Mathematical Problems in Engineering

Volume 2015, Article ID 194568, 10 pages

http://dx.doi.org/10.1155/2015/194568

## Improvement on Load-Induced Cascading Failure in Asymmetrical Interdependent Networks: Modeling and Analysis

School of Aeronautics and Astronautics Engineering, Air Force Engineering University, Xi’an 710038, China

Received 2 June 2015; Revised 1 August 2015; Accepted 4 August 2015

Academic Editor: Xiaobo Qu

Copyright © 2015 Haiyan Han and Rennong Yang. 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

Many real-world systems can be depicted as interdependent networks and they usually show an obvious property of asymmetry. Furthermore, node or edge failure can trigger load redistribution which leads to a cascade of failure in the whole network. In order to deeply investigate the load-induced cascading failure, firstly, an asymmetrical model of interdependent network consisting of a hierarchical weighted network and a WS small-world network is constructed. Secondly, an improved “load-capacity” model is applied for node failure and edge failure, respectively, followed by a series of simulations of cascading failure over networks in both interdependent and isolated statuses. The simulation results prove that the robustness in isolated network changes more promptly than that in the interdependent one. Network robustness is positively related to “capacity,” but negatively related to “load.” The hierarchical weight structure in the subnetwork leads to a “plateau” phenomenon in the progress of cascading failure.

#### 1. Introduction

Complex network theory has been a successful tool in modeling and analysis of modern systems [1]. A variety of network models have been proposed to approximate realistic systems such as power grids [2], transportation systems [3], communication networks [4], and other systems. Most of them focus on single and isolated systems. Recently,* interdependent network* [5] provides a new insight in understanding the structure [6], percolation [7], spreading processes [8], evolution games [9], and robustness [10, 11] of complex systems. This kind of structure usually consists of two or more subnetworks, with the subnetworks working dependently on each other [12, 13]. For example, power grids and computer systems must depend on each other because computers require power grids to supply electricity. Power grids, in turn, rely on computer systems to control power transmission process.

In 2010, Buldyrev et al. [14] found that interdependent networks have become significantly more vulnerable than their noninteracting counterparts under random attack. Gao et al. [5] reviewed the connectivity properties of “networks of networks” formed by interdependent random networks. Hu et al. [15] constructed a partially coupled network with both interdependent and interconnecting links. The interconnections satisfy “one to one” condition. They found that the change of interconnecting links leads to the change of the phase transition from second order to first order through hybrid phase transition. Huang et al. [16] studied the robustness of interdependent networks under targeted attack on high or low degree nodes. It provided a routine method to study the degree-based targeted attack problems in both single networks with dependency links [17, 18] and other general randomly connected and uncorrelated interdependent networks. Moreover, Parshani et al. [19] described the dynamic process of cascading failures on two partially interdependent networks.

Currently, researchers focus on the problem of modelling an interdependent network model for analyzing the progress of cascading failure. Based on classical network models such as ER [20], WS [21], and BA [22], researchers studied well some symmetrical and asymmetrical networks [19, 23, 24], such as ER-ER, WS-WS, BA-BA, ER-WS, BA-ER, and BA-WS; but these models differ greatly from real-world systems [25, 26]. Taking interdependent networks like ground transportation network and airline network as an example, ground transportation network displays a hierarchical property. That is, stations at provincial level have higher capacity and more importance than municipal stations. However, the airline network can be seen as a single level network with small world property [27]. So, some other researchers [28–30] made their trials to construct framework of asymmetrical interdependent networks which are more authentic to reality.

Moreover, load such as cargoes transported in the transportation network and electric stream in power grids can trigger cascading failures [31]. The load carried by the failed nodes or edges will not disappear but will flow to the remaining part of the network, which, in a possible way, will cause further failures. As far as we are concerned, not enough attention has been paid to the cascading failure induced by load redistribution in interdependent networks.

In order to investigate the load-induced cascading failure in interdependent networks, we propose asymmetrical interdependent networks model in Section 2 and the load-induced cascading failure model in Section 3. Section 4 simulates the cascading failure in the proposed model and compares it with a WS-WS symmetrical network when nodes and edges suffer from intentional attacks, respectively. Section 5 summarizes the contribution of this paper and identifies future research needs.

#### 2. Network Model

Figure 1 shows an asymmetrical interdependent network model. has a hierarchical and weighted structure, where nodes are assigned with weights and distributed into different levels, where is a single level network. The numbers of nodes in and are equal. The coupling proportion is set to . Nodes in different subnetworks randomly construct “one to one” connections; for example, node in merely couples with in .