I Introduction
The Full Model PSO can be modeled using combinations of Ring and Star topologies in combination with Synchronous and Asynchronous Particle Updates. The four types of Particle Swarm Optimization (PSO) algorithm are the Full Model, Cognition Model, Social Model, and Selfless Model. The Full Model learns from itself and others , . The Cognition Model learns from itself , . The Social Model learns from others , . The Selfless Model learns from others , , except for the best particle in the swarm, which learns from changing itself randomly () [4].
There are two types of PSO topologies: Ring and Star. The star topology is dynamic, but the ring topology is not. For the star neighborhood topology, the social component of the particle velocity update reflects information obtained from all the particles in the swarm [1].
There are two types of particle update methods: asynchronous and synchronous. The asynchronous method updates the particles one at a time, while the synchronous method updates the particles all at ones. The asynchronous update method is similar to the SteadyState Genetic Algorithm update method, while the synchronous update method is similar to the Generational Genetic Algorithm update method. The Asynchronous Particle Update Method allows for newly discovered solutions to be used more quickly
[4]. Synchronous updates are done separately from particle position updates. Asynchronous updates calculate the new best positions after each particle position update and have the advantage of being given immediate feedback about the best regions of the search space. Feedback with synchronous updates is only given once per iteration. Carlisle and Dozier reason that asynchronous updates are more important for lbest PSO where immediate feedback will be more beneficial in loosely connected swarms, while synchronous updates are more appropriate for gbest PSO [1].Having the algorithm terminate when a maximum number of iterations, or function evaluations, has been exceeded is useful when the objective is to evaluate the best solution found in a restricted time period [1].
Ii Methodology
In PSO, the vectors are
, , and , where represents the particle and represents the dimension. The xvector represents the current position in search space. The pvector represents the location of the best solution found so far by the particle. The vvector represents the gradient (direction) that the particle will travel if undisturbed [4].The Fitness Values are and . The xfitness records the fitness of the xvector. The pfitness records the fitness of the pvector [4].
Iia Ring Topology with Synchronous Particle Update PSO
Ring Topology with Synchronous Particle Update PSO (RS PSO) is used for sparsely connected population so as to speed up convergence. In this case the particles have predefined neighborhood based on their location in the topological space. The connection between the particles increases the convergence speed which causes the swarm to focus on the search for local optima by exploiting the information of solutions found in the neighborhood. Synchronous update provides feedback about the best region of the search space once every iteration when all the particles have moved at least once from their previous position.
IiB Ring Topology with Asynchronous Particle Update PSO
The Ring Topology with Asynchronous Particle Update PSO (RA PSO) has information move at a slower rate through the social network, so convergence is slower, but larger parts of the search space are covered compared to the star structure. This provides better performance in terms of the quality of solutions found for multimodal structures than those found using the star structure. Asynchronous updates provide immediate feedback about the best regions of the search space, while synchronous updates only provide feedback once per iteration.
IiC Star Topology with Synchronous Particle Update PSO
The Star Topology with Synchronous Particle Update PSO (SS PSO) uses a global neighborhood with the star topology. Whenever searching for the best particle, it checks every particle in the swarm instead of just the neighborhood of three used in a ring topology. The synchronous update only provides feedback once each cycle, so all the particles in the swarm will update their positions before more feedback is provided, instead of checking to see if one of the recently updated particles has a better fit than the particle deemed best fit at the beginning of the cycle.
IiD Star Topology with Asynchronous Particle Update PSO
The Star Topology with Asynchronous Particle Update PSO (SA PSO) has particles moving all at once in the search space, which allows for newly discovered solutions to be used more quickly. The Star Topology uses a global neighborhood, meaning that the entire swarm can communicate with one another and each particle bases its search off of the global best particle known to the swarm. The benefit of using a global neighborhood is that it allows for quicker convergence since the best known particle is communicated to all the particles in the swarm.
Iii Experiment
The experiment consists of four instances of a Full Model PSO with a cognition learning rate, , and a social learning rate, , equal to 2.05. To regulate the velocity and improve the performance of the PSO, the constriction coefficient implements to ensure convergence.
The inertia weight, , is also implemented to control the exploration and exploitation abilities of the swarm. Both topologies in this experiment use an value of 1.0, in order to facilitate exploration and increase diversity. The particles in this experiment are updated in two different ways: synchronously, and asynchronously.
Asynchronous Particle Update is a method that updates particles one at a time and allows newly discovered solutions to be used more quickly, while Synchronous Particle Update is a method that updates all the particles at once. The four instances of the PSO are variations of the two Particle Update methods, and the two topologies described.
With these four instances of the PSO, a population of 30 particles is evolved and each particle’s fitness is evaluated; this is done 30 times for each PSO. The number of function evaluations is observed after each population of 30 is evolved, and these 30 best function evaluation values for 30 runs are used to perform ANOVA tests and TTests to determine the equivalence classes of the four instances of the PSO.
Iiia Ring Topology with Synchronous Particle Update PSO
The RS PSO updates synchronously at the end of every iteration. It uses ring topology to compare and select the best solution within the neighborhood of three.
IiiB Ring Topology with Asynchronous Particle Update PSO
The RA PSO updates asynchronously, which allows for quick updates, and uses ring topology to compare solutions within a neighborhood of three.
IiiC Star Topology with Synchronous Particle Update PSO
The SS PSO updates synchronously, which only allows for one update per iteration, and uses star topology to compare solutions with a global neighborhood.
IiiD Star Topology with Asynchronous Particle Update PSO
The SA PSO updates asynchronously, which allows for quicker updates on newly discovered solutions. The star topology uses a global neighborhood to compare solutions, which allows for quicker convergence.
Iv Results



