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Abstract

In practical application, the generation and evolution of many real networks always do not follow rigorous mathematical model, making network topology optimization a great challenge in the field of complex networks. In this research, we optimize the topology of non-scale-free networks by turning it into scale-free networks using a nonlinear preferential rewiring method. For different kinds of original networks generated by Watts and Strogatz model, we systematically demonstrate the optimization process and the modified networks to verify the performance of nonlinear preferential rewiring. We conduct further researches to explore the effect of nonlinear preferential rewiring’s parameters on performance. Simulation results show that various non-scale-free networks with different network topologies generated by WS model, including random networks and various networks between regular and random, are turned into scale-free networks perfectly by nonlinear preferential rewiring method.

Keywords

Complex networks topology optimization scale-free nonlinear preferential rewiring

Introduction

The interdisciplinary field of complex networks is a young and attractive area of scientific research. It is generally accepted that network science is an important tool to model and explain many real complex systems in the world, for instance, the World Wide Web,^1,2 power grid,^3–5 social collaborations,^6–8 and protein–protein interaction.^9–11 However, practical networks are always difficult to be described accurately and completely. One of the great challenges in understanding the structure and function of networks is to propose effective mathematical models, which at least capture the key properties of the entire networks.

Erdős–Rényi (ER) random graph model is a simple and useful model to generate random graphs which is based on probabilistic theory.^12–14 ER model assumes that all pairs of nodes are connected with the same probability. ER graphs tend to have low clustering coefficients and short average path length. However, though the importance of ER model should not be underestimated, many practical networks do not follow the ER model because of its oversimplified assumption. Networks generated by ER model do not capture two important properties which are generally observed in real networks: the local clustering and the formation of hubs. The collaboration graph of film actors and the electrical power grid are shown to have short average path lengths as well as high clustering coefficients, which is always called “small-world” networks according to the six degrees of separation theory.^15–17 Watts and Strogatz (WS) model is proposed to generate graphs with small-world properties. It is generally accepted that most practical networks are neither regular nor random graphs, and WS model can produce between random and regular networks by random rewiring from a regular lattice. The degree distribution of graphs produced by WS model converges to a Poisson distribution. In contrast, real complex networks often have non-homogeneous structures that demonstrate a power-law degree distribution, and these networks are often called scale-free networks.^18–20 Barabási–Albert (BA) model is proposed for the first time to generate scale-free networks according to two critical mechanisms: growth and preferential attachment. In BA model, a node is more likely to connect to nodes with higher degrees. In addition, there are several mathematical models have been proposed based on BA model, such as the local-world evolving network model,^21,22 and the model of growing scale-free networks with tunable clustering.^23,24

While BA model and its modifications and extensions do capture the property of power-law degree distribution, there are still several limitations in technological application. For instance, when constructing sensor networks,^25,26 there are usually several sensors need to be added simultaneously rather than one-by-one as proposed in BA model. In addition, it is possible that these sensors waiting to be added already have links between them. The demand for global information to implement preferential attachment in constructing process is also unrealized. In total, it is difficult to construct a scale-free sensor network practically by applying BA model. However, scale-free networks have an excellent property—the robustness to failure, and this property is critical to sensor networks. As a result, it is important to optimize network topology by turning non-scale-free networks into scale-free networks. To the best of our knowledge, there is little attention has been paid to this field. It is shown that in socio-ecological systems, rewiring random networks with bounded rationality would lead to scale-free networks in the process of converging toward Nash equilibria.²⁷ This method optimizes the topology of random networks at the expense of high complexity. Furthermore, it is shown that by preferential rewiring, complex networks can reach a stationary state, even with a power-law distribution.^28,29 However, the existing preferential rewiring methods are sensitive to parameter selection and demonstrate a poor performance in power-law distribution when degree is large.^28,29 Herein, we propose a novel method to turn non-scale-free networks into scale-free networks by a new nonlinear preferential rewiring (NPR) method.

The rest of this research is organized in the following manner. Section “NPR method” introduces the mechanism of NPR method. Section “Modification of networks by NPR method” describes the experiments which are conducted to test the performance of NPR in network topology optimization. Section “Exploring the effect of NPR’s parameters” explores the effect of NPR’s parameters on the optimization performance. The conclusions and discussions of this study are presented in section “Conclusions and discussions.” Dynamic features of whole rewiring process are presented in Supplemental information.

