**5. Co-Occurrence and Timeline Analyses of Keywords**

This section analyzes the co-occurrence of author-keywords and presents the timeline view. In the 1980 s, the co-occurrence analysis was first provided and has been widely applied in bibliometrics analyses [31]. When more than or equal to two keywords appear in the same paper, this can be called keywords occurrence [32]. Through the co-occurrence analysis, we can identify the research hotspots of the journal.

There are 11,731 author-keywords of publications in the journal from 2009 to 2019, according toVOSviewer. Figure 7 presents the author-keywords co-occurrence network, where there are 164 author-keywords, by setting the threshold of minimum occurrences to five and giving the closest relationship. They are classified into 16 clusters marked in different colors. The node presents an author-keyword; its size denotes the citations. The bigger the node is, the more citations the keyword has. A link between two nodes means the co-occurrence of the two keywords. In Figure 7, symmetry has the greatest citations; therefore, it has the most links with other clusters. Then, we present the top 20 most frequent author-keywords and their frequencies in Table 11.

research topics.

*Symmetry* **2020**, *12*, x FOR PEER REVIEW 13 of 19

**Figure 7.** The co-occurrence network of author-keywords. **Figure 7.** The co-occurrence network of author-keywords.


The frequency of symmetry is 92 and ranks first. Following, decision making (37), deep learning **Table 11.** The top 20 most frequent author-keywords in *Symmetry*.

5 machine learning 27 15 cloud computing 16 6 internet of things 22 16 image processing 16 7 symmetry breaking 20 17 clustering 15 The frequency of symmetry is 92 and ranks first. Following, decision making (37), deep learning (30), fixed point (29), and machine learning (27) rank in the second to fifth place, respectively, which explain the research topics of the journal. The frequencies of all the keywords in Table 11 are higher than 10, and seven of them have frequencies greater than 20.

4 fixed point 29 14 fluctuating asymmetry 17

8 neural network 19 18 aggregation operator 14 9 neutrosophic set 19 19 classification 14 10 supersymmetry 19 20 data mining 14 Table 12 explains the top 20 keywords with the strongest citation bursts. Most of them are close to present (2019), such as fuzzy set, decision making, aggregation operator, deep learning, etc., which denote that the research contents in the publications of *Symmetry* closely follow hot topics and leading Table 12 explains the top 20 keywords with the strongest citation bursts. Most of them are close to present (2019), such as fuzzy set, decision making, aggregation operator, deep learning, etc., which denote that the research contents in the publications of *Symmetry* closely follow hot topics and leading issues. Comparing Table 11 with Table 12, we can find several of the same keywords, i.e., symmetry, decision making, symmetry breaking, neutrosophic set, and deep learning. This phenomenon explains that these are the research hotspots of the journal from 2009 to 2019; at the same time, the research fever of these may continue. Thus, the scholars interested in the journal can start from these research topics.

issues. Comparing Table 11 with Table 12, we can find several of the same keywords, i.e., symmetry, decision making, symmetry breaking, neutrosophic set, and deep learning. This phenomenon explains that these are the research hotspots of the journal from 2009 to 2019; at the same time, the research fever of these may continue. Thus, the scholars interested in the journal can start from these

**Table 12.** The top 20 keywords with the strongest citation bursts.

**Keywords Year Strength Begin End 2009–2019**  1 fuzzy set 2009 18.3082 2017 2019 ▂▂▂▂▂▂▂▂▃▃▃ 2 aggregation operator 2009 11.0627 2018 2019 ▂▂▂▂▂▂▂▂▂▃▃


**Table 12.** The top 20 keywords with the strongest citation bursts. 11 TOPSIS 2009 5.7079 2018 2019 ▂▂▂▂▂▂▂▂▂▃▃ 12 network 2009 5.6572 2017 2019 ▂▂▂▂▂▂▂▂▃▃▃

