Hi Conn community,
I'm a newbie to the world of graph theory, so apologize in advance for the simplicity of these questions. I'm trying to wrap my head around some of basic terms in a practical way, and was wondering if someone can please confirm whether I'm understanding correctly, as well as answer a few questions.

If I'm properly understanding, nodes represent ROIs and edges represent significant r-to-Z correlations between two ROIs. Here are my questions:

1. It seems like most measures are agnostic of the magnitude of the edge (e.g., degree is the number of edges from one node, or number of significant correlations between one ROI and other ROIs in the graph). Do any metrics account for correlation magnitude?
2. Several resources described path length as the "distance" between two regions. I understand how this translates to the topology of a graph visually, but haven't been able to figure out what this means in terms of r-to-Z values. For example, would a shorter path length equate to greater r-to-Z magnitude between two ROIs without accounting for the influence of additional ROIs?
3. Some articles described a clustering coefficient as a measure of possible "triangles" around a node. Although this explanation again makes sense visually, I'm unsure of the pragmatic interpretation related back to correlation values. Do "triangles" represent ROIs with at least 3 significant r-to-Z correlations with nodes included in the model (so not accounting for relative correlation magnitude, just whether or not it is significant)?

I've read some seminal articles on this topic (e.g., Rubinov & Sporns, 2010; Bullmore & Sporns, 2012; Fallani et al., 2014; Wang et al., 2010), but would be grateful for additional resources that others have found helpful!

Thank you,
Janelle

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

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:

Thank you so much for this great, detailed explanation! If I'm interpreting what you said correctly, it sounds like path length for fcMRI data represents magnitude of correlations in a weighted graph, but is not the case in a binarized graph. I looked at the Rubinov and Sporns appendix, but again got an interpretation that generally describes path length as the distance between two nodes for an unweighted graph, which is still difficult for me to conceptualize in the arbitrary functional space, especially if magnitudes are binarized.

From looking at some output from Conn, the adjacency matrices only have values of 0 and 1. Does this indicate a binarized graph? If so, I read that 1 represents presence of an edge (so significant correlation between 2 ROIs) and 0 represents no edge (no significance achieved). For path length in this case then, would we say that a shorter path length means a value of 1 for the ROI pair (kind of like a direct association), whereas an increased path length would indicate a value of 0 between the direct ROI pair of interest, with higher values based on how many additional ROI-ROI comparisons were conducted until the initial ROIs had a common ROI with a significant correlation (indirect association)?

Again, thank you so much for taking the time to respond to my questions. It is incredibly helpful!!

Best,
Janelle

Hi Janelle,

Consider an adjacency matrix as follows:

A B C D
0 1 0 1
1 0 1 1
0 1 0 0
1 1 0 0

You could then calculate a distance matrix which would calculate the number of edges you would have to pass through in order to go from one to another:


A B C D
0 1 2 1
1 0 1 1
2 1 0 2
1 1 2 0

You can interpret it as follows:

In order to walk from A to B, you need to pass through 1 edge
In order to walk from A to C, you need to pass through 2 edges (because A and C are not directly connected; you would have to walk via B)
In order to walk from A to D, you need to pass through 1 edge
and so on.

So in a binary network, you could think of it as the nodes being connected directly or indirectly. If you go by the assumption that information flows via shortest routes in a network, then shortest path lengths could be a useful index to quantify how information is flowing in a network. There are studies which have shown that in certain disorders, the mean shortest path length (also known as characteristic path length) are increased as compared to healthy brains. For a weighted network, you would typically transform the weighted adjacency matrix into a connection matrix having some sort of mapping between correlation values and lengths. For example, an inverse mapping such that higher correlation values would be shown as shorter paths. The interpretation of path lengths in a weighted network is rather ambiguous. You could still use the hop counting method (as in binarized case) to calculate path lengths or else sum the edge weights. I suggest having a look at Chapter 7 of 'Fundamentals of Brain Network Analysis' by Fornito, Zalesky, and Bullmore for thoughts on interpreting path lengths for weighted and unweighted network.

A value of 0 and 1 in your adjacency matrix would mean binarized graph. And indeed, a value of 1 would mean a direct connection between the two nodes while a value of 0 would mean that a direct connection between the two nodes is absent. Since you are working with functional data rather than anatomical data, you should think of them as functional connections (for example, two brain regions which may or may not be anatomically directly connected will work together when performing a task). When calculating path length in such a case, a path length of 1 would mean that the areas are directly connected and a value more than 1 would mean that the areas are indirectly connected (consequently you have to move through several more brain areas in order to reach the destination; see example above where A and C are not directly connected and thus have a higher path length). Functionally speaking, you could think of them as exchanging information directly or indirectly. For example, A and C need to communicate indirectly because the output of A needs to be further processed by B before it is sent to C. As a crude analogy, think of A as the input device of a computer, B as the processing unit, and C as the output unit. So A and C are indirectly (functionally) connected because information from A can only go to C after B has processed it. Depending on the nature of the task being performed, it might make sense to think of the brain regions doing some segregated processing. Similarly, it may make sense to think of processing happening in modules which are connected by paths. 


Hope the above helps.



Best
Pravesh

Such a great explanation again! Ok, I think I've wrapped my head around it now, and your explanation would make sense in the context of the output that I have. I will read over the chapter that you suggested from Fornito et al. Again, I appreciate the resources and all of your help/patience!

Thanks so much,
Janelle