NPR method

WS model is used to generate non-scale-free networks here.¹⁷ There are three critical parameters in WS model: the desired number of nodes (N), the mean degree (K, assumed to be an even integer), and rewiring probability $β$ $(0 \leq β \leq 1)$ . WS model first constructs a regular ring lattice with N nodes and each connects to K neighbors and then rewires every pair edge (n_i, n_j) by edge (n_i, n_k) with the rewiring probability $β$ . In addition, the kth node is chosen uniformly from all other possible nodes except the ith node itself (k ≠ i). $β = 0$ and $β = 1$ result in regular and random networks, respectively. As a result, the uniform rewiring mechanism is used in WS model and no global information is introduced in topology modification process.

For a given complex network $G_{0} (N, E)$ with $N$ nodes and $E$ edges, we can get a modified network by NPR, without changing the number of nodes and edges. NPR is implemented according to the following way, and the flow chart of NPR method is presented in Supplemental Figure 1:

For every node Node₁, if its degree is equal or greater than two, randomly choose a node Node₂ that is connected to the selected node Node₁.

Choose a node Node₃ with a probability that is an increasing function of the number of links that the existing nodes already have.

A rewiring is implemented by replacing the link between Node₁ and Node₂ with a link between Node₂ and Node₃ if the following three conditions are satisfied: the degree of Node₁ and Node₃ is not equal; Node₃ is a different node from Node₁ and Node₂; and there is no link between Node₂ and Node₃.

After all nodes are processed, a modified network is generated. In order to generate a complex network with scale-free degree distribution, the processes 1, 2, and 3 should be repeated several times till a termination condition is satisfied. In addition, the function about choosing $Nod e_{3}$ and the termination condition is clarified in the following text.

First, we reorder all the nodes by sorting their degree in ascending order. Resorted nodes and their corresponding degree are denoted by $N_{k}$ and $D_{k}$ , respectively, where $k = 1, 2, \dots, N$ and $D_{i} \leq D_{j}$ if $i \leq j$ . The cumulative sum of $D_{k}$ can be presented as a vector $C_{1}$ . The global information is introduced into topology optimization process using the cumulative sum of sorted node degree

C_{1} = {C_{1} (1), C_{1} (2), \dots, C_{1} (N)}

(1)

where $C_{1} (k)$ is the sum of degree $D_{1}, D_{2}, \dots, D_{k}$

C_{1} (k) = \sum_{i = 1}^{k} D_{k}

(2)

here, we introduce a positive constant $α$ , which can be regarded as preferential rewiring index (PRI), in order to choose $Nod e_{3}$ . A modified vector of $C_{1}$ is presented as $C_{2}$

C_{2} = {C_{2} (0), C_{2} (1), C_{2} (2), \dots, C_{2} (N)}

(3)

C_{2} (k) = {(\frac{C_{k} (k)}{C_{1} (N)})}^{α}

(4)

where $C_{2} (0) = 0$ .

The probability of choosing node $N_{k}$ as $Nod e_{3}$ is calculated as follows

P (N_{k}) = C_{2} (k) - C_{2} (k - 1)

(5)

If these processes are repeated $M$ iterations, we would get a series of modified networks

G = {G_{1} (N, E), G_{2} (N, E), \dots, G_{M} (N, E)} .

(6)

Each network can be represented by a symmetric adjacent matrix

G_{m} (N, E) = [\begin{matrix} G_{m} (1, 1) & \dots & G_{m} (1, N) \\ ⋮ & ⋱ & ⋮ \\ G_{m} (N, 1) & \dots & G_{m} (N, N) \end{matrix}]

(7)

where $m = 0, 1, 2, \dots, M$ . The elements of the adjacent matrix indicate whether pairs of nodes are adjacent or not in the network.

