancement strategies are proposed. First, factor sets are established according to resilience stages, including rapidity before disturbance, robustness, redundancy, resourcefulness, and rapidity after disturbance. Using the G1 method, the weight matrix for each factor is calibrated, and a membership degree matrix is determined based on their affiliation with the review set. Multiplying the weight matrix and membership degree matrix yields the resilience value. We apply these steps to a representative station with the assistance of Anylogic simulation in calculating the hard-to-obtain data, yielding a peak-hour resilience value of 0.3425, which indicates a “poor” rating in the review set. By combining the peak-hour resilience with resilience curves under different multiples of peak-hour flows, an enhancement prioritization strategy is proposed for the station, which can act as a reference for the management of URT transfer stations.

The application of the game theory combination weighting-normal cloud model to the quality evaluation of surrounding rocks

The status of surrounding rocks dramatically influences the safety of construction workers, so the quality assessment of surrounding rocks has great significance. The uniaxial saturated compressive strength of rock (X1), the quality index of surrounding rock (X2), the frictional coefficient of the structural surface (X3), the joint spacing (X4), the state of groundwater(X5), and the integrity coefficient (X6) are selected as the initial evaluation index. Then, the game theory combination weighting-normal cloud model is introduced. Second, the certainty degree matrix of each index is established, and the weight coefficients of assessment indexes are determined based on the game theory combination weighting method. Finally, the quality level of surrounding rocks is judged. Compared with the traditional methods, the proposed model solves the fuzziness and randomness of different indexes, improves the reliability of the assessment process, and enhances the predictive accuracy of assessment results. In addition, it can provide a solution scheme for the evaluation indicators, which are difficult to quantify, and reduce the influence of human factors. The results obtained from the suggested model are consistent with the current specification. Its accuracy approaches 100%, and the method is feasible for the quality level assessment of surrounding rocks, providing a new technique and approach to assessing the risk level of surrounding rocks.

Ordering of graphs with fixed size and diameter by A α-spectral radii

The A α -matrix of a graph G is defined as the convex linear combination of the adjacency matrix A ( G ) and the diagonal matrix of degrees D ( G ) , i.e. A α ( G ) = αD ( G ) + ( 1 − α ) A ( G ) with α ∈ [ 0 , 1 ] . The maximum modulus among all A α -eigenvalues is called the A α -spectral radius. In this paper, we order the connected graphs with size m and diameter (at least) d from the second to the ( ⌊ d 2 ⌋ + 1 ) th regarding to the A α -spectral radius for α ∈ [ 1 2 , 1 ) . As by-products, we identify the first ⌊ d 2 ⌋ largest trees of order n and diameter (at least) d in terms of their A α -spectral radii, and characterize the unique graph with at least one cycle having the largest A α -spectral radius among graphs of size m and diameter (at least) d. Consequently, the corresponding results for signless Laplacian matrix can be deduced as well.

Using Fuzzy C-Means Clustering to Determine First Arrival of Microseismic Recordings.

Accurate and automatic first-arrival picking is one of the most crucial steps in microseismic monitoring. We propose a method based on fuzzy c-means clustering (FCC) to accurately divide microseismic data into useful waveform and noise sections. The microseismic recordings' polarization linearity, variance, and energy are employed as inputs for the fuzzy clustering algorithm. The FCC produces a membership degree matrix that calculates the membership degree of each feature belonging to each cluster. The data section with the higher membership degree is identified as the useful waveform section, whose first point is determined as the first arrival. The extracted polarization linearity improves the classification performance of the fuzzy clustering algorithm, thereby enhancing the accuracy of first-arrival picking. Comparison tests using synthetic data with different signal-to-noise ratios (SNRs) demonstrate that the proposed method ensures that 94.3% of the first arrivals picked have an error within 2 ms when SNR = -5 dB, surpassing the residual U-Net, Akaike information criterion, and short/long time average ratio approaches. In addition, the proposed method achieves a picking accuracy of over 95% in the real dataset tests without requiring labelled data.

Maxima of the [formula omitted]-index of graphs with given size and domination number

The Aα-matrix of a graph G was defined by Nikiforov in 2017 as Aα(G)=αD(G)+(1−α)A(G), where α∈[0,1], D(G) and A(G) are the diagonal matrix of degrees and the adjacency matrix respectively. The largest eigenvalue of Aα(G) is called Aα-index of G. In this paper, we completely determine the extremal graphs with maximal Aα-index among all graphs with size m , domination number γ and no isolated vertices for α∈[12,1).

