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Questions on node-weighted centrality or hybrid centrality

 

Dear Prof. Borgatti and all,

?

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:

?

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.

?

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:

?

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.

?

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.

?

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.

?

To recap, I list the four questions as follows:

?

1. Whether we can account for weights of nodes when calculating different types of centrality, especially in population software like R.

?

2. Whether we can use individual-level attribute like wealth, educational level or personality variables to indicate the weight.

?

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.

?

4. Whether we can also understand the centrality based on the product of two matrix as one form of hybrid centrality.

?

I look forward to any suggestions from all of you in the list. Thanks in advance!

?

Best,

Chuding

?

References

?

Borgatti, S., Everett, M., Johnson, J., & Agneessens, F. (2022). Analyzing social networks using R. SAGE Publications.

?

Everett, M. G., & Borgatti, S. P. (2014). Networks containing negative ties. Social Networks, 38, 111–120.

?

Singh, A., Singh, R. R., & Iyengar, S. R. S. (2020). Node-weighted centrality: A new way of centrality hybridization. Computational Social Networks, 7, 6.

?

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. 1–27). Springer.

 

Hi
See my answers below.

Martin

On Saturday 10 August 2024 at 07:31:23 BST, CHUDING LING <lingchuding@...> wrote:


Dear Prof. Borgatti and all,

?

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:

?

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.

?

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:

?

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.

?

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.

?

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.

?

To recap, I list the four questions as follows:

?

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.??

?

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

?

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.

?

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.

?

I look forward to any suggestions from all of you in the list. Thanks in advance!

?

Best,

Chuding

?

References

?

Borgatti, S., Everett, M., Johnson, J., & Agneessens, F. (2022). Analyzing social networks using R. SAGE Publications.

?

Everett, M. G., & Borgatti, S. P. (2014). Networks containing negative ties. Social Networks, 38, 111–120.

?

Singh, A., Singh, R. R., & Iyengar, S. R. S. (2020). Node-weighted centrality: A new way of centrality hybridization. Computational Social Networks, 7, 6.

?

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. 1–27). Springer.

 

Dear Prof. Everett,

Thanks for your prompt and detailed responses! I will read the sections you recommend again.

?

Best,

Chuding



meverett61 via <meverett61=yahoo.com@groups.io> 于2024年8月11日周日 19:19写道：

Hi
See my answers below.

Martin

On Saturday 10 August 2024 at 07:31:23 BST, CHUDING LING <lingchuding@...> wrote:


Dear Prof. Borgatti and all,

?

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:

?

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.

?

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:

?

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.

?

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.

?

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.

?

To recap, I list the four questions as follows:

?

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.??

?

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

?

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.

?

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.

?

I look forward to any suggestions from all of you in the list. Thanks in advance!

?

Best,

Chuding

?

References

?

Borgatti, S., Everett, M., Johnson, J., & Agneessens, F. (2022). Analyzing social networks using R. SAGE Publications.

?

Everett, M. G., & Borgatti, S. P. (2014). Networks containing negative ties. Social Networks, 38, 111–120.

?

Singh, A., Singh, R. R., & Iyengar, S. R. S. (2020). Node-weighted centrality: A new way of centrality hybridization. Computational Social Networks, 7, 6.

?

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. 1–27). Springer.

 

Dear Prof. Everett (and all),

?

After reading the sections recommended again, I agree with that it would be clearer in concept if we only weight either edges or nodes rather than both of them. Otherwise, if we get a large score, we cannot determine whether this is because of the strong tie strength or a high level of node attribute (e.g., wealth). In addition to this, I have three follow-up questions.

?

1. From the perspective mentioned above, it seems that we had better use binary network when calculating PN centrality, though Everett and Borgatti (2014) indicates that the algorithm can be applied to directed valued network and the xPNCentrality also supports. Is this understanding right?

?

2. For measures of alter composition, the current content in the book is mainly about degree, which focuses on alters directly connected with the ego. I think maybe it is not easy to depict the composition of alters which the ego indirectly connected with. This is because we cannot confirm the scope of the indirect ties that we need to include in. After all, we can reach everyone in the world through six steps (Milgram, 1967). As such, if we include the nodes which the ego indirectly connected with when calculating composition of alter, probably the variance of this variable would be small because every ego can reach all the others ultimately. Therefore, I think we cannot test idea like: Beta centrality × Composition of alters accounted for calculating Beta → Dependent variable. And this conclusion can be applied to all the centrality measures accounting for indirect ties like Eigenvector, Beta, k-step reach, Beta reach. Is this understanding right?

?

3. However, I am not sure whether we can calculate the composition of alters involved in calculating closeness and betweenness, which do not involve indirect ties (if my understanding is correct). I am especially interested in the latter because closeness “is of limited use as a centrality measure” (Borgatti et al., 2022). And if yes, how shall we determine the alters and then calculate the composition of them?

?

I look forward to hearing from you. Thanks again in advance!

?

Best,

Chuding



meverett61 via <meverett61=yahoo.com@groups.io> 于2024年8月11日周日 19:19写道：

Hi
See my answers below.

Martin

On Saturday 10 August 2024 at 07:31:23 BST, CHUDING LING <lingchuding@...> wrote:


Dear Prof. Borgatti and all,

?

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:

?

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.

?

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:

?

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.

?

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.

?

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.

?

To recap, I list the four questions as follows:

?

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.??

?

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

?

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.

?

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.

?

I look forward to any suggestions from all of you in the list. Thanks in advance!

?

Best,

Chuding

?

References

?

Borgatti, S., Everett, M., Johnson, J., & Agneessens, F. (2022). Analyzing social networks using R. SAGE Publications.

?

Everett, M. G., & Borgatti, S. P. (2014). Networks containing negative ties. Social Networks, 38, 111–120.

?

Singh, A., Singh, R. R., & Iyengar, S. R. S. (2020). Node-weighted centrality: A new way of centrality hybridization. Computational Social Networks, 7, 6.

?

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. 1–27). Springer.

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