The change between network $G_{m - 1} (N, E)$ and network $G_{m} (N, E)$ is denoted by $H_{m} (N, E)$

H_{m} (N, E) = G_{m - 1} (N, E) - G_{m} (N, E)

(8)

H_{m} (N, E) = [\begin{matrix} G_{m - 1} (1, 1) - G_{m} (1, 1) & \dots & G_{m - 1} (1, N) - G_{m} (1, N) \\ ⋮ & ⋱ & ⋮ \\ G_{m - 1} (N, 1) - G_{m} (N, 1) & \dots & G_{m - 1} (N, N) - G_{m} (N, N) \end{matrix}]

(9)

H_{m} (N, E) = [\begin{matrix} H_{m} (1, 1) & \dots & H_{m} (1, N) \\ ⋮ & ⋱ & ⋮ \\ H_{m} (N, 1) & \dots & H_{m} (N, N) \end{matrix}]

(10)

$H_{m} (a, b) = 1$ denotes that $Nod e_{a}$ is connected to $Nod e_{b}$ in network $G_{m - 1} (N, E)$ , whereas the link is removed in network $G_{m} (N, E)$ . $H_{m} (a, b) = - 1$ denotes that there is no link between $Nod e_{a}$ and $Nod e_{b}$ in network $G_{m - 1} (N, E)$ , whereas $Nod e_{a}$ is connected to $Nod e_{b}$ in network $G_{m} (N, E)$ . $H_{m} (a, b) = 0$ denotes that the relationship between $Nod e_{a}$ and $Nod e_{b}$ is not changed from network $G_{m - 1} (N, E)$ to network $G_{m} (N, E)$ .

The number of changed links is denoted by $K_{m} (N, E)$

K_{m} (N, E) = \frac{1}{4} \sum_{a = 1}^{N} \sum_{b = 1}^{N} | H_{m} (a, b) |

(11)

A rate describing the changed links in the transformation from network $G_{m - 1} (N, E)$ to network $G_{m} (N, E)$ is defined as follows

R_{m} (N, E) = \frac{K_{m} (N, E)}{N}

(12)

The value range of $R_{m} (N, E)$ is $[0, 1]$ . $R_{m} (N, E)$ reflects the changes in mth iteration in topology optimization process. A termination condition rate (TCR) is set as the threshold of $R_{m} (N, E)$ . NPR method finishes when the rule of $R_{m} (N, E) \leq \tilde{R}$ is satisfied. The number of iterations is effected by original network and parameters of PRI and TCR. The complexity of topology optimization for each iteration is O(n²), where n is the number of nodes in network.

Modification of networks by NPR method

WS model is used to generate non-scale-free networks. Two different experiments are implemented to show the process of this method and the modification of networks. In order to visually demonstrate the change in the topology between original non-scale-free networks and the networks modified by NPR method, the number of nodes is set relatively small. Each experiment is repeated 30 times independently. The parameters are set according to Table 1. The visualization of networks is organized by Gephi software.³⁰

Table 1.

Parameters of WS model and NPR method.

Experiment no.	Repeated times	WS model			NPR method		Power-law exponent $γ$
Experiment no.	Repeated times	N	K	$β$	PRI(α)	$TCR (\tilde{R})$	Power-law exponent $γ$
1	30	200	4	1.0	6	0.2	–1.795
2	30	200	4	0.1	6	0.2	–1.717

WS: Watts and Strogatz; NPR: nonlinear preferential rewiring; PRI: preferential rewiring index.

The results of the first experiment are illustrated in Figure 1. The original complex network is a non-scale-free network, actually a random network, generated by WS model with parameters of N = 200, K = 4, and $β = 1$ , whose topology is shown in Figure 1(a). The network modified by NPR method with parameters of $α = 6$ , $\tilde{R} (N, E) = 0.2$ , is shown in Figure 1(b). In both Figure 1(a) and (b), the size of nodes is proportional to its degree. Both complex networks have 200 nodes and 400 links. In the process of preferential rewiring, some links in the original non-scale-free network are removed and some links are added at the same time. A part of the modifications of network are illustrated in Figure 1(c), where 50 nodes are shown. Red dots denote the links removed from original network, and Green dots denote the links added in the network modified by NPR method. The average degree distributions of 30 independent experiments are illustrated in Figure 1(d). The degree distribution of the original network follows a passion distribution, while the degree distribution of the modified network follows a scale-free distribution, with a power-law exponent $γ = - 1.795$ as denoted by the fitting line.

Figure 1.

