BioMed Research International

Volume 2019, Article ID 4979582, 14 pages

https://doi.org/10.1155/2019/4979582

## Finding Community Modules of Brain Networks Based on PSO with Uniform Design

Correspondence should be addressed to Xiaoshu Zhu; nc.ude.usc@uhzsx and Fang-Xiang Wu; ac.ksasu.liam@143waf

Received 26 December 2018; Revised 11 August 2019; Accepted 28 September 2019; Published 17 November 2019

Academic Editor: Nasimul Noman

Copyright © 2019 Jie 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

The brain has the most complex structures and functions in living organisms, and brain networks can provide us an effective way for brain function analysis and brain disease detection. In brain networks, there exist some important neural unit modules, which contain many meaningful biological insights. It is appealing to find the neural unit modules and obtain their affiliations. In this study, we present a novel method by integrating the uniform design into the particle swarm optimization to find community modules of brain networks, abbreviated as UPSO. The difference between UPSO and the existing ones lies in that UPSO is presented first for detecting community modules. Several brain networks generated from functional MRI for studying autism are used to verify the proposed algorithm. Experimental results obtained on these brain networks demonstrate that UPSO can find community modules efficiently and outperforms the other competing methods in terms of modularity and conductance. Additionally, the comparison of UPSO and PSO also shows that the uniform design plays an important role in improving the performance of UPSO.

#### 1. Introduction

Graph theory is a very helpful mathematical tool in the field of brain network analysis [1–3]. A brain can be represented as a modular network [4, 5], which is composed of some important neural unit modules. They can provide us rich and useful information and exhibit small-world properties of brain networks [6]. These modules are known as community modules. In brain networks, each vertex denotes a region of interest (ROI) [7], and each edge and its weight represent the connectivity and its strength, respectively [8–10].

Community detection methods are frequently used to find community modules. Girvan and Newman proposed the concept of modularity [11–13], which is the widely used and best known metric. A larger modularity represents a better community partition. Modularity-based community detection methods find the best community modules by seeking the maximum modularity. Namely, when the modularity is maximal, the methods terminate. Therefore, community detection methods can be addressed by means of optimization methods. The FastQ [14] community detection method uses a greedy optimization to maximize modularity. It repeatedly joins communities together in pairs by choosing the join that results in the maximum alteration of modularity in each step. Danon et al.’s [15] community detection method is a modification of FastQ, in which the communities of different sizes are treated equally. The Louvain [16, 17] community detection method firstly calculates the gain of modularity by exchanging a node to its neighbor nodes. Then, the neighbor node obtaining the maximum gain replaces the node.

Particle swarm optimization (PSO) [18–20], as one of the swarm intelligent optimization algorithms, was first put forward by Eberhart and Kennedy [21, 22]. It simulates the foraging process of birds. Each bird (particle) may search the feasible solution space individually and share its individual optimal information to the other bird (particle). The swarm can obtain the global optimal solution by comparing the best solutions of all birds (particles) in the swarm. PSO can obtain the optimal solution quickly. However, it has the drawback of premature convergence [23].

The uniform design belongs to the category of the pseudo-Monte Carlo method. It can generate the solutions scattered uniformly over the vector space, and the solutions are independent of each other [24–26]. The uniform design can be applied to many problems, including bio-inspired intelligent optimizations. Zhang et al. [27] combined the uniform design and artificial bee colony to find the community of brain networks. Zhang et al. [26] introduced the uniform design into association rule mining and presented a multiobjective association rule mining algorithm based on the attribute index and the uniform design. Leung and Wang [24] integrated the uniform design and the multiobjective genetic algorithm to obtain the Pareto optimal solutions uniformly over the Pareto frontier. Zhu et al. [28] combined the uniform design and PAM to find the Pareto optimal solutions of the multiobjective particle swarm optimization. Dai and Wang [29] presented a new decomposition-based evolutionary algorithm with the uniform design. Liu et al. [30] proposed a hybrid genetic algorithm based on the variable grouping and the uniform design for global optimization problems. Tan et al. [31] adopted the uniform design to set the aggregation coefficient vectors of the subproblems and proposed the uniform design multiobjective evolutionary algorithm based on decomposition. Feng et al. [32] presented a uniform dynamic programming to alleviate the dimensionality problem of dynamic programming by means of introducing a uniform dynamic to dynamic programming.

There are only a few reports on community detection in brain networks in the literature. Liao et al. [33] utilized U-Net-based deep convolutional networks to identify and segment the brain tumor. Williams et al. [34] utilized both Louvain [16] and Infomap [35] community detection algorithms to identify modules in noisy or incomplete brain networks. Zhang et al. [27] utilized the artificial bee colony with the uniform design to detect community modules of brain networks. Wang et al. [36] used the multiview nonnegative matrix factorization to detect modules in multiple biological networks.

This study presents a novel method to find community modules of brain networks by integrating PSO with the uniform design. PSO is used to maximize modularity, while the uniform design is used to alleviate premature convergence of PSO by generating sampled points scattered evenly over the vector space.

The rest of this study is organized as follows: Section 2 describes the preliminaries of UPSO. Section 3 introduces two evaluation metrics. The dataset and the preprocessing method to be used are described in Section 4. The details of UPSO are shown in Section 5. The comparison between UPSO and several competing algorithms is illustrated in Section 6. The conclusion and future work are described in Section 7.

#### 2. Preliminaries

In this section, we describe PSO and the uniform design.

##### 2.1. Particle Swarm Optimization

In a *d*-dimensional search space, the position and velocity of the *i*-th particle are, respectively, represented as and , where , in which denotes the population size. The optimal solution of the *i*-th particle is called the individual optimum, while the optimal solution of the whole swarm is called the global optimum. They, respectively, are denoted as and . The following formulas are utilized to update the velocity and position of each particle in the swarm [21, 22], respectively:where ; is called the inertia weight coefficient reflecting the ability to track the previous speed; *c*_{1} and *c*_{2} are called the acceleration coefficients of the individual and the global optimum, respectively, and are commonly set as 2; and are two random numbers distributed uniformly in (0, 1).

From the theoretical analysis of a PSO algorithm, the trajectory of a particle converges to the mean of and . Whenever the particle converges, it “flies” to the individual best position and the global best position [37]. According to formulas (1) and (2), the individual optimum position of each particle gradually moves closer to the global optimum position. Therefore, all the particles may converge to the global optimum position.

##### 2.2. Uniform Design

The uniform design is an experimental design method. Its main objective is to sample a small set of points from a given set of points such that the sampled points are uniformly scattered.

Let be the number of factors and be the number of levels per factor. When *n* and *q* are given, the uniform design selects *q* combination from all *q*^{n} possible combinations such that these combinations are scattered uniformly over the space of all possible combinations. The selected *q* combinations are expressed in a uniform array , where is the level of the *l*_{2}-th factor in the *l*_{1}-th combination and can be calculated by the following formula [24–26, 28, 29, 38]:where is a parameter given in Table 1.