Degree distribution and robustness of cooperative communication network with scale-free model

With the requirements of users enhanced for wireless communication, the cooperative communication will become a development trend in future. In this paper, a model based on complex networks with both preferential attachment is researched to solve an actual network CCN （Cooperative Communication Network）. Firstly, the evolution of CCN is given by four steps with different probabilities. At the same time, the rate equations of nodes degree are presented to analyze the evolution of CCN. Secondly, the degree distribution is analyzed by calculating the rate equation and numerical simulation. Finally, the robustness of CCN is studied by numerical simulation with random attack and intentional attack to analyze the effects of degree distribution and average path length. The results of this paper are more significant for building CCN to programme the resource of communication.

Degree distribution of random birth-and-death network with network size decline

In this paper,we provide a general method to obtain the exact solutions of the degree distributions for random birthand-death network（RBDN） with network size decline.First,by stochastic process rules,the steady state transformation equations and steady state degree distribution equations are given in the case of m ≥ 3 and 0 〈 p 〈 1/2,then the average degree of network with n nodes is introduced to calculate the degree distributions.Specifically,taking m = 3 for example,we explain the detailed solving process,in which computer simulation is used to verify our degree distribution solutions.In addition,the tail characteristics of the degree distribution are discussed.Our findings suggest that the degree distributions will exhibit Poisson tail property for the declining RBDN.

THE STUDY ON DEGREE DISTRIBUTION PROPERTY OF COMPLEX NETWORK BASED ON COOPERATIVE COMMUNICATION

Complex networks have been widely studied. Recently,many results show that the degree distributions of some large networks follow the form of power-law and these networks possess better robustness against random nodes failure. As an effective technology on combating the channel fading,wireless cooperative communication is becoming one of the most important methods to improve the wireless communication performances. In this paper,the complex network models based on cooperative communication and non-cooperative communication are established; and the degree distribution properties for them are studied. The simulation results show that the degree distributions of these networks also follow the form of power-law,which means that the addition of cooperative communi-cation links will not change the property of degree distribution and then these networks will possess better robustness against random nodes failure as well.

NEW METHOD TO ESTIMATE SCALING EXPONENTS OF POWER-LAW DEGREE DISTRIBUTION AND HIERARCHICAL CLUSTERING FUNCTION FOR COMPLEX NETWORKS

A new method and corresponding numerical procedure are introduced to estimate scaling exponents of power-law degree distribution and hierarchical clustering function for complex networks. This method can overcome the biased and inaccurate faults of graphical linear fitting methods commonly used in current network research. Furthermore, it is verified to have higher goodness-of-fit than graphical methods by comparing the KS （Kolmogorov-Smirnov） test statistics for 10 CNN （Connecting Nearest-Neighbor） networks.

Performance of degree distribution based HARQ scheme for LDPC-coded OFDM system

This article studies the degree distribution property of low density parity check （LDPC） codes by Gaussian approximation （GA） and presents an efficient hybrid automatic repeat quest （HARQ） scheme for LDPC-coded orthogonal frequency division multiplexing （OFDM） system. In the scheme, the important bits with large degrees have high retransmission priorities and are mapped to the sub-carriers with better channel quality indicator （CQI） levels in the OFDM system. The new scheme provides more protection to the bits with large degrees and thus contributes more to the decoding process by offering more transmission power. In this way the system performance would be improved. The statistics and simulation results also prove the new scheme.

Markov Chain-based Degree Distributions of Evolving Networks

In this paper, we study a class of stochastic processes, called evolving network Markov chains, in evolving networks. Our approach is to transform the degree distribution problem of an evolving network to a corresponding problem of evolving network Markov chains. We investigate the evolving network Markov chains, thereby obtaining some exact formulas as well as a precise criterion for determining whether the steady degree distribution of the evolving network is a power-law or not. With this new method, we finally obtain a rigorous, exact and unified solution of the steady degree distribution of the evolving network.

Degree Distribution of a Scale-free Random Graph Model

In this paper, we consider the degree distribution of a general random graph with multiple edges and loops from the perspective of probability. Based on the first-passage probability of Markov chains, we give a new and rigorous proof to the existence of the network degree distribution and obtain the precise expression of the degree distribution. The analytical results are in good agreement with numerical simulations.

The Degree Distribution of the Random Multigraphs

In this paper, as a generalization of the binomial random graph model, we define the model of multigraphs as follows： let G（n; {pk}） be the probability space of all the labelled loopless multigraphs with vertex set V = {v1, v2, ..., vn }, in which the distribution of tvi,vj, the number of the edges between any two vertices vi and vj is P{tvi,vj =k}=Pk, k=0, 1,2,...and they are independent of each other. Denote by Xd = Xd（G）,Yd = Yd（G）, Zd = Zd（G） and Zcd = Zcd（G） the number of vertices of G with degree d, at least d, at most d and between c and d. In this paper, we discuss the distribution of Xd, Yd, Zd and Zcd in the probability space G（n; （Pk））.