Run 
RS  RA  SS  SA 
1 
4000  77  129  75 
2 
4000  71  57  72 
3 
82  82  82  65 
4 
62  60  4000  71 
5 
4000  72  49  56 
6 
72  4000  48  4000 
7 
95  4000  83  189 
8 
45  4000  4000  4000 
9 
71  54  4000  4000 
10 
61  68  91  4000 
11 
4000  66  38  89 
12 
50  4000  71  4000 
13 
4000  4000  4000  4000 
14 
4000  72  4000  4000 
15 
4000  65  4000  4000 
16 
4000  57  4000  146 
17 
54  69  58  4000 
18 
76  81  65  53 
19 
58  77  47  4000 
20 
4000  95  4000  4000 
21 
55  4000  89  56 
22 
90  65  51  4000 
23 
4000  72  4000  4000 
24 
4000  4000  4000  73 
25 
90  4000  55  52 
26 
55  4000  4000  4000 
27 
4000  58  61  40 
28 
65  4000  47  4000 
29 
62  4000  110  4000 
30 
68  4000  68  64 
Average 
1640.3667  1642.0333  1509.9667  2170.0333 



Groups 
Count  Sum  Average  Variance 
RS 
30  49211  1640.3667  2840033.757 
RA 
30  49261  1642.0333  3834547.482 
SS 
30  45299  1509.9667  3713758.378 
SA 
30  65101  2170.0333  3959879.413 





SS  df  MS  F  Pvalue  F crit  
Between 
7721005  3  2573668  0.7  0.6  2.7  
Within 
445098352  116  3837055  
Total 
452819357  119  

The results place all four algorithms in the same equivalence class using both the ANOVA and Student Ttests. When the ANOVA test and TTest are performed, the ANOVA test of the four algorithms yields a pvalue of 0.57, so the FTest is then performed to determine which twotailed twosample TTest to use. In each comparison between algorithms, the TTest results in a t Stat value that is smaller than the t Critical value, therefore the null hypothesis is accepted. The data set used is shown in Table
I, while the ANOVA test results are shown in Tables II and III. Representative Ttests are shown in Tables IV and V.Iva Ring Topology with Synchronous Particle Update PSO




RS  SS 
Mean 
1640.367  1509.967 
Variance 
3840033.757  3713758.37 
Observations 
30  3 
Pooled Variance 
3776896.068  
Hypothesized Mean Difference 
0  
df 
58  
t Stat 
0.2599  
P(Tt) onetail 
0.3979  
t Critical onetail 
1.6716  
P(Tt) twotail 
0.7959  
t Critical twotail 
2.0017  