*Symmetry* **2020**, *12*, x FOR PEER REVIEW 14 of 19

3 number 2009 10.6203 2018 2019 ▂▂▂▂▂▂▂▂▂▃▃ 4 symmetry 2009 9.8647 2009 2014 ▃▃▃▃▃▃▂▂▂▂▂ 5 similarity measure 2009 9.2729 2018 2019 ▂▂▂▂▂▂▂▂▂▃▃ 6 information 2009 8.9273 2017 2019 ▂▂▂▂▂▂▂▂▃▃▃ 7 decision making 2009 8.7959 2017 2019 ▂▂▂▂▂▂▂▂▃▃▃ 8 operator 2009 8.3778 2018 2019 ▂▂▂▂▂▂▂▂▂▃▃

Figure 8 shows the keyword clusters visualizations by CiteSpace and summarizes nine clusters for all keywords of publication in *Symmetry*, which are "exact solution", "decision making", "aromaticity", "aesthetic", "deep learning", "cyclic twin", "fixed point", "quantum theory", and "drosophila melanogaster" in order. such as "exact solution", "framework", "three-dimensional space", "neural network", "equational simulation", "fluctuating asymmetry", etc. For the next phase, the publication preferred to occur keywords like "operator", "convolutional neural network", and "support vector machine". We can see that keywords continually change over time, and scholars expanded and deepened the research content on the journal.

occurred most from 2010 to 2013. Next, from 2013 to 2016, a large number of research topics emerged,

**Figure 8.** The keywords clusters visualizations. **Figure 8.** The keywords clusters visualizations.

Then, based on the timeline view of keywords, the development and the research trend of the hotspots from 2009 to 2019 are presented. In Figure 9, there are four stages from the time perspective. Specifically, it focused more on the "symmetry property", "simulation", "aromaticity", "model", and so on, between 2009 and 2010. The keywords of "tilling aperiodicity" and "symmetry group of knot" occurred most from 2010 to 2013. Next, from 2013 to 2016, a large number of research topics emerged, such as "exact solution", "framework", "three-dimensional space", "neural network", "equational simulation", "fluctuating asymmetry", etc. For the next phase, the publication preferred to occur keywords like "operator", "convolutional neural network", and "support vector machine". We can see that keywords continually change over time, and scholars expanded and deepened the research content on the journal.

.

*Symmetry* **2020**, *12*, x FOR PEER REVIEW 15 of 19

Moreover, this paper also retrieves the papers published in 2020 and obtains 691 papers. Figure 10 shows the author-keywords co-occurrence network of these papers by setting the minimum number of occurrences of a keyword to two and displaying the closest network, which is related to 138 keywords. The side of the nodes denotes the frequency of co-occurrence. We can see that machine learning has the greatest frequency (24), followed by symmetry (20), deep learning (9), fixed point (9), asymmetry (8), dark matter (8), classification (7), dark energy (7), fuzzy logic (7), etc. The tenth to twentieth author-keywords are internet of things, particle swarm optimization, sustainability, conservation laws, convergence, hermite-hadamard inequality, stability analysis, ahp, artificial intelligence, bioconvection, and cloud computing. Moreover, this paper also retrieves the papers published in 2020 and obtains 691 papers. Figure 10 shows the author-keywords co-occurrence network of these papers by setting the minimum number of occurrences of a keyword to two and displaying the closest network, which is related to 138 keywords. The side of the nodes denotes the frequency of co-occurrence. We can see that machine learning has the greatest frequency (24), followed by symmetry (20), deep learning (9), fixed point (9), asymmetry (8), dark matter (8), classification (7), dark energy (7), fuzzy logic (7), etc. The tenth to twentieth author-keywords are internet of things, particle swarm optimization, sustainability, conservation laws, convergence, hermite-hadamard inequality, stability analysis, ahp, artificial intelligence, bioconvection, and cloud computing.

**Figure 10.** Author-keywords co-occurrence network of publications in 2020. **Figure 10.** Author-keywords co-occurrence network of publications in 2020.