Impact of Distribution Fairness Degree and Entanglement Degree on Cooperation

We explore the impact of distributional fairness degree and entanglement degree on the cooperation between different players by investigating a modified prisoner＇s dilemma game. We not only introduce a new concept of distributional fairness degree, but also obtain the cooperation conditions for overcoming dilemma in terms of fairness and entanglement inequalities. It is demonstrated that distributional fairness can be of fundamental importance to promote cooperation with the help of quantum entanglement.

THE MAXIMUM AND MINIMUM DEGREES OF RANDOM BIPARTITE MULTIGRAPHS

In this paper the authors generalize the classic random bipartite graph model, and define a model of the random bipartite multigraphs as follows：let m = m（n） be a positive integer-valued function on n and ζ（n,m;{pk}） the probability space consisting of all the labeled bipartite multigraphs with two vertex sets A ={a_1,a_2,...,a_n} and B = {b_1,b_2,...,b_m}, in which the numbers t_（ai）,b_j of the edges between any two vertices a_i∈A and b_j∈ B are identically distributed independent random variables with distribution P{t_（ai）,b_j=k}=pk,k=0,1,2,...,where pk ≥0 and ∞Σk=0 pk=1. They obtain that X_（c,d,A）, the number of vertices in A with degree between c and d of G_（n,m）∈ζ（n, m;{pk}） has asymptotically Poisson distribution, and answer the following two questions about the space ζ（n,m;{pk}） with {pk} having geometric distribution, binomial distribution and Poisson distribution, respectively. Under which condition for {pk} can there be a function D（n） such that almost every random multigraph G_（n,m）∈ζ（n,m;{pk}） has maximum degree D（n）in A？ under which condition for {pk} has almost every multigraph G（n,m）∈ζ（n,m;{pk}） a unique vertex of maximum degree in A？

Degree of polarization based on the three-component pBRDF model for metallic materials

An expression of degree of polarization（DOP） for metallic material is presented based on the three-component polarized bidirectional reflectance distribution function（p BRDF） model with considering specular reflection, directional diffuse reflection and ideal diffuse reflection. The three-component p BRDF model with a detailed reflection assumption is validated by comparing simulations with measurements. The DOP expression presented in this paper is related to surface roughness, which makes it more reasonable in physics. Test results for two metallic samples show that the DOP based on the three-component p BRDF model accords well with the measurement and the error of existing DOP expression is significantly reduced by introducing the diffuse reflection. It indicates that our DOP expression describes the polarized reflection properties of metallic surfaces more accurately.

Uncertainty entropy-based exploratory evaluation method and its applications on missile effectiveness evaluation

Some attributes are uncertain for evaluation work because of incomplete or limited information and knowledge.It leads to uncertainty in evaluation results.To that end,an evaluation method,uncertainty entropy-based exploratory evaluation(UEEE),is proposed to guide the evaluation activities,which can iteratively and gradually reduce uncertainty in evaluation results.Uncertainty entropy(UE)is proposed to measure the extent of uncertainty.First,the belief degree distributions are assumed to characterize the uncertainty in attributes.Then the belief degree distribution of the evaluation result can be calculated by using uncertainty theory.The obtained result is then checked based on UE to see if it could meet the requirements of decision-making.If its uncertainty level is high,more information needs to be introduced to reduce uncertainty.An algorithm based on the UE is proposed to find which attribute can mostly affect the uncertainty in results.Thus,efforts can be invested in key attribute(s),and the evaluation results can be updated accordingly.This update should be repeated until the evaluation result meets the requirements.Finally,as a case study,the effectiveness of ballistic missiles with uncertain attributes is evaluated by UEEE.The evaluation results show that the target is believed to be destroyed.

The Evolving Bipartite Network and Semi-Bipartite Network Models with Adjustable Scale and Hybrid Attachment Mechanisms

The bipartite graph structure exists in the connections of many objects in the real world, and the evolving modeling is a good method to describe and understand the generation and evolution within various real complex networks. Previous bipartite models were proposed to mostly explain the principle of attachments, and ignored the diverse growth speed of nodes of sets in different bipartite networks. In this paper, we propose an evolving bipartite network model with adjustable node scale and hybrid attachment mechanisms, which uses different probability parameters to control the scale of two disjoint sets of nodes and the preference strength of hybrid attachment respectively. The results show that the degree distribution of single set in the proposed model follows a shifted power-law distribution when parameter r and s are not equal to 0, or exponential distribution when r or s is equal to 0. Furthermore, we extend the previous model to a semi-bipartite network model, which embeds more user association information into the internal network, so that the model is capable of carrying and revealing more deep information of each user in the network. The simulation results of two models are in good agreement with the empirical data, which verifies that the models have a good performance on real networks from the perspective of degree distribution. We believe these two models are valuable for an explanation of the origin and growth of bipartite systems that truly exist.

