Several major concerns have been raised in the article "The State of Social Science Research on COVID-19" (Liu et al., 2022): (1) no framework was used to highlight the overall research; (2) author weighted schemes (AWS) (Chien et al., 2019; Kan et al., 2020) were not applied to fairly quantify the contributions made by entities to articles; (3) the size of nodes in social network analysis (SNA) was inconsistently defined in keyword co-occurrence relations and coauthorship relations. A node's size, for example, represents both the occurrence counts in the keyword co-occurrence network (or the number of links between terms) in Figs. 1 to 3 of the study (Liu et al., 2022).

A study framework

In the study (Liu et al., 2022) relation-based analyses were employed, including three categories of analysis: (1) citation relations, (2) word co-occurrence, and (3) coauthorship relations (Li et al., 2021). Table 1 illustrates a framework excluding the parts indicated by the ■ symbol, but √ representing the existence in this article (Liu et al., 2022). Due to the lack of quantification of the credits in article bylines, we assumed that the original analyses were based on the first author in countries/institutions.

Table 1 Study structure of methods and representations for each topical entity
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The AWS applied to bibliometrics

Using the AWS, Table 2 illustrates four articles with ten connections, including one single author (#1) and three coauthors (#2 to 4). The size of the nodes represents the number of occurrences in the keyword (replaced by country here) co-occurrence network defined in the study (Liu, Yuan, & Zhu, 2021). In this case, there are 3, 2, 2, and 2 occurrences in the US, China, UK, and India. Based on the size of the nodes, the total number of links for the US, China, UK, and India is 4, 6, 5, and 5, respectively; see Table 3. It appears that the results of the two computations are different.

Table 2 Author-affiliated countries in four articles
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Table 3 Relations on the traditional SNA
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When the Equal AWA, defined in Eq. 1, is applied, the weights are endorsed in Table 4 and the co-occurrences are shown in Table 5. Total centrality degree (CD) equals 3 (i.e., the number of articles excluding the single-author article (#1).

Table 4 Weights on the equal AWS
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Table 5 Relations on the Equal AWS
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$${W}_{i}=\frac{1}{L},$$
(1)

In Table 4, L represents the number of elements in the article, and Wi represents the weight assigned to each element.

Similar to the above, if the nonequal AWA is applied, the weights are endorsed in Table 6, and the co-occurrences are shown in Table 7. As a result, the total centrality degree (CD) equals 3 (= the number of articles, excluding the single-author article (#1)).

Table 6 Weights on the non-equal AWS
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Table 7 Relations on the non-equal AWS
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A consistent scheme applied to bibliometrics

According to Table 8, the total centrality degree (CD) will equal 4 (the number of articles) if the single-author article (#1) is included using the nonequal AWS. According to Table 9, if each article contains ten citations, the total centrality degree (CD) equals 40 (= the total number of articles).

Table 8 Publications on the Non-Equal AWS including one single-author article (#1)
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Table 9 Citations on the non-equal AWS including one single-author article (#1)
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Discussions

All weights equal article number

Traditionally, CD is defined by the number of co-occurring entities in an article. It is possible that the sum of CD across all entities in all articles is not equal to the number of articles in the study. In order for all weighted CD (WCD) in Eq 2 to equal the number of articles via Eq, a specific adjustment must be made via Eqs. 1, 2, 3.

$$ \begin{aligned} WCD\;in\;an\;article\; & = \left( {\sum\nolimits_{{i = 1}}^{{L - 1}} {\sum\nolimits_{{j = i + 1}}^{L} {(W_{i} + W_{j} )} } } \right) \div (L - 1) \\ = \;2 \times W_{i} \times \frac{{L \times (L - 1)}}{2} \div (L - 1) \\ = 2 \times \frac{1}{L} \times \frac{{L \times (L - 1)}}{2} \div (L - 1) = 1, \\ \end{aligned} $$
(2)

In an article byline, L indicates the number of co-occurring entities (e.g., authors or countries). The Wi is the equal weight for an entity in an article byline; see Eq. 1 (or the unequal weight in Table 5) (Chien et al., 2019; Kan et al., 2020). According to Eq. 2, Wi equals Wj. For each entity in an article, the WCD is equal to 1.0. In contrast to traditional SNA, which ignores the WCD equal to the number of articles, the total WCD is equal to the number of articles (see the panel IV in Fig. 1). The network would look similar to that shown in Fig. 2 when SNA was performed using Pajek with codes in Fig. 2.

Fig. 1
figure 1

Four scenarios are visually displayed

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Fig. 2
figure 2

An example of the scenario with an equal weight assigned to each term based on word co- occurrence in Table 2.

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For instance, there are two keywords A and B in an article. The WCD = \(2 \times \frac{1}{2} \times \frac{2 \times (2 - 1)}{2} \div (2 - 1) = 1\). In a similar manner, WCD equals 1 if L exceeds 2. For a specific keyword k, the WCD is the sum of Wi in all articles in Eq. 3.

$${WCD}_{k}=\sum_{j=1}^{n}{W}_{j},$$
(3)

whereas n = 4 in our study. The WCDk is thus sized by a bubble in SNA.

All citation weights equal article number multiplied by citations

In the case of article citations (denoted by the symbol IF), the weights of a specific entity (e.g., WCD in Eq. 2) are combined with IF to calculate the IFWCDk by employing the following equations.

$$ \begin{aligned} {\text{IFWCD}}\;{\text{in}}\;{\text{an}}\;{\text{article}}\; & = IF \times \left( {\sum\nolimits_{{i = 1}}^{{L - 1}} {\sum\nolimits_{{L = i + 1}}^{L} {(W_{i} + W_{L} )} } } \right) \div (L - 1) \\ = IF \times 2 \times W_{i} \times \frac{{L \times (L - 1)}}{2} \div (L - 1) \\ = \;IF \times 2 \times \frac{1}{L} \times \frac{{L \times (L - 1)}}{2} \div (L - 1) = IF, \\ \end{aligned} $$
(4)
$$ IFWCD_{k} = (\sum\nolimits_{j = 1}^{n} {IFWCD_{j} ) \div WCD_{k} } $$
(5)
$$Citations=\sum_{k=1}^{n}{IFWCD}_{k},$$
(6)

where \({IFWCD}_{k}\) is the mean IF of entity k. All citations in articles are composed of individual \({IFWCD}_{k}\) in Eq. 6. Accordingly, the bubble can be sized by \({IFWCD}_{k}\) for each entity.

Conclusion

While VOSviewer (van Eck & Waltman, 2010, 2017) was used to construct and visualize the scientometric networks, the counts within each topical entity were not consistently integrated into a common scheme. Several adjustments should be applied before feeding the relations to VOSviewer (e.g., Tables 3, 5, 7, 8, 9) in SNA, or to other SNA software, such as Pajek (de Nooy et al., 2011) providing the upload files to node sizes and edge values.

If this is the case, all weights in SNA are equal to the number of publications (or citations). Bubbles (or nodes) should be of a reasonable size. With relation-based analyses, the three categories (citation relations, word co-occurrences, and coauthorship relations) could be compared fairly and reasonably more than ever before in traditional SNA.