Original and modified networks. (a) Topology of non-scale-free network generated by WS model with parameters of N = 200, K = 4, and $β = 1$ . (b) Topology of the network modified by NPR method. (c) Modifications of network. (d) Degree distribution of the original network and modified network.

The results of the second experiments are shown in Figure 2. The original and modified networks are shown in Figure 2(a) and (b), respectively. The topology of the original network is between regular and random graph as illustrated in Figure 2(a), and it is a non-scale-free network and demonstrates high clustering coefficients. A part of the modifications of network are illustrated in Figure 2(c), where the removed links (red dots) tend to gather together as a result of the WS model with little value of $β$ (β= 0.1). The average degree distribution denotes that the modified network follows a power-law distribution, with a power-law exponent $γ = - 1.717$ as denoted by the fitting line.

Figure 2.

Original and modified networks. (a) Topology of non-scale-free network generated by WS model with parameters of N = 200, K = 4, and $β = 0.1$ . (b) Topology of the network modified by NPR method. (c) Modifications of network. (d) Degree distribution of the original network and modified network.

Exploring the effect of NPR’s parameters

The following experiments are implemented to explore the effects of NPR’s parameters on the optimization of network topology. The performance of NPR method in turning non-scale-free networks into scale-free networks is denoted by the extent to which the degree distribution of modified network follows the power-law distribution. In order to demonstrate the degree distribution better, complex networks with enormous nodes are considered here. The original networks in the following experiments are generated by WS model with different values of $β$ (0.1, 0.4, 0.7, 1.0). In addition, the other parameters of WS model are set as follows: N = 3000, K = 4. As a result, four different kinds of non-scale-free networks are generated. When $β = 1.0$ , the original network is a random network whose degree distribution follows a passion distribution. When $β = 0.1, 0.4, 0.7$ , the topology of original networks is between a regular lattice and a random graph. Each kind of experiment is implemented 30 times independently. A fitting line of distribution with $PRI (α) = 6$ and $TCR (\tilde{R}) = 0.2$ is shown in each panel in Figures 3 and 4. Power-law exponent $γ$ of networks with $β = 0.1$ , $β = 0.4$ , $β = 0.7$ , and $β = 1.0$ is −1.748, −1.773, −1.778, and −1.774, respectively.

Figure 3.

Degree distribution of original and modified networks with different PRI. Original networks generated by WS model with different values of $β$ : (a) $β = 0.1$ , (b) $β = 0.4$ , (c) $β = 0.7$ , and (d) $β = 1.0$ .

Figure 4.

Degree distribution of original and modified networks with different TCR. Original networks generated by WS model with different values of $β$ : (a) $β = 0.1$ , (b) $β = 0.4$ , (c) $β = 0.7$ , and (d) $β = 1.0$ .

PRI

Average degree distributions of original and modified networks are illustrated in Figure 3 and Table 2. Three different PRIs are considered for each kind of experiment: PRI(α) = 3, 6, 9. The results of topology optimization of original networks with $β = 0.1$ , $β = 0.4$ , $β = 0.7$ , and $β = 1.0$ are shown in Figure 3(a)–(d), respectively. The value of parameter $TCR (\tilde{R})$ is set as 0.2. For all four kinds of experiment, the degree distributions of original networks follow different functions. However, all these distributions do not follow a scale-free distribution. The topology optimization is not sufficient when PRI(α) = 3 because the degree distributions of modified network follow power-law distribution poorly as illustrated in Figure 3. Furthermore, it is shown that PR method has a better performance with PRI(α) = 6 than PRI(α) = 9. PRI is an empirical parameter. PRI controls rewiring mechanism and the proper value of PRI should be chosen according to the results of simulation. Too small and too big of the PRI may result in redundant and inadequate rewiring, respectively (Figure 3 and Table 2). Linear fittings are implemented in log–log systems with the weight of 1/(1 + x)2 in order to balance the non-uniformity of sample density in log–log systems.

Table 2.

Linear fitting results in log–log systems with different PRIs.

