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

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.


Introduction
Graph theory is a very helpful mathematical tool in the field of brain network analysis [1][2][3]. A brain can be represented as a modular network [4,5], which is composed of some important neural unit modules. ey can provide us rich and useful information and exhibit small-world properties of brain networks [6]. ese 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][9][10].
Community detection methods are frequently used to find community modules. Girvan and Newman proposed the concept of modularity [11][12][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. erefore, community detection methods can be addressed by means of optimization methods. e 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. e Louvain [16,17] community detection method firstly calculates the gain of modularity by exchanging a node to its neighbor nodes. en, the neighbor node obtaining the maximum gain replaces the node.
Particle swarm optimization (PSO) [18][19][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). e 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]. e 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][25][26]. e 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.
ere 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.
is 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. e rest of this study is organized as follows: Section 2 describes the preliminaries of UPSO. Section 3 introduces two evaluation metrics. e dataset and the preprocessing method to be used are described in Section 4. e details of UPSO are shown in Section 5. e comparison between UPSO and several competing algorithms is illustrated in Section 6. e conclusion and future work are described in Section 7.

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

Particle Swarm Optimization.
In a d-dimensional search space, the position and velocity of the i-th particle are, respectively, represented as where i � 1, 2, . . . , N pop , in which N pop denotes the population size. e 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.
ey, respectively, are denoted as P best i � [p i,1 , p i,2 , . . . , p i,d ] and G best � [p g,1 , p g,2 , . . . , p g,d ]. e following formulas are utilized to update the velocity and position of each particle in the swarm [21,22], respectively: where i � 1, 2, . . . , N pop ; ω 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 r 1 and r 2 are two random numbers distributed uniformly in (0, 1). From the theoretical analysis of a PSO algorithm, the trajectory of a particle x i converges to the mean of P best i and G best . 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. erefore, all the particles may converge to the global optimum position.

Uniform Design.
e 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 n be the number of factors and q 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. e selected q combinations are expressed in a uniform array U(n, q) � [U l 1 ,l 2 ] q×n , where U l 1 ,l 2 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 σ is a parameter given in Table 1.
Based on the uniform design, a crossover operator is as follows [24]. It quantizes the solution space defined by two parents into a finite number of points and then applies the uniform design to select a small sample of uniformly scattered points as the potential offspring.
Consider two parents x 1 � (x 1,1 , x 1,2 , . . . , x 1,d ) and . e minimal and maximal values of each dimension for x 1 and x 2 can generate a novel solution space [l parent , u parent ], denoted as follows: Each domain of [l parent , u parent ] is quantized into Q 1 levels β i,1 , β i,2 , . . . , β i,Q 1 , where Q 1 is a predefined prime number and β i,j is given as follows: en, the uniform design is applied to select a sample point as the potential offspring. e crossover operator of two parents x 1 and x 2 can acquire Q 1 offsprings, which are scattered evenly over the vector space spanned by x 1 and x 2 . More details of the algorithm can be obtained from references [24,25].

Evaluation Metrics
ere exist many evaluation metrics for community modules of complex brain networks. In this study, we adopt the following metrics. [27]. e modularity metric is a statistic that quantifies the degree to which the network may be divided into such clearly delineated groups [39,40]. Newman et al. introduced the modularity function and modularity matrix to avoid the influences of random factors so as to obtain the better divisions of the community structure [12,13,41]. e modularity Q is the number portion of edges falling within communities minus the expected number portion in an equivalent network with edges placed at random. e modularity Q can be expressed as follows [42,43]:

Modularity
where δ(i, j) � 1 if vertices i and j belong to the same community or δ(i, j) � 0 otherwise. m � ( i k i )/2 denotes the number of edges in the network, k i is the degree of the vertex i, and a ij is the weight in the adjacent matrix A. Let B � a ij − (k i k j /2m), which is called the modularity matrix. Formula (6) can be rewritten in the matrix format as follows: where the assignment matrix X � (x ih ), in which x ih � 1 if vertex i belongs to the community h or x ih � 0 otherwise. e function Trace( ) denotes the sum of diagonal elements of a matrix.
A high modularity indicates a better partitioning of the graph. e search for optimal modularity Q is an NP-hard problem [44,45] because the space of possible partitions grows faster than any power of system size. [27]. e conductance of a cut is a metric that compares the size of a cut (i.e., the number of edges cut) and the number of edges in either of the two subgraphs induced by that cut. e conductance ϕ(G) of a graph is the minimum conductance value between all its clusters.

