Scientific Programming

Volume 2016, Article ID 5659687, 7 pages

http://dx.doi.org/10.1155/2016/5659687

## Optimal Design on Robustness of Scale-Free Networks Based on Degree Distribution

^{1}School of Electrical Engineering and Automation, Jiangsu Normal University, Xuzhou 221116, China^{2}School of Government, Beijing Normal University, Beijing 100875, China^{3}People’s Government of Wuduan Town, Peixian, Xuzhou 221638, China

Received 3 April 2016; Revised 7 June 2016; Accepted 5 July 2016

Academic Editor: Meng Guo

Copyright © 2016 Jianhua Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper uses 2-norm degree and coefficient of variation on degree to analyze the basic characteristics and to discuss the robustness of scale-free networks. And we design two optimal nonlinear mixed integer programming schemes to investigate the optimal robustness and analyze the characteristic parameters of different schemes. In this paper, we can obtain the optimal values of the corresponding parameters of optimal designs, and we find that coefficient of variation is a better measure than 2-norm degree and two-step degree to study the robustness of scale-free networks. Meanwhile, we discover that there is a tradeoff among the robustness, the degree, and the cost of scale-free networks, and we find that when average degree equals 6, this point is a tradeoff point between the robustness and cost of scale-free networks.

#### 1. Introduction

Complex networks have become more and more important to our daily life [1–5], such as biology system, electrical power grid, transportation network, and pipeline networks. The security and reliability of these networks have been the concern of more and more scientists. Since the discovery of small-world networks, the small-word property [2] and scale-free property [3] have attracted continuous attention from all over the world. It has been recognized that many networks have scale-free property which means that the degree distribution follows a power-law distribution [4, 5].

Recently, there are many novel concepts and approaches in many subjects, such as information science, control science, statistical and nonlinear physics, and mathematics and social science which are used to investigate the characteristics in many fields, especially complex network [6, 7]. The critical point is that the information flow between topological nodes and other physical quantities is important to network security, so we must keep the information exchange unimpeded. Because of the ubiquity of scale-free networks in natural and manmade systems, the security and reliability of these networks have attracted great interest [8, 9]. The work by Albert demonstrated that scale-free networks possess the robust yet fragile property and he found that it is robust against random failures of nodes but fragile to intentional attacks [3–5].

Cascading failure can occur in many complex systems; an intuitive thinking suggests that the possibility of breakdown of networks triggered by attacks or failures cannot be ignored in scale-free networks. Avalanche of breakdown is a serious threat to the network when nodes and links are sensitive to overloading. The removal of nodes which resulted from random breakdown or intentional attack can change the balance of flows and lead to redistributing loads all over the network; sometimes the redistribution of loads cannot be tolerated and might trigger a cascade of overload failure [10]; finally the network would be collapsing. But it can also propagate and cut down information transmission of the whole network in some cases [11]. Therefore the robustness is an important aspect to investigate the characteristics of scale-free network.

Based on the statistical property of computer network, it is known that computer network is a scale-free network. Supercomputer is a computer with a high-level computational capacity compared to a general-purpose computer. With the developments of supercomputers, supercomputer networks will also be formed in the future; therefore supercomputer networks also possess scale-free property, and the reliability and robustness of supercomputer network become more and more important. The optimal design of supercomputer networks can improve the reliability and robustness of supercomputer networks and also enhance speed of the information transmission and exchange of supercomputer networks. Hence the optimal design can present the theoretical and practical significances for the design and construction of supercomputer networks.

This paper is organized as follows. Section 2 discusses the basic characteristics by analyzing the characteristic parameters of scale-free networks. Two-norm degree and its optimal programming of degree distribution are studied in Section 3. Section 4 investigates the coefficient of variation and its optimal design of degree distribution. Finally, a conclusion is presented in Section 5.

#### 2. Basic Characteristics of Scale-Free Network

Recently, many researchers have constructed several schemes to measure the robustness of scale-free networks, such as the average degree, average two-step degree () [12], and entropy [13, 14]. In this paper, we construct two models defined as the average 2-norm degree () and coefficient of variation () to discuss the robustness of scale-free network. From Motter and Lai [11], we know that the node of complex network has a load capacity as follows:where is the tolerance parameter, is the initial load of node , and is the total number of vertices.

The network is designed as , is the set of vertices, and is the set of edges which can exchange information from vertices to others. According to [12, 13, 15], one knows that the robustness can be improved with the increase of the heterogeneity, and the larger the average degree is, the better the robustness is. Many complex networks are scale-free networks, and the degree distribution of nodes is power-law distribution [3–5], the density function being as follows:

Based on the discrete property of degree and continuous approximation principle, we can declare where and are the minimal and the maximum degree in finite network. From the literature [16], we can get the maximum degree of scale-free networks:Substituting (4) into (3), we can obtainAccording to (2) and (5), we can obtain the density function of degree distribution as follows:

Formula (6) introduces the characteristics of the density function on degree of scale-free networks. We know that the scale-free network is heterogeneous network, and we observe the cumulative probability on degree and investigate the robustness of scale-free networks; formula (7) presents the cumulative probability of scale-free network:

Figure 1 introduces the properties of the cumulative probability on degree distribution of the scale-free network, and we can declare that the cumulative probability decreases with the increase of the parameter ; meanwhile, with the increase of , the cumulative probability decreases. And we find that the cumulative probability is very small when ; that is to say, there are few vertices which are degree surpassing ; this phenomenon reflects the heterogeneity of scale-free networks.