#### **6. Discussions 6. Discussions**

This journal explores symmetry phenomena related to mathematical, physics, interdisciplinary fields, etc. After the bibliometric analysis, we further discussed the possible reasons and given future suggestions. According to analyses of *Symmetry*, the basic characteristics, citation structures, and productive objects are summarized as follows: This journal explores symmetry phenomena related to mathematical, physics, interdisciplinary fields, etc. After the bibliometric analysis, we further discussed the possible reasons and given future suggestions. According to analyses of *Symmetry*, the basic characteristics, citation structures, and productive objects are summarized as follows:

(1) The most frequent type of publication is the article, occupying 94.11% of the publications. There were more high-quality papers published in 2020, 2017, and 2018, in the view of TC and Hindex. The trend of the publication-year distribution increased from 2013 to 2019. As of June, 2020, the publications received the most citations (3324) in 2018. The publications in 2019 were cited 2234 times; this year not only had the greatest number of papers, but also ranks second place in terms of citation frequency, the reason for which may be that an increasing amount of scholars are paying attention to the journal. (1) The most frequent type of publication is the article, occupying 94.11% of the publications. There were more high-quality papers published in 2020, 2017, and 2018, in the view of TC and H-index. The trend of the publication-year distribution increased from 2013 to 2019. As of June, 2020, the publications received the most citations (3324) in 2018. The publications in 2019 were cited 2234 times; this year not only had the greatest number of papers, but also ranks second place in terms of citation frequency, the reason for which may be that an increasing amount of scholars are paying attention to the journal.

(2) Since most of the top 15 cited publications are cooperative, communication plays a key role in improving the level of publications. From the cooperation network, China receives the largest number of cooperation and is at the core place; at the same time, it is the country with the greatest number of publications. (2) Since most of the top 15 cited publications are cooperative, communication plays a key role in improving the level of publications. From the cooperation network, China receives the largest number of cooperation and is at the core place; at the same time, it is the country with the greatest number of publications.

(3) Publications cover 100 countries/regions, and the most influential countries/regions are mainly from Asia and Europe. In terms of TP, TC, and H-index, China led, which shows that publications in *Symmetry* from China have higher influence, followed by the USA. From the top 10 most cited references/sources/authors: (a) the publications cite reference universally; (b) the cited (3) Publications cover 100 countries/regions, and the most influential countries/regions are mainly from Asia and Europe. In terms of TP, TC, and H-index, China led, which shows that publications in *Symmetry* from China have higher influence, followed by the USA. From the top 10 most cited references/sources/authors: (a) the publications cite reference universally; (b) the cited sources are

mostly in the field of physics and fuzzy mathematics; (c) the researchers can pay close attention to the papers of the top cited authors and sources.

(4) Combined with the strong citation burst analyses, we can find that the top 10 lists between Tables 8 and 9 are different. This phenomenon explains that the author citation bursts at various times, and especially the emergence of some new authors, including Torra V. and Wang H. For the same authors, the ranks also exist discrepancy. The reason may be that the research topics of the journal are constantly enriched and there are diverse focuses, i.e., from physics to comprehensive discipline, including decision making, fuzzy mathematic, and deep learning.

(5) Considering the top 20 most frequent author-keywords and the top 20 keywords with the strongest citation bursts, in recent years, the main research contents of *Symmetry* focus on fuzzy set, aggregation operation, etc. The burst detection and the timeline view analyses of keywords show the knowledge structure and research trends in the journal. According to the results, the following topics can be considered in the future: (a) to process the complex and diverse raw data, and investigate new operators; (b) to study the symmetry phenomena in the artificial intelligence; (c) to excavate the symmetry nature in matching problems, aimed at solving more social management problems; (d) to predict the possible time change trends and their weights in dynamic issues; (e) to study the intelligent algorithms and promote their stability and reliability.

Due to the characteristics of publications, the most influential objects, and the co-occurrence analyses of author-keywords, regarding the publications from 2009 to 2019 in *Symmetry*, we find that, although the papers were first published in 2009, the numbers of publications have been increasing until the present. From 2015, the annual number of publications always exceeded 100. This suggests that the journal has constructed its influence on multidisciplinary theory and practice. Especially, authors from Asia and Europe pay more attention to this journal. In the future, it can expand its influence through cooperation. Then, the analyses results suggest that scholars should investigate advanced techniques (such as neural network, data mining, fuzzy decision-making, etc.) to keep pace with the times and solve the practical problems. Besides, with the uncertainty and diversification of the environment, enriching the research contents of the journal, at the same time, promoting the robustness of theory methods, etc. are also challenges for future scholars.