Conductance
Consider a cut that divides G into k nonoverlapping clusters C 1 , C 2 , . . ., C k . e conductance of any given cluster ϕ(C i ) is given by the following formula [43,46]: where K denotes the number of clusters, a ij is the weight in the adjacent matrix, C k represents the k-th cluster (k � 1, 2, . . . , K), and a(C i ) is the number of edges with at least one endpoint in C i . is ϕ(C i ) represents the cost of one cut that bisects G into two vertex sets C i and C i (the complement of C i ). Since we want to find a number k of clusters, we will need k− 1 cuts to achieve that number. e conductance for the whole clustering is the average value of those k− 1ϕ cuts. e conductance metric can evaluate how difficultly a random walk is that leaves a cluster [40]. e more difficultly a random walk leaves a cluster is, the more compact cluster is. A low conductance indicates a better partitioning of the graph. e conductance metric usually ranges from 0 to 1, while 0 is the optimal score, which means that each cluster corresponds to a maximal strongly connected component of the network.

Dataset.
A network is a mathematical representation of a real-world complex system and is determined by a collection of nodes (vertices) and links (edges) between pairs of nodes. Brain connectivity datasets comprise networks of brain regions connected by anatomical tracts or by functional associations. Nodes in brain networks usually represent ROIs, while links represent anatomical, functional, or effective connection [40]. A connectivity matrix (CM) is used to store the connectivity strength between all pairs of ROIs in a brain network [47].
In this study, 79 rsfMRI brain networks (42 ASD and 37 TD) are utilized to test the proposed algorithm.

Data Preprocessing.
In this study, we conduct the following preprocessing steps for the above dataset: (1) Reverse z-transformation is performed on the original CM to acquire the PCC connectivity matrix (PCM) according to the following formula: where x ∈ CM and x ′ ∈ PCM denote the original and new values, respectively. (2) e negative data in the PCM signify that the correlation among the vertices is negative correlation. In this study, these negative elements are taken as 0 to get rid of negative correlation. After conducting the above two steps, all data in the PCM are in [0, 1], and the PCM turns into a symmetric and nonnegative matrix. (3) To eliminate data noise, this study adopts the thresholding method to remove all edges with the weight less than a specific value θ. Namely, if x < θ, then x � 0. In the later numerical experiment, θ � 0.2.

The Proposed Algorithm
In this study, we propose a novel algorithm for finding community modules of brain networks by integrating PSO with the uniform design (abbreviated as UPSO). Its coding and detailed steps are described as follows.

Coding. A brain network G can be represented as
where a ij denotes the weight between vertices i and j. From the above-mentioned dataset and data processing, we can see that A is a symmetric and nonnegative matrix. e number of community modules and centroid of a community module are denoted by K and CC k � (cc k1 , . . . , cc kN ), where k � 1, 2, . . . , K, respectively. In PSO, the position coding x i of a particle is expressed as where x i is a K * N-dimensional row vector and i � 1, . . . , N pop (the population size in PSO).

