Correlation of centrality measure random vectors [closed]

Question

Let's assume that we have 2 random vectors A=(a1,a2,a3) and B=(b1,b2,b3). Each of these elements is a centrality measure of a network. For instance a1 and b1 are the centrality measures of the same network for nodes A and B respectively. a2 and b2 are the centrality measure of another network for nodes A and B and so forth.

Now, I want to obtain a single value between these two vectors that would show the correlation between them, but the issue is that I don't know how to construct it to actually be meaningful. I thought of using Pearson correlation coefficient but I'm not sure how to interpret it, so I'm looking for a tool that would give me insight about the correlation of these two random vectors. Please refer any papers, resources, ... that you think are applicable.

Ps. This is just a simple example, I have 20 vectors each with 50 elements.

Thank you in advance.

Answer 1

Not sure if I fully understood what you meant by "how to construct it to actually be meaningful", but this might be somewhere to start with and to think about what type of correlation you're interested in capturing (e.g., linear vs. non-linear