A note on products of idempotents in the ring of upper triangular infinite matrices

It is known that every singular matrix of a finite degree defined over a field can be written as a product of a finite number of idempotent matrices. Insipired by this result we investigate the same problem for N × N matrices. It turns out that in this case the analogous theorem does not hold.

Algebraic spectral analysis of the locating degree matrix of a graph

In this article, we present a new graph matrix based on locating degree, we call it the locating degree matrix. The spectrum and energy related to this new matrix are established for certain classes of graphs. Bounds of the new related energy for a special type of regular graph are obtained.

Sparse random hypergraphs: non-backtracking spectra and community detection

Abstract We consider the community detection problem in a sparse $q$-uniform hypergraph $G$, assuming that $G$ is generated according to the Hypergraph Stochastic Block Model (HSBM). We prove that a spectral method based on the non-backtracking operator for hypergraphs works with high probability down to the generalized Kesten–Stigum detection threshold conjectured by Angelini et al. (2015, Spectral detection on sparse hypergraphs. In: 2015 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton). IEEE, pp. 66–73). We characterize the spectrum of the non-backtracking operator for the sparse HSBM and provide an efficient dimension reduction procedure using the Ihara–Bass formula for hypergraphs. As a result, community detection for the sparse HSBM on $n$ vertices can be reduced to an eigenvector problem of a $2n\times 2n$ non-normal matrix constructed from the adjacency matrix and the degree matrix of the hypergraph. To the best of our knowledge, this is the first provable and efficient spectral algorithm that achieves the conjectured threshold for HSBMs with $r$ blocks generated according to a general symmetric probability tensor.

The application of gray system-variable fuzzy sets coupling theory on the susceptibility assessment of debris flow hazards

The occurrence of debris flow hazards has significant destructive effects people’s safety and the safety of their property, so the risk assessment of debris flow hazards has great significance. The maximum amount of debris flow at one time (X1), the reserves of loose substance (X2), the watershed area (X3), the length of the ditch (X4), the relative height difference (X5), the gradient of side slope (X6), the longitudinal slope of main ditch (X7), the length ratio of replenishment section (X8), the daily maximum rainfall (X9), and Faults developed in the watershed (X10) are selected as the evaluation index at first. Then the gray system-variable fuzzy sets coupling model is introduced; secondly, the relative membership degree matrix about the fuzzy variable sets is established, and the weight coefficients considering the uncertainty of assessment indices are determined based on the entropy weight method; finally, the susceptibility level of debris flow hazards is determined using the mean ranking feature value. The conclusion is drawn that the results obtained based on the coupling technique are consistent with the current specification; its accuracy arrives at 83%, and the method is feasible for the susceptibility level assessment of debris flow hazards. Compared to the traditional variable fuzzy sets model, it avoids the uncertainty of magnitude in the standard analysis, it can ensure the objectivity and accuracy of the calculation result, and its calculation process is simple and efficient. In total, the findings of the proposed model provide an alternate way to assess the susceptibility level of debris flow hazards and improve the evaluation accuracy in the future.

Counting roots of fully triangular polynomials over finite fields

Let Fq be a finite field with q elements, f∈Fq[x1,…,xn] a polynomial in n variables and let us denote by N(f) the number of roots of f in Fqn. In this paper we consider the family of fully triangular polynomials, i.e., polynomials of the formf(x1,…,xn)=a1x1d1,1+a2x1d1,2x2d2,2+…+anx1d1,n⋯xndn,n−b, where di,j>0 for all 1≤i≤j≤n. For these polynomials, we obtain explicit formulas for N(f) when the augmented degree matrix of f is row-equivalent to the augmented degree matrix of a linear polynomial or a quadratic diagonal polynomial.

Some extremal problems on [formula omitted]-spectral radius of graphs with given size

Nikiforov defined the Aα-matrix of a graph G as Aα(G)=αD(G)+(1−α)A(G), where α∈[0,1], D(G) and A(G) are the diagonal matrix of degrees and the adjacency matrix respectively. The largest eigenvalue of Aα(G) is called the Aα-spectral radius of G, denoted by ρα(G). In this paper, we first give an upper bound on ρα(G) of a connected graph G with fixed size m≥3k and maximum degree Δ≤m−k, where k is a positive integer. For two connected graphs G1 and G2 with size m≥4, employing this upper bound, we prove that ρα(G1)>ρα(G2) if Δ(G1)>Δ(G2) and Δ(G1)≥2m3+1. As an application, we determine the graph with the maximal Aα-spectral radius among all graphs with fixed size and girth. Our theorems generalize the recent results for the signless Laplacian spectral radius of a graph.