Detailed
Steps. e proposed algorithm UPSO utilizes the uniform design to obtain the sampled points scattered evenly over the solution space. e initial method based on the uniform design can generate a group of suitable initial particles scattered evenly over the solution space. e crossover operator based on the uniform design can acquire the offspring scattered uniformly over the space spanned by two crossover parents. UPSO iteratively tries to improve a candidate solution in terms of modularity. It integrates the uniform design and PSO to find community modules of brain networks. It can not only obviate the shortcoming of premature convergence in PSO but also acquire the solutions scattered evenly over the solution space. It can find out community modules from brain networks without knowing the number of community modules. Its flow chart is illustrated in Figure 1.
e detailed steps of the proposed algorithm UPSO are described as follows.
Step 1. (generating a temporary initial swarm). e following operations are performed one after another: (1) Let K � K + 1, where K denotes the number of community modules, and its minimal and maximal values are, respectively, 1 and N (the number of vertices). K � 1 to N is to acquire the fittest number of community modules. (2) According to the swarm size N pop , the number of subintervals S and the swarm size of a subinterval Q 0 are determined such that S × Q 0 ≥ N pop , where S can be taken as 2, or 2 2 , or 2 3 , etc.; Q 0 is one of the prime numbers in the first column of Table 1. Here, any combination satisfying S × Q 0 ≥ N pop can be chosen. (3) e generation algorithm of initial population based on the uniform design described in reference [28] is implemented to generate a temporary initial swarm Tmp_pop in terms of K, in which each element x i contains K community centroids.
Step 2. (calculating the fitness of the temporary initial swarm). For each particle p i in Tmp_pop, the following operations are performed in sequence: (1) K community centroids CC k are separated from x i . For each element a ij in the adjacent matrix A described in Section 5.1, the distances are calculated between a ij and each CC k . (2) Each vertex is assigned to the closest community C k to obtain its community affiliation IDX i and K community modules. (3) e modularity Q of K community modules is calculated using formula (6) or (7) in terms of IDX i , and it is taken as the fitness f(x i ) of x i .
Step 3. (generating the initial swarm from the temporary initial swarm). According to the acquired fitness of each particle in the temporary initial swarm, the best N pop ones of the Q 0 * S particles are selected as the initial swarm pop.
Step 4. (regulating each community module). For each particle position x i in pop, the following operations are performed in sequence. e centroid of the community C k is updated according to the following formula: where a i � (a i1 , a i2 , . . . , a iN ), i � 1, 2, . . . , N and N is the number of vertices; k � 1, 2, . . . , K; n k is the number of vertices which belong to the community C k ; and CC k is the new community centroid of C k . K new community centroids (CC 1 , CC 2 , . . . , CC K ) form a new position, marked as KC i , whose fitness and community affiliation are f(KC i ) and KC_IDX i , respectively. If Step 5. (initializing the velocity s i , individual optimal P best i , and global optimal G best ). e velocity s i and individual optimal P best i of the particle p i are initialized as its position x i , and the fitness of P best i is set as e maximal value in all P best i is taken as the global optimal G best , which stores the best x i and Q of the swarm. e community affiliation G best is stored into IDX.
Step 6. (increasing iterations and judging terminal conditions). Let t � t + 1, then judge whether terminal conditions are satisfied or not, where t denotes the t-th iteration and its initial value is 0. If K is known, and any of the terminal conditions is satisfied, the algorithm terminates and outputs the optimal solution and its community affiliation; otherwise, the algorithm moves to Step 7. Terminal conditions are described in Section 6.1.
Step 7. (computing the weight coefficient w in PSO). e weight coefficient w in PSO utilizes a linear decreasing strategy [48,49] indicated in the following formula: where w max and w min are the maximal and minimal values of w and t max is the maximal number of iterations. In the later numerical experiment, w min � 0.1 and w max � 1.
Step 8. (updating the velocity and position of each particle).
To guide the moving trajectory of a particle by KC i , formula (1) is modified into the following formula: e velocity and position of each particle in the pop are updated in terms of formulas (13) and (2), respectively.
Step 9. (calculating the fitness and regulating each community module). e fitness of each particle in the pop is calculated according to the operations in Step 2, and Step 4 is implemented to regulate each community module.
Step 10. (updating P best i , G best , and IDX).
Step 11. (implementing the crossover operator based on the uniform design). For each particle in the pop, the following operations are performed in sequence: (1) e crossover operator based on the uniform design is implemented on x i and P best i to acquire the Q 1 offspring scattered uniformly over the space spanned by them and also on x i and G best to acquire another Q 1 offspring. (2) e fitness of the 2 * Q 1 offspring is calculated, and the best one of them is marked as O best . e fitness and community affiliation of O best are expressed as f(O best ) and I DX O best , respectively.
G best � O best , and I DX � I DX O best .
Step 12. e algorithm is returned to Step 6.
Step 13. If K < N, the best Q and community affiliation IDX are saved and then the algorithm returns to Step 1; otherwise, the algorithm outputs the optimal solution G best , the community affiliation IDX, and the fittest K.

Numerical Results
In this study, we select four competing community detection algorithms to compare the performances of UPSO. ey include the spectral clustering [50], FastQ [14], Danon et al. [15], and Louvain [16] algorithms. FastQ, Danon, and Louvain algorithms are three commonly used community detection methods. Among five algorithms, UPSO and the spectral clustering are stochastic search algorithms, while FastQ, Danon, and Louvain algorithms are deterministic search algorithms.
e parameter values of UPSO and the numerical results obtained by UPSO and four competing algorithms are described as follows.

Parameter Values.
In this study, the parameters of UPSO are described as follows.

Parameters for PSO.
e minimal and maximal inertia weight coefficients are w min � 0.1 and w max � 1 (the recommended values in PSO); the acceleration coefficients c 1 , c 2 , and c 3 are all equal to 2 (the recommended values in PSO); the population size Npop � 100; the maximal number of iterations t max � 100.

