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Global static routing is one kind of important routing algorithms for complex networks, especially in large communication networks. In this paper, we propose a heuristic global static routing algorithm to mitigate traffic congestion on two-layer complex networks. The proposed routing algorithm extends the relevant static weighted routing algorithm in the literature [Y. Zhou, Y. F. Peng, X. L. Yang and K. P. Long, Phys. Sci.84, 055802 (2011)]. Our routing path is constructed from a proper assignment of edge weights by considering the static information of both layers and an adjustable parameter α. When this routing algorithm is adopted on BA–BA two-layer networks with an appropriate parameter α, it can achieve the maximum network traffic capacity compared with the shortest path (SP) routing algorithm and the static weighted routing algorithm.

The Beijing road transportation network (BRTN), as a large-scale technological network, exhibits very complex and complicate features during daily periods. And it has been widely highlighted that how statistical characteristics (i.e. average path length and global network efficiency) change while the network evolves. In this paper, by using different modeling concepts, three kinds of network models of BRTN namely the abstract network model, the static network model with road mileage as weights and the dynamic network model with travel time as weights — are constructed, respectively, according to the topological data and the real detected flow data. The degree distribution of the three kinds of network models are analyzed, which proves that the urban road infrastructure network and the dynamic network behavior like scale-free networks. By analyzing and comparing the important statistical characteristics of three models under random attacks and intentional attacks, it shows that the urban road infrastructure network and the dynamic network of BRTN are both robust and vulnerable.

The network topology structure has great impact on its traffic capacity. Most of the biological and technological networks have been proved to exhibit scale-free feature. In real networks, the hub links, which usually bear large traffic loads, have restricted transport efficiency of the complex networks. In this paper, we focus on the bandwidth congestion and propose a rewiring link strategy to optimize traffic dynamics on scale-free networks. Two link rewiring strategies are compared and the simulation results on scale-free networks show that our strategy has more advantages, which makes the degree distribution of the network more uniform, reducing the average path length of different nodes, balancing the resources of the network and increasing the traffic capacity of the network. This work will be beneficial for designing network topology and optimizing network performances.

In this paper, we aim to study the role of hubs in the network coherence quantified by the Laplacian spectra and choose two families of unicyclic and bicyclic networks with the same network size as our network models. In order to investigate the influence of adding links on the coherence, we construct four types of bicyclic networks with the same average degree. Using the network’s regular structures and matrix theories, we obtain analytical solutions of the coherences regarding the degrees of hub nodes. Based on these exact results for the coherence, the network with one hub displays higher coherence compared to the network with two hubs. We then obtain exact relations for the coherences of the bicyclic networks with the same average degree and show that different adding links and hub’s positions are responsible for distinct performance of the consensus algorithms. Finally, we show that the coherence and average path length behave in a linear way meaning that smaller average path length results in better coherence.

It was generally believed that scale-free networks would be small-world. In this paper, two models, named Model A and Model B, are proposed to show that certain scale-free networks can be linear-world instead of small-world. By linear-world, it means that the average path length L of the network grows linearly with the total number of nodes N, i.e., L~N. Model A generates a deterministic scale-free network with high assortativity and numerical simulations demonstrate that the network is linear-world when it satisfies degree exponent λ>1. Model B constructs a partially deterministic scale-free network, which is connected by identical small scale-free networks following certain rules. Analytical arguments and numerical simulations both yield L~N which suggests that it is also linear-world. It is further discussed in this paper that the network generated by Model Bcould be either correlated or uncorrelated. This suggests that, inconsistent with the results in related works, uncorrelated scale-free networks can also be linear-world.

Dendrimer has wide number of important applications in various fields. In some cases during transport or diffusion process, it transforms into its dual structure named Husimi cactus. In this paper, we study the structure properties and trapping problem on a family of generalized dual dendrimer with arbitrary coordination numbers. We first calculate exactly the average path length (APL) of the networks. The APL increases logarithmically with the network size, indicating that the networks exhibit a small-world effect. Then we determine the average trapping time (ATT) of the trapping process in two cases, i.e., the trap placed on a central node and the trap is uniformly distributed in all the nodes of the network. In both case, we obtain explicit solutions of ATT and show how they vary with the networks size. Besides, we also discuss the influence of the coordination number on trapping efficiency.

Considering the link congestion based traffic model, which can more accurately model the traffic diffusing process of many real complex systems such as the Internet, we propose an efficient weighted routing strategy in which each link's weight is assigned with the edge betweenness of the original un-weighted network with a tunable parameter α. As the links with the highest edge betweenness are susceptible to traffic congestion, our routing strategy efficiently redistribute the heavy traffic load from central links to noncentral links. The highest traffic capacity under this new routing strategy is achieved when compared with the shortest path routing strategy and the efficient routing strategy. Moreover, the average path length of our routing strategy is much smaller than that of the efficient routing strategy. Therefore, our weighted routing strategy is preferable to other routing strategies and can be easily implemented through software method.

