Dear
Prof. Borgatti and all,
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I am a
reader of your book ¡°Analyzing social networks using R¡± (Borgatti et al.,
2022). Recently I notice that there is a stream of
literature called node-weighted centrality or hybrid centrality (Singh et al., 2020; Singh,
2022). This literature points out that previous
measures of centrality account for weight on edges (i.e., valued network) or
length (e.g., eigenvector or beta centrality). However, these previous measures
treat nodes as uniform. But it is very likely that different alters which an
ego connects with are different in terms of a particular attribute (i.e., has different
levels of importance). Let me use an example from Singh et al. (2020) to further illustrate:
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For a
friendship network, nodes represent persons and edges represent the friendship
relationship between the considered set of persons. Here, weights on the
nodes can be understood as a mapping of wealth, power, education level, or some
other attribute of persons. It is notable that existence of two persons
with identical attributes is highly unlikely and therefore all person¡¯s
attributes can be mapped to different real values based on the application
specific mapping.
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Therefore,
my first question is whether we can account for such weights of nodes when
calculating different types of centrality. Of note, I have quickly gone
through the literature on node-weighted centrality or hybrid centrality,
including 18 methodological or statistical research cited by Singh and
colleagues (Singh
et al., 2020; Singh, 2022), though I do not fully understand the mathematical
parts. However, it seems that all of them use another
type of centrality as the weight when calculate a particular type of centrality.
For example, they use betweenness centrality as the weight when calculating
degree centrality. From this perspective, your work on PN centrality might be
also understood as a hybrid centrality (Everett & Borgatti, 2014). Particularly, in Everett and Borgatti (2014), you state:
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Note
that this equation weights everyone the same, but in many cases we might
prefer a measure in which actors with high centrality affect the centrality
score of the actors they are connected to more than those with a low score.
In other words, we might want to capture the notion that to have negative
connections with actors who have a low centrality score is better than having
negative connections to actors who have a high score.
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Nevertheless,
all the literature aforementioned do not fully capture the notion in the
example from Singh
et al. (2020) that I cited. That is, they use
relational attributes to obtain the weight. Therefore, my second question is
whether we can use individual-level attribute like wealth, educational level or
personality variables to indicate the weight.
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Related
to the PN centrality in Everett and Borgatti (2014), I have the third question.
That is, whether we can multiply the positive network and negative network
using the matrix multiplication function in R first, and then calculate a
particular type of centrality to obtain the weighted centrality. According
to your example in Chapter 2 of Borgatti et al. (2022), maybe we can
indicate the first matrix as the negative network and the second matrix as the
positive network, then the product of the two networks indicates the extent to
which an ego is negatively connected to alters who are positively connected to
others. Therefore, conceptually it may also capture ¡°the notion that to have
negative connections with actors who have a low centrality score is better than
having negative connections to actors who have a high score.¡± However,
after experimenting with the example data provided in PN centrally section, I
find the results are significantly different. Nevertheless, my fourth
question is whether we can also understand the centrality based on the product
of two matrix as one form of hybrid centrality.
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To recap,
I list the four questions as follows:
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1. Whether
we can account for weights of nodes when calculating different types of
centrality, especially in population software like R.
First I refer to the R book so you can read what we say. I suggest you look at section 8.4.2. In the book we deal with degree and weighted attributes and these are in the chapter about node level measures. We do not (mainly through lack of space) discuss other centrality measures. There would be a vast array of possible measures. If the edges do not have weights the simplest way is to use the node attribute weight on the edge. If the edges do have weights then some method of combining node attribute weight and edge weight would need to be done. What you do needs to depend on exactly what is being measured. For example if you are in an organisation and the edges are number of times an advice interaction takes place in a week and the node attributes are tenure you could take the product but this assumes this is meaningful in some way as these are two very different quantities.
In summary this is about research questions and data so yes it is possible but personally to enable easyy interpretation I would stick with the degree type example we have in the book.??
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2. Whether
we can use individual-level attribute like wealth, educational level or
personality variables to indicate the weight.
Yes this is done all the time
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3. Whether
we can multiply the positive network and negative network using the matrix
multiplication function in R first, and then calculate a particular type of
centrality to obtain the weighted centrality.
Yes this is possible but again it is about interpretation. See section 2.6. After multiplication the network you have is a count of composite relations. Suppose P is friend and N is enemy then the entries of PN(i,j) count the number of i's friends who have j as an enemy. If on PN we ran weighted degree centrality then the raw score would be a count of the total number of friends who have enemies relation. Note NP is different to PN.
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4. Whether
we can also understand the centrality based on the product of two matrix as one
form of hybrid centrality.
No this is really covered in the PN answer above as this is the same thing in essence.
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I look
forward to any suggestions from all of you in the list. Thanks in advance!
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Best,
Chuding
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References
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Borgatti, S., Everett, M.,
Johnson, J., & Agneessens, F. (2022). Analyzing social networks using R.
SAGE Publications.
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Everett, M. G., & Borgatti,
S. P. (2014). Networks containing negative ties. Social Networks, 38,
111¨C120.
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Singh, A., Singh, R. R., &
Iyengar, S. R. S. (2020). Node-weighted centrality: A new way of centrality
hybridization. Computational Social Networks, 7, 6.
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Singh, R. R. (2022). Centrality
measures: A tool to identify key actors in social networks. In A. Biswas, R.
Patgiri, & B. Biswas (Eds.), Principles of Social Networking: The New
Horizon and Emerging Challenges (pp. 1¨C27). Springer.
