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Link prediction is a significant and fundamental research issue in the field of network science. Numerous similarity-based algorithms have been widely applied due to low computational cost and high prediction accuracy. The topological features of networks like community structure are beneficial to link prediction. In this paper, we first introduce a Weighted Two-level Neighborhood Similarity (WTNS) index that integrates the resource allocation index and local path index. Then we define the Strength of Community Link (SCL) as a quantitative index to evaluate the close relationship among communities. Based on this, the connection likelihood between two nodes can be calculated and a new link prediction algorithm called SCL-WTNS is presented. The proposed algorithm has been compared with nine popular similarity methods on 12 real-world networks to verify the performance. SCL-WTNS is also compared with two groups of community-based link prediction methods. Experiments indicate that the proposed algorithm is better than comparison algorithms both in prediction accuracy and stability.

Community detection can not only help people understand organizational structure and function of complex networks, but also attributes to many potential applications including targeted advertising and customer relationship management. Due to the low time complexity, the label propagation algorithm is widely used, but there is still room to improve the community quality and the detection stability. Inspired by resource allocation and local path similarity, we first give a new two-level neighbourhood similarity measure called TNS, and on this basis we propose an improved label propagation algorithm for community detection. In this new algorithm, the minimum distance and local centrality index are considered to select the initial community centers, to ensure that they are both important and far away from each other. In the process of forming initial community, we employ the new similarity measure and an optimization strategy of asynchronously updating labels according to node importance. To further improve the accuracy of community division, we introduce the label influence based on the new similarity measure to further optimize the community division of networks. The experimental results on both the artificial network and ten real-world networks show that our proposed algorithm has better comprehensive performance than several existing algorithms in terms of modularity, normalized mutual information and adjusted rand index.

In this paper, by introducing new traveling wave transformations in specific nonlinear forms, a variety of new multiwave interaction solutions for two higher dimensional nonlinear evolution equations with time-dependent coefficients are investigated. These new kinds of multiwave solutions can enrich solutions of nonlinear evolution equations with variable coefficients.

The identification of influential nodes is one of the most significant and challenging research issues in network science. Many centrality indices have been established starting from topological features of networks. In this work, we propose a novel gravity model based on position and neighborhood (GPN), in which the mass of focal and neighbor nodes is redefined by the extended outspreading capability and modified k-shell iteration index, respectively. This new model comprehensively considers the position, local and path information of nodes to identify influential nodes. To test the effectiveness of GPN, a number of simulation experiments on nine real networks have been conducted with the aid of the susceptible–infected–recovered (SIR) model. The results indicate that GPN has better performance than seven popular methods. Furthermore, the proposed method has near linear time cost and thus it is suitable for large-scale networks.

Identifying multiple key spreaders in a network can effectively control the diffusion of information and optimize the use of available resources, which can be also called the influence maximization problem in sociology domains. In order to maximize collective influence in complex networks, multiple spreaders must have both large single influence and small overlapping influence, but it is rather difficult to satisfy these two conditions simultaneously. In this paper, we try to achieve the best compromise between importance and dispersibility for multiple spreaders through clustering. The cluster centers are surrounded by nodes with lower influence, and the distance among different cluster centers is relatively far. In addition, the initial centers selection directly affects the efficiency of clustering and the realization of global optimization. Consequently, we present an initial centers selection algorithm combining H-index and minimum distance. The experimental results on four actual datasets show that the proposed method has better performance than the traditional benchmark methods in terms of transmission speed, diffusion scale and structural characteristics.

How to identify key nodes is a challenging and significant research issue in complex networks. Some existing evaluation indicators of node importance have the disadvantages of limited application scope and one-sided evaluation results. This paper takes advantage of multiple centrality measures comprehensively, by regarding the identification of key nodes as a multi-attribute decision making (MADM) problem. Firstly, a new local centrality (NLC) measure is put forward through considering multi-layer neighbor nodes and clustering coefficients. Secondly, combining the grey relational analysis (GRA) method and the susceptible-infectious-recovered (SIR) model, a modified dynamic weighted technique for order preference by similarity to ideal solution (TOPSIS) method is proposed. Finally, the effectiveness of the NLC is illustrated by applications to nine actual networks. Furthermore, the experimental results on four actual networks demonstrate that the proposed method can identify key nodes more accurately than the existing weighted TOPSIS method.

Identifying the influential nodes in complex networks is a challenging and significant research topic. Though various centrality measures of complex networks have been developed for addressing the problem, they all have some disadvantages and limitations. To make use of the advantages of different centrality measures, one can regard influential node identification as a multi-attribute decision-making problem. In this paper, a dynamic weighted Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is developed. The key idea is to assign the appropriate weight to each attribute dynamically, based on the grey relational analysis method and the Susceptible–Infected–Recovered (SIR) model. The effectiveness of the proposed method is demonstrated by applications to three actual networks, which indicates that our method has better performance than single indicator methods and the original weighted TOPSIS method.

An algorithm is presented to prove the Painleve integrability of parameterized nonlinear evolution equations such that one can filter out Painleve integrable models from nonlinear equations with general forms. Then two well known nonlinear models with physical interests illustrate the effectiveness of this algorithm. Some new results are reported for the first time.

By applying the Painleve test, the Kadomtsev–Petviashvili equation and Jimbo–Miwa equation in (3 + 1)-dimensions are shown to be non-integrable. Through the obtained truncated Painleve expansions, two bilinear equations are constructed. In addition, starting from the bilinear equations, one soliton, two soliton and dromion solutions are also derived. The analysis of the dromions shows that the interactions of the dromions for the (3 + 1)-dimensional equations may be elastic or inelastic.

An elliptic equation method is presented for constructing new types of elliptic function solutions of nonlinear evolution equations. The key idea of this method is to use solutions of an elliptic equation involving four real distinct roots to construct solutions of nonlinear evolution equations. The (3+1)-dimensional modified KdV–ZK equation and Whitham–Broer–Kaup equation are chosen to illustrate the application of the elliptic equation method. Consequently, new elliptic function solutions of rational forms are derived that are not obtained by the previously known methods.

It is proven that generalized coupled higher-order nonlinear Schrodinger equations possess the Painleve property for two particular choices of parameters, using the Weiss–Tabor–Carnevale method and Kruskal’s simplification. Abundant families of periodic wave solutions are obtained by using the Jacobi elliptic function expansion method with the assistance of symbolic manipulation system, Maple. It is also shown that these solutions exactly degenerate to bright soliton, dark soliton and mixed dark and bright soliton solutions with physical interests.

In this paper, an interesting fact is found that the auxiliary equation method is also applicable to a coupled system of two different equations involving both even-order and odd-order partial derivative terms. Furthermore, singular travelling wave solutions can also be obtained by considering other types of exact solutions of auxiliary equation. The Whitham–Broer–Kaup and the (2 + 1)-dimensional Broer–Kaup–Kupershmidt equations are chosen as examples to illustrate the effectiveness of the auxiliary equation method.