Considering the heterogeneous structure of scale-free networks causing low traffic capacity of network, we propose to improve the network transport efficiency by rewiring a fraction of edges for the network. In this paper, six edge rewiring strategies are discussed and extensive simulations on Barabási–Albert (BA) scale-free networks confirm the effectiveness of these strategies. From another perspective, rewiring edges for scale-free networks directly reuse the removed edges under some edge-removal strategies [Z. Liu, M. B. Hu, R. Jiang, W. X. Wang and Q. S. Wu, Phys. Rev. E76 (2007) 037101; G. Q. Zhang, D. Wang and G. J. Li, Phys. Rev. E76 (2007) 017101], and can significantly enhance the traffic capacity of the network at the expense of increasing a little average path length. After the edge rewiring process, the network structure becomes significantly homogeneous. This work is helpful for network design and network performance optimization.

Considering that many real networks do not have strict self-similarity property, compared with deterministic evolutionary fractal networks, networks with random sequence structure may be more in accordance with the properties of real networks. In this paper, we generate a hierarchical network by a random sequence based on BRV model. Using the encoding method, we present a way to judge whether two nodes are neighbors and calculate the total path length of the network. We get the degree distribution and limit formula of the average path length of a class of networks, which are obtained by analytical method and iterative calculation.

Dürer’s pentagon is known to the artist Albrecht Dürer, whose work has produced an effect on modern telecommunication. In this paper, we consider directed networks generated by Dürer-type polygons, which is based on an $M$-sided polygon where $M=4m$ and $m\ge 2$. This object is quite different from what we previously studied when $M$ is not a multiple of 4. We aim to study some properties of these networks, such as degree distribution, clustering coefficient and average path length. We show that such networks have the scale-free effect, but do not have the small-world effect. It is expected that our results will provide certain theoretical support to further applications in modern telecommunication.

This paper proposes a novel complex network with disassortative property based on multicenter networks. The average path length and clustering coefficient of the network are calculated, and the impact on the network topology is investigated. A simple dynamic system established on the proposed network is used to analyze how the disassortative property of the network affects synchronization.

Recently, the assortative mixing of complex networks has received much attention partly because of its significance in various social networks. In this paper, a new scheme to generate an assortative growth network with given degree distribution is presented using a Monte Carlo sampling method. Since the degrees of a great number of real-life networks obey either power-law or Poisson distribution, we employ these two distributions to grow our models. The models generated by this method exhibit interesting characteristics such as high average path length, high clustering coefficient and strong rich-club effects.

Many real systems behave similarly with scale-free and small-world structures. In this paper, we generate a special hierarchical network and based on the particular construction of the graph, we aim to present a study on some properties, such as the clustering coefficient, average path length and degree distribution of it, which shows the scale-free and small-world effects of this network.

Dürer’s pentagon is known to the artist Albrecht Dürer. In this paper, we consider directed networks generated by Dürer-type polygons. We aim to present a study on some properties of these networks, such as degree distribution, clustering coefficient and average path length. We show that such networks have the scale-free effect, but do not have the small-world effect.

In this paper, we consider the networks modeled by several self-affine sets based on the Bedford–Mcmullen carpet. We calculate three properties of the networks, including the cumulative degree distribution, the average clustering coefficient and the average path length. We show that such networks have scale-free and small-world effects.

The average path length of a network serves to elucidate the network’s fluency and coherence by providing insights into how efficiently and effectively information or interactions can traverse the network and it is extensively studied in the context of network science. The Fermat distance among three nodes $i$, $j$, and $k$, denoted as $f(i,j,k)$, was defined as the shortest total path length between a node $p$ and nodes $i$, $j$, and $k$. The corresponding average Fermat distance also plays an important role in describing the connectedness of a network.

In this paper, we study a class of polygon networks with pseudo-fractal structure and analyze the average path length. Moreover, we derive the average Fermat distance in two ways. Interestingly, we find the ratio of asymptotic average Fermat distance to asymptotic average path length is exactly 3/2 and these metrics grow linearly with the order of the polygon networks.

Path length calculation is a frequent requirement in studies related to graph theoretic problems such as genetics. Standard method to calculate average path length (APL) of a graph requires traversing all nodes in the graph repeatedly, which is computationally expensive for graphs containing large number of nodes. We propose a novel method to calculate APL for graphs commonly required in the studies of genetics. The proposed method is computationally less expensive and less time-consuming compared to standard method.

On the basis of Koch networks constructed using Koch fractals, we propose a family of Koch networks with novel features including an initial state that is a complete graph with an arbitrary number of nodes as a generalization of a triangle. In the subsequent evolving steps, existing nodes create finite complete graphs. The analytical expressions for some topological and dynamical properties are obtained, including degree distribution, clustering coefficient, average path length, degree correlations, first passage time, average receiving time, mean return time, and mean first passage time. This family of Koch networks follows a power-law distribution, and has a large clustering coefficient, a small average path length, and degree correlations with rich dynamical properties. The mean return time of the new emerging nodes is identical and increases with some network parameter values. Moreover, the average sending time grows with the network order. In addition, the mean first passage time grows linearly with the number of nodes in the large limit of the network order and increases with the dimension of the polygon in the network.