Spectral clustering eigenvector selection of hyperspectral image based on the coincidence degree of data distribution

Spectral clustering is a well-regarded subspace clustering algorithm that exhibits outstanding performance in hyperspectral image classification through eigenvalue decomposition of the Laplacian matrix.However,its classification accuracy is severely limited by the selected eigenvectors,and the commonly used eigenvectors not only fail to guarantee the inclusion of detailed discriminative information,but also have high computational complexity.To address these challenges,we proposed an intuitive eigenvector selection method based on the coincidence degree of data distribution(CDES).First,the clustering result of improved k-means,which can well reflect the spatial distribution of various types was used as the reference map.Then,the adjusted Rand index and adjusted mutual information were calculated to assess the data distribution consistency between each eigenvector and the reference map.Finally,the eigenvectors with high coincidence degrees were selected for clustering.A case study on hyperspectral mineral mapping demonstrated that the mapping accuracies of CDES are approximately 56.3%,15.5%,and 10.5%higher than those of the commonly used top,high entropy,and high relevance eigenvectors,and CDES can save more than 99%of the eigenvector selection time.Especially,due to the unsupervised nature of k-means,CDES provides a novel solution for autonomous feature selection of hyperspectral images.

The complexity nature of large-scale software systems

In software engineering, class diagrams are often used to describe the system＇s class structures in Unified Modelling Language （UML）. A class diagram, as a graph, is a collection of static declarative model elements, such as classes, interfaces, and the relationships of their connections with each other. In this paper, class graphs axe examined within several Java software systems provided by Sun and IBM, and some new features are found. For a large-scale Java software system, its in-degree distribution tends to an exponential distribution, while its out-degree and degree distributions reveal the power-law behaviour. And then a directed preferential-random model is established to describe the corresponding degree distribution features and evolve large-scale Java software systems.

A novel evolving scale-free model with tunable attractiveness

In this paper, a new evolving model with tunable attractiveness is presented. Based on the Barabasi-Albert （BA） model, we introduce the attractiveness of node which can change with node degree. Using the mean-field theory, we obtain the analytical expression of power-law degree distribution with the exponent γ∈ （3, ∞）. The new model is more homogeneous and has a lower clustering coefficient and bigger average path length than the BA model.

Time-varying networks based on activation and deactivation mechanisms

A class of models for activity-driven networks is proposed in which nodes vary in two states： active and inactive. Only active nodes can receive links from others which represent instantaneous dynamical interactions. The evolution of the network couples the addition of new nodes and state transitions of old ones. The active group changes with activated nodes entering and deactivated ones leaving. A general differential equation framework is developed to study the degree distribution of nodes of integrated networks where four different schemes are formulated.

Irregular and regular LDPC codes with high spectrum efficiency modulation in image transmission over fading channel

If the degree distribution is chosen carefully, the irregular low-density parity-check (LDPC) codes can outperform the regular ones. An image transmission system is proposed by combining regular and irregular LDPC codes with 16QAM/64QAM modulation to improve both efficiency and reliability. Simulaton results show that LDPC codes are good coding schemes over fading channel in image communication with lower system complexity. More over, irregular codes can obtain a code gain of about 0.7 dB compared with regular ones when BER is 10 -4. So the irregular LDPC codes are more suitable for image transmission than the regular codes.

A difference equation approach to statistical mechanics ofcomplex networks

In this paper, we propose a difference equation approach to the estimation of the degree distributions in growing networks after having analyzed the disadvantages of some existing approaches. This approach can avoid logic conflicts caused by the continuum of discrete problems, and does not need the existence assumption of the stationary degree distribution in the network analysis. Using this approach, we obtain a degree distribution formula of the Poisson growth and preferential attachment network. It is rigorously shown that this network is scale-free based on the Poisson process theory and properties of F-distribution.

The distribution of degree of crystallinity-New concept in polymer science

Due to the structure characteristics of huge macromolecular size and the very low motion feature of the polymer chain which is very difficult to arrange regularly their long chain into the three dimensional space,thus the polymer will incompletely crystallize except for the macroscopic single crystal of polydiacetylene polymerized by means of solid state crystalline polymerization.