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help > RE: Pragmatic graph theory definitions
Sep 21, 2017 06:09 PM | Pravesh Parekh - National Institute of Mental Health and Neurosciences
RE: Pragmatic graph theory definitions
Hi Janelle,
One way to think about the graphs is to think of a matrix with the nodes corresponding to the rows and columns and the edges being the values contained therein. For example, consider a matrix with two rows and two columns. Then, there are four entries in this matrix:
r1c1 ; r1c2
r2c1 ; r2c2
This can be thought of as a matrix with 2 nodes (say A and B) and the entries of the matrix are the connections or edges. Therefore, r1c1 would be a connection from A-A and r2c2 would be a connection from B-B (self connections). A connection between A-B would be the entries r1c2 and r2c1. If it is an undirected graph, then these values would be the same. A directed graph would have a value for the edge A->B (A to B) and B->A (B to A).
Furthermore, these edges can have a weight or else be a binary value. If it carries a weight (such as the Fisher transformed correlation coefficient), it is a weighted graph and if the edges are simply 0 and 1, it is an unweighted graph (a value of 1 would mean that the connection exists; 0 would mean that the connection does not exist).
Now to come to your questions: yes, nodes are the ROIs and the edges are the Fisher transformed correlation coefficients.
1. There are several metrics which take into account the actual edge value. However, it also depends on whether you are using unweighted graph or weighted graph. Most of the times, the edge weights are discarded and people tend to analyze these "binarized" graph (by setting, say, a value above a threshold to be equal to 1 and all others to 0). However, for most of the metrics, there is a method to take into account the actual weight or value of the edge too. For example, weighted degree, weighted path length, etc. You may want to check Appendix A of the Rubinov and Sporns (2010) paper.
2. You could think of the path length (in case of a binarized graph) as the number of nodes you need to hop in order to travel from a particular source node to a destination node. Say for example, you would like to calculate the path length between two ROIs. You would work out the minimum number of nodes that you need to move to in order to reach the destination ROI. This makes more sense if you consider an anatomical network; shorter paths could then be interpreted as reflecting a stronger potential for functional integration. For a functional network, interpretation of path lengths are not that straightforward. You may want to take a look at few papers to get additional information on that; as far as weighted path lengths go, usually an inverse mapping is done between the correlation coefficient and the path length such that stronger weights or correlation coefficient would mean smaller path lengths. At some level this makes intuitive sense since areas which are functionally highly correlated with each other would be closer to each other (small path length) in some arbitrary "functional space" (they need not be geographically close to each other).
3. Another simpler way to think of clustering coefficient is the fraction of the node's neighbours which are also neighbours of each other. For example, you start with a node for which you want to calculate the clustering coefficient. You check its neighbours and calculate if the neighbours are also connected to each other. If they are, the clustering coefficient is higher in that area. The clustering coefficient tries to quantify how segregated the network is. This is related to the idea of identifying modules in the brain where specialized processing may happen. There are weighted measures of clustering coefficient present too.
When using weighted networks, another thing to keep in mind is what do you do about the negative values. Would a negative value mean that the path length is increasing or decreasing? Is the network becoming more integrated or less...similarly how do you interpret negative weights when calculating clustering coefficients? Are negative weights leading to increased segregation or are they disrupting the modules? Most often, negative weights are discarded but there are studies which have also accounted for the same. If you are interested in weighted network metrics, I would suggest going through Rubinov and Sporns (2011) paper ["Weight-conserving characterization of complex functional brain networks"].
I am hoping that this would provide some preliminary insight into what's going on...perhaps others can contribute?