Parameters for the Uniform Design.
As the abovementioned each rsfMRI imaging is a 264 × 264 CM, we set the number of subintervals S as 4 (S can be 2 1 , 2 2 , 2 3 , ......); the number of sample points or the swarm size of each subinterval Q 0 is set as 31 because Q 0 can be any values in Table 1 and the product of Q 0 and S must be larger than the population size N pop , namely, (Q 0 * S � 31 * 4 � 124) > (N pop � 100). e parameter Q 1 is set as 5 in order to only generate 5 offsprings in uniform cross to decrease time consumption.

Terminal Conditions
(1) e number of iterations t > t max (2) e number of fitness remains unchanged, t no , and is larger than or equal to 30% of t max When any of the above two terminal conditions is satisfied, the algorithm terminates.
It is worth noting that the above parameter values are not fixed and can be changed according to different datasets. e above parameter values are only one of the suitable values, and they do not need to be fine tuned.
As the spectral clustering needs to preestimate the number of community modules, it uses the identical number of community modules to UPSO. FastQ, Danon, and Louvain algorithms do not necessarily need to estimate the number of community modules; therefore, they use their default parameters.

Comparisons of Evaluation Metrics.
All the 79 rsfMRI brain networks are utilized to test the performance of five algorithms. Five algorithms independently performed 20 runs to compare their average values. For stochastic search algorithms, UPSO and the spectral clustering, we also compare their standard deviations (the values in parentheses in Tables 2 and 3). Tables 2 and 3, respectively, show the results of modularity and conductance metrics obtained by five algorithms.
From Table 2, it can be obviously observed that, for all 79 rsfMRI brain networks, the modularity metrics obtained by UPSO are all the best among five algorithms.
is fully demonstrates that the proposed algorithm outperforms other four competing algorithms in terms of modularity. e main reasons for UPSO obtaining good results are explained as follows: Firstly, UPSO is a heuristic optimization algorithm, so it can search a good solution as much as possible. Secondly, as UPSO is a swarm intelligent optimization algorithm, it can use all the individuals in a swarm to search   the optimal solution, while the other four algorithms can use only one individual. Last but not the least, UPSO can use the uniform design to obtain the solutions scattered evenly over the feasible solution space. In five algorithms, the gaps of the results obtained by UPSO and Louvain algorithm are much less than those by UPSO and other three algorithms, and even UPSO and Louvain algorithm obtain the identical results for ASD90B and ASD95 brain networks. us, the Louvain algorithm is the most competing in the other four algorithms.
We can also see from Table 2 that the standard deviations obtained by UPSO and the spectral clustering are all very small compared to the average values obtained by them. is demonstrates that UPSO and the spectral clustering are both relatively stable in terms of modularity for 79 rsfMRI brain networks. Meanwhile, we can also observe that, for 65 of 79 brain networks, the standard deviations obtained by UPSO are less than or equal to those obtained by the spectral clustering. is demonstrates that UPSO has higher stability than the spectral clustering in terms of modularity.
From Table 3, we can clearly see that UPSO obtains the best conductance metrics for most brain networks, but not for all 79 brain networks. is is because that the evaluation perspectives of two metrics are different. However, UPSO obtains the best conductance metrics for 50 brain networks and accounts for about 63% of 79 brain networks. is manifests that UPSO is superior to other competing algorithms in terms of conductance. Meanwhile, this also demonstrates that UPSO can acquire better conductance metrics while ensuring the best modularity metrics. e number of brain networks in that the spectral clustering, FastQ, Danon, and Louvain algorithms obtained the best conductance metrics is 1, 6, 20, and 2, respectively. We can also clearly observe that the best modularity metrics obtained by UPSO and Louvain algorithm are relatively close, but the number of brain networks in that the Louvain algorithm obtained the best conductance metrics is just 2. is fully demonstrates that UPSO outperforms the Louvain algorithm in terms of conductance. From Table 3, we can also see that the standard deviations obtained by UPSO and the spectral clustering are all very small compared to the average values obtained by them.
is is also similar to data in Table 2. Namely, for 79 rsfMRI brain networks, UPSO and the spectral clustering are relatively stable in terms of both modularity and conductance metrics. Meanwhile, we can also observe that, for 42 of 79 brain networks, the standard deviations obtained by UPSO are less than or equal to those obtained by the spectral clustering. is demonstrates that UPSO has higher stability than the spectral clustering in terms of conductance. is conclusion is also similar to that concluded from Table 2.