The RS PSO results is the better compared to SA PSO as observed from the Ttest. It provides comparable quality solutions to RA PSO but is slower than RA PSO as it waits for all the particles to be updated. The SS PSO outperforms RS PSO by a significant margin as it has an appreciably lower mean than RS PSO when subjected to Ttest. The Ttest is shown in Table IV.
IvB Ring Topology with Asynchronous Particle Update PSO
The RA PSO results in better quality solutions than the SA PSO, since larger parts of the search space are covered compared to the star structure. Using the RA PSO, solutions are found more quickly than when using a RS PSO. The SS PSO is relatively more slow of an algorithm and results in solutions of lesser quality.
IvC Star Topology with Synchronous Particle Update PSO
The SS PSO is found to be in the same equivalence class as all the other algorithms in the experiment. However, the mean value of the SS PSO is slightly smaller than the mean values of the other three algorithms. It appears that it is able to find solutions slightly more quickly than the algorithms using the ring topology as it compares solutions using a global neighborhood allowing for quicker convergence. The Ttest is shown in Table V.
IvD Star Topology with Asynchronous Particle Update PSO




SS  SA 
Mean 
1509.967  2170.033 
Variance 
3713758.378  3959879.413 
Observations 
30  30 
Pooled Variance 
3836818.895  
Hypothesized Mean Difference 
0  
df 
58  
t Stat 
1.305  
P(Tt) onetail 
0.0985  
t Critical onetail 
1.6716  
P(Tt) twotail 
0.1970  
t Critical twotail 
2.0017  

The SA PSO is found to be in the same equivalence class as all of the other algorithms in this experiment. The mean value of the SA PSO is larger than the mean values of the other three algorithms; and the F value is found to be larger than the F crit value when comparing the SA PSO to each of the other algorithms as well, so the TTest: Two Sample Assuming Equal Variances is performed. In each comparison of the SA PSO to the other three algorithms, the TTest results in a t Stat value that is smaller than the t Critical twotail value, therefore the null hypothesis is accepted that the hypothesized mean difference is zero, since the t Stat value is less than the t Critical twotail value.
IvE Comparison of Run Times
A comparison of the run times for each algorithm shows that the asynchronous algorithms run more quickly than the synchronous algorithms, and the ring algorithms result in longer run times than the star algorithms. The SA PSO algorithm has an average runtime of 22.56 seconds, based on 30 runs of the algorithm. The RS PSO algorithm has an average runtime of 18.07 seconds, based on 30 runs of the algorithm. The SS PSO algorithm has an average runtime of 7.07 seconds, based on 30 runs of the algorithm. The SA PSO algorithm has an average runtime of 4.65 seconds, based on 30 runs of the algorithm.
The SA PSO algorithm has the smallest run time, while the RS PSO algorithm has the longest run time.
V Conclusions
The four different types of PSO are significant in their own way and have different applications. The Ring and Star topologies determine the scope of feedback whereas the synchronous or asynchronous method choice decides the nature of feedback.
The results indicate that all four algorithms are in the same equivalence class, so there is no statistically significant difference in their performance. The Ttests indicate that the best quality solutions are provided by Star Synchronous algorithm.
The SS PSO algorithm is the quickest algorithm, while the RS PSO algorithm is the slowest. These results are as expected and show that the asynchronous algorithms are quicker than the synchronous algorithms and the star algorithms have a significantly smaller run time than that of the ring algorithms.
Vi Breakdown of the Work
Alison Jenkins  RA PSO and Introduction, Methodology, (Introduction). RA PSO part in Methodology, Experiment, and Results sections of LaTeX report.
Vinika Gupta  RS PSO and Methodology (Modification and Conclusion). RS PSO part in Methodology, Experiment, and Results sections. Full editing and modification of LaTeX report.
Alexis Myrick  SS PSO and Result. SS PSO part in Methodology, Experiment, and Results sections of LaTeX report.
Mary Lenoir  SA PSO and Experiment. SS PSO part in Methodology, Experiment, and Results sections of LaTeX report.
References
 [1] Engelbrecht, Andries P. Computational Intelligence: An Introduction. John Wiley & Sons, 2007.
 [2] Joseph, Anthony D., et al. Adversarial Machine Learning. Cambridge University Press, 2018.
 [3] Sarkar, Dipanjan. Text Analytics with Python. Apress, 2016.
 [4] Dozier, J. Computational Intelligence and Adversarial Machine Learning: Particle Swarm Optimization. Powerpoint Presentation, COMP6970  Computational Intelligence and Adversarial Machine Learning class, Auburn University, 2019.
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