Best
Pravesh
Originally posted by Janelle Letzen:
One way to think about the graphs is to think of a matrix with the nodes corresponding to the rows and columns and the edges being the values contained therein. For example, consider a matrix with two rows and two columns. Then, there are four entries in this matrix:
r1c1 ; r1c2
r2c1 ; r2c2
This can be thought of as a matrix with 2 nodes (say A and B) and the entries of the matrix are the connections or edges. Therefore, r1c1 would be a connection from A-A and r2c2 would be a connection from B-B (self connections). A connection between A-B would be the entries r1c2 and r2c1. If it is an undirected graph, then these values would be the same. A directed graph would have a value for the edge A->B (A to B) and B->A (B to A).
Furthermore, these edges can have a weight or else be a binary value. If it carries a weight (such as the Fisher transformed correlation coefficient), it is a weighted graph and if the edges are simply 0 and 1, it is an unweighted graph (a value of 1 would mean that the connection exists; 0 would mean that the connection does not exist).
Now to come to your questions: yes, nodes are the ROIs and the edges are the Fisher transformed correlation coefficients.
1. There are several metrics which take into account the actual edge value. However, it also depends on whether you are using unweighted graph or weighted graph. Most of the times, the edge weights are discarded and people tend to analyze these "binarized" graph (by setting, say, a value above a threshold to be equal to 1 and all others to 0). However, for most of the metrics, there is a method to take into account the actual weight or value of the edge too. For example, weighted degree, weighted path length, etc. You may want to check Appendix A of the Rubinov and Sporns (2010) paper.
2. You could think of the path length (in case of a binarized graph) as the number of nodes you need to hop in order to travel from a particular source node to a destination node. Say for example, you would like to calculate the path length between two ROIs. You would work out the minimum number of nodes that you need to move to in order to reach the destination ROI. This makes more sense if you consider an anatomical network; shorter paths could then be interpreted as reflecting a stronger potential for functional integration. For a functional network, interpretation of path lengths are not that straightforward. You may want to take a look at few papers to get additional information on that; as far as weighted path lengths go, usually an inverse mapping is done between the correlation coefficient and the path length such that stronger weights or correlation coefficient would mean smaller path lengths. At some level this makes intuitive sense since areas which are functionally highly correlated with each other would be closer to each other (small path length) in some arbitrary "functional space" (they need not be geographically close to each other).
3. Another simpler way to think of clustering coefficient is the fraction of the node's neighbours which are also neighbours of each other. For example, you start with a node for which you want to calculate the clustering coefficient. You check its neighbours and calculate if the neighbours are also connected to each other. If they are, the clustering coefficient is higher in that area. The clustering coefficient tries to quantify how segregated the network is. This is related to the idea of identifying modules in the brain where specialized processing may happen. There are weighted measures of clustering coefficient present too.
When using weighted networks, another thing to keep in mind is what do you do about the negative values. Would a negative value mean that the path length is increasing or decreasing? Is the network becoming more integrated or less...similarly how do you interpret negative weights when calculating clustering coefficients? Are negative weights leading to increased segregation or are they disrupting the modules? Most often, negative weights are discarded but there are studies which have also accounted for the same. If you are interested in weighted network metrics, I would suggest going through Rubinov and Sporns (2011) paper ["Weight-conserving characterization of complex functional brain networks"].
I am hoping that this would provide some preliminary insight into what's going on...perhaps others can contribute?
Best
Pravesh
Originally posted by Janelle Letzen:
Hi again Conn experts,
I just wanted to follow-up on whether someone could please provide an interpretation of "path length" in terms of the Fisher's r-to-Z transformed correlation values.
Thank you!
Janelle
I just wanted to follow-up on whether someone could please provide an interpretation of "path length" in terms of the Fisher's r-to-Z transformed correlation values.
Thank you!
Janelle
Threaded View
| Title | Author | Date |
|---|---|---|
| Janelle Letzen | Sep 13, 2017 | |
| Janelle Letzen | Sep 21, 2017 | |
| Pravesh Parekh | Sep 21, 2017 | |
| Janelle Letzen | Sep 22, 2017 | |
| Pravesh Parekh | Sep 22, 2017 | |
| Janelle Letzen | Sep 22, 2017 | |