Comparisons of Other Perspectives.
Besides the abovementioned comparisons, we also evaluate the performances of UPSO from other perspectives, such as influences of the uniform design, comparisons with other heuristic algorithms, and complexity analysis.
To show the benefit of hybridizing the uniform design in PSO, we modify UPSO by removing the uniform design from UPSO. Namely, the initialization (Steps 1, 2, and 3) uses the random initialization method instead of the generation algorithm of the initial population based on the uniform design, and the crossover operator based on the uniform design (Step 11) is not performed. For brevity, the modified algorithm is called PSO. We compare the  Table 4.
To verify the performance of UPSO, we also compare it with ABC (artificial bee colony). Similar to PSO, ABC is also a heuristic algorithm. Table 4 also shows the results obtained by ABC.
From Table 4, we can clearly see that, for 67 of 79 brain networks, the modularity metrics obtained by UPSO are larger than those obtained by PSO. In comparison, there are just 4 brain networks for which the modularity metrics obtained by UPSO are less than those obtained by PSO. is fully demonstrates that the influence of the uniform design on improving the performance of UPSO is significant. Figures 2 and 3 in the next section also obviously illustrate the benefit of the uniform design.
By comparison of the modularity metrics obtained by UPSO and those obtained by ABC, it can be clearly seen from Table 4 that, for 79 brain networks, the modularity metrics of UPSO are all larger than those of ABC. is fully demonstrates that UPSO significantly outperforms ABC in terms of modularity. A comparison of PSO and ABC is the same as the comparison of UPSO and ABC. Namely, for 79 brain networks, the modularity metrics of PSO are all larger than those of ABC. It follows from the above that PSO is also superior to ABC for 79 rsfMRI brain networks even without the uniform design.
By a detailed analysis of the proposed algorithm UPSO, its computational complexity is obtained as follows: if the number of community modules K is pregiven or preestimated, the time complexity of UPSO is O(t max * N pop ); otherwise, the time complexity of UPSO is O(t max * N pop * N), where t max , N pop , and N, respectively, denote the maximal number of iterations, the population size, and the number of vertices in brain networks. us, unless it is absolutely necessary, UPSO often uses the pregiven K or the same K as that of the other methods to decrease its computational complexity.

Representative Brain Networks.
According to different cases of the modularity and conductance metrics in Tables 2  and 3, two representative brain networks are chosen to demonstrate the performance of UPSO. TD86C Brain Network. For the TD86C brain network, the best modularity and conductance metrics are both obtained by UPSO. Figure 4 illustrates the plot of the modularity metrics obtained by UPSO and PSO. e community plot of the TD86C brain network is illustrated in Figure 2.
From Figure 4, it can be seen that the modularity metrics obtained by UPSO and PSO both converge to a stable state when the number of iterations increases. Meanwhile, we can also clearly see that the plot of UPSO is always above that of PSO after the third iteration. is obviously illustrates that the uniform design plays an important role in improving the performance of UPSO. ASD104 Brain Network. For the ASD104 brain network, the best modularity is obtained by UPSO, while the best conductance metric is obtained by the Danon algorithm. Figure 3 illustrates the changing process of the modularity metrics obtained by UPSO and PSO with the number of iterations. Figure 5 illustrates the community plot of ASD104 brain networks.
We can clearly observe from Figure 3 that the plots of UPSO and PSO both go up when the number of iterations increases, which show the processes of searching the optimal solution. However, the plot of UPSO is above or overlapping that of PSO in the whole iterating process. is fully illuminates that the influence of the uniform design is considerable.

Conclusions and Future Work
In this study, we design a particle swarm algorithm with the uniform design (UPSO) for finding the community modules in brain networks. We conduct UPSO and several competing algorithms on 79 rsfMRI brain networks. e obtained results demonstrate that UPSO can find community modules with maximal modularity and obviously outperforms other competing methods in terms of modularity. e comparison of UPSO and PSO shows that the uniform  design plays an important role in improving the performance of UPSO. e comparison of PSO and ABC shows that PSO is superior to ABC for 79 rsfMRI brain networks. e proposed algorithm UPSO does not apply to very high-dimensional problems because it more likely needs long execution time. To solve the limitations, UPSO can be designed as a parallel algorithm and implemented in the cloud computing platform. In addition, our proposed algorithm is going on for further improvement, such as designing more efficient coding to speed up its converging rate and stability.
Data Availability e data we used can be publicly available at http://umcd. humanconnectomeproject.org.

Conflicts of Interest
e authors declare no conflicts of interest regarding the publication of this paper.