$PRI (α)$	$β$
$PRI (α)$	0.1	0.4	0.7	1.0
3	R² = 0.9451	R² = 0.9382	R² = 0.9404	R² = 0.9392
6	R² = 0.9783	R² = 0.9787	R² = 0.9795	R² = 0.9777
9	R² = 0.9546	R² = 0.9602	R² = 0.9595	R² = 0.9606

TCR

Average degree distributions of original and modified networks are illustrated in Figure 4 and Table 3. Three different TCRs are considered for each kind of experiment: $TCR (\tilde{R}) = 0.2, 0.5, and 0.8$ . The results of topology optimization of original networks with $β = 0.1$ , $β = 0.4$ , $β = 0.7$ , and $β = 1.0$ are shown in Figure 4(a)–(d), respectively. The value of parameter PRI(α) is set as 6 according to our experimental results in Figure 3 and Table 2. For the original network with $β = 0.1$ , The topology optimization is not sufficient when $TCR (\tilde{R}) = 0.8$ because the degree distributions of modified network follow power-law distribution poorly as illustrated in Figure 4(a). In total, it is shown that NPR method has a better performance with $TCR (\tilde{R}) = 0.2$ than $TCR (\tilde{R}) = 0.5 and 0.8$ for all four experiments (Figure 4 and Table 3). TCR is also a critical parameter which controls the termination of NPR method. The smaller of TCR, the more iterations NPR method requires. Too big of TCR may result in insufficient optimization of network structure. However, too small of TCR would lead to useless iterations which denote increased complexity and time (Figure 4 and Table 3). As a result, a proper value of TCR is a balance of the topology optimization and the complexity of NPR method.

Table 3.

Linear fitting results in log-log systems with different TCRs.

$TCR (\tilde{R})$	$β$
$TCR (\tilde{R})$	0.1	0.4	0.7	1.0
0.8	R² = 0.1489	R² = 0.9761	R² = 0.9660	R² = 0.9669
0.5	R² = 0.9777	R² = 0.9769	R² = 0.9771	R² = 0.9766
0.2	R² = 0.9783	R² = 0.9787	R² = 0.9795	R² = 0.9777

TCR: termination condition rate.

Conclusion and discussion

The main objective of this research is topology optimization of existing complex networks, and our study resulted in promising findings. Usually, many practical networks such as sensor networks do not follow the generation process of rigorous mathematical models. As a result, many practical networks need to be optimized. For example, topology optimization of wireless networks plays an important role in regulating transmission power of the entire network in wireless communication area.^31,32 In addition, the topology optimization problem of wireless network is usually a dynamic process and evolves to efficient architecture temporarily.³³ Network topology optimization is a new challenge which is even difficult than constructing a totally new network, where not only the network topology is considered, but dynamical evolution and optimization expenses are also taken into account.

Preferential attachment is the key mechanism in BA model referring to generate scale-free network.²⁰ In addition, rewiring is also a critical point when generating small-world network in WS model.¹⁷ BA model exhibits growth processes by adding new nodes and edges, while WS model exhibits topology modification processes by rewiring. As a result, it is reasonable to combine preferential attachment process with rewiring process in network topology optimization problem. In this study, we proposed a NPR method to optimize network topology, by turning non-scale-free networks into scale-free networks. The mechanism of this NPR method was introduced, and several experiments were implemented to demonstrate the effect of this method on many kinds of non-scale-free networks, including random networks and several networks between regular and random. Conclusively, simulation results showed that regardless of the original networks, the networks modified by our method demonstrated power-law degree distribution perfectly. At technological level, such nonlinear preferential attachment method will help developing smart complex networks which are able to adjust to different conditions in real time to improve efficiency.

Supplemental Material

Supplemental_information – Supplemental material for Network topology optimization by turning non-scale-free networks into scale-free networks using nonlinear preferential rewiring method

Supplemental material, Supplemental_information for Network topology optimization by turning non-scale-free networks into scale-free networks using nonlinear preferential rewiring method by Feng Su, Peijiang Yuan, Yuanwei Liu and Shuangqian Cao in International Journal of Distributed Sensor Networks

Footnotes

Acknowledgements

The authors thank Dr Li jianmin of Tsinghua University for beneficial discussions and valuable comments on this work. Feng Su and Peijiang Yuan contributed equally to this work.

Handling Editor: Amiya Nayak

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (No. 61375085) and the Foundation of Shanghai Aircraft Manufacturing Co., Ltd (no. 32284001).

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Supplementary Material

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