Computational and Mathematical Methods in Medicine

Volume 2018 (2018), Article ID 8950794, 9 pages

https://doi.org/10.1155/2018/8950794

## Novel Model for Cascading Failure Based on Degree Strength and Its Application in Directed Gene Logic Networks

Correspondence should be addressed to Maoxian Zhao and Kebo Lv

Received 12 November 2017; Revised 15 January 2018; Accepted 18 January 2018; Published 19 February 2018

Academic Editor: Xiaoqi Zheng

Copyright © 2018 Yulin 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

A novel model for cascading failures in a directed logic network based on the degree strength at a node was proposed. The definitions of in-degree and out-degree strength of a node were initially reconsidered, and the load at a nonisolated node was proposed as the ratio of in-degree strength to out-degree strength of the node. The cascading failure model based on degree strength was applied to the logic network for three types of cancer including adenocarcinoma of lung, prostate cancer, and colon cancer based on their gene expression profiles. In order to highlight the differences between the three networks by the cascading failure mechanism, we used the largest-scale cascades and the cumulative cascade probability to depict the damage. It was found that the cascading failures caused by hubs are usually larger. Furthermore, the result shows that propagations against the networks were correlated with the structures motifs of connected logical doublets. Finally, some genes were selected based on cascading failure mechanism. We believe that these genes may be involved in the occurrence and development of three types of cancer.

#### 1. Introduction

Over the past few decades, many scientists focused on the study of cascading failures in different networks, such as the electrical power networks [1–3], traffic networks [4, 5], Internet networks [6], social networks [7, 8], and even biological networks [9, 10]. The various models of cascading failures and their mechanisms, as well as their prevention, have been proposed. For instance, Motter and Lai [11] proposed a load-capacity cascading failure model and simulated an arbitrary power exponent of scale-free networks. The results showed that loads would redistribute among the nodes, and intentional attacks would lead to a cascade of overload failures, which could cause the entire part of the network to collapse. Wang and Xu [12] investigated cascading failures in coupled map lattices with different topologies and found that cascading failures are much easier to occur in small-world and scale-free coupled map lattices than in globally coupled map lattices. Crucitti et al. [13] presented a simple model for cascading failures based on the dynamical redistribution of the flow in the network, showing that the breakdown of a single node is sufficient to reduce the efficiency of the entire system if the node is among those with the largest load.

Recently, some researchers focused on the cascading failure mechanisms for directed networks. Fang et al. [14] proposed the cascading failure model in the context of directed complex networks. They used two attack strategies including minimum in-degree and the maximum out-degree attack strategy, which were compared with random attack strategy through simulations. Numerical results show that the cascading failure propagation in directed complex networks is highly dependent on the attack strategies and the directionality of the network. Jin et al. [15] built the load-capacity cascading failure model of the directed and weighted network. They applied the models to two typical real networks, namely, the Poisson distribution network and power law distribution network. Through simulation analyses, they concluded that the average weight and the average in-degree should be increased, respectively, for enhancing the resistibility of overloading and short-loading failures. Smart et al. [9] investigated the relationship between structure and robustness in the metabolic networks of* Staphylococcus aureus* and so on using a cascading failure model based on a topological flux balance criterion.

Despite this success, few studies have attempted to identify the cascading failure mechanism in a directed gene logic network. In this study, we investigate a load-capacity cascading failure model based on the degree strength of nodes and identify the influence of cascading failures on the gene logic networks. The directed network is constructed. The definitions of in-degree, out-degree, and degree strength are refined for different regulation types of second-order logical relationships. Then a novel algorithm for cascading failure based on load-capacity model is investigated. The load at a node is defined as the ratio of the in-degree strength to the out-degree strength of the node. The capacity of a node is the interval from the minimum load to the maximum load that the node can handle. By removing a particular gene node initially, the corresponding number of cascading failure nodes generated is noted. This process is repeated for each gene node in the network. The parameters, that is, the probability that a gene node will yield damage greater or equal to , as well as the largest size ratio of cascading failure, are used to detect the relationship between network structure and robustness. Applying the model to gene expression profiles data for adenocarcinoma of lung, prostate cancer, and colon cancer, we find that hubs connected with other nodes by logical motif are more likely to break down. The study of cascading failure for gene networks may provide useful information underlying the biological mechanism of the formation and the development of cancers.

#### 2. Methods

##### 2.1. In-Degree and Out-Degree in the Logic Network

Bowers et al. [16, 17] proposed the logic analysis of phylogenetic profiles (LAPP) and demonstrated the benefits of identifying the relationships among gene triplets, as they have a greater likelihood of yielding the network organization of the interactions among gene triplets which forms the gene logical network. In fact, it can be considered as a weighted and directed graph that deciphers different logic interactions among gene node, including first-order and second-order logical relationships by the uncertainty coefficient at some thresholds (for details about the gene logical network, see Wang et al. [18] and Zhang et al. [19]).

In the first-order logical relationship, taking , its uncertainty coefficient is defined aswhich measures the probability that gene regulates gene , where and are the Shannon entropies for vectors and , respectively, and is the joint entropy of and . This regulatory relationship is denoted as a weighted and directed edge. Figure 1 gives three topologies for in-degree and out-degree for node of 1st-order logical relationship. Obviously both the in-degree of and the out-degree of are increased by one for . A second-order logical relationship as shown in Figure 2, for example, , has an uncertainty coefficient denoted as that measures the probability of existence of this second-order logical relationship. In this formula, is the logical function. The uncertainty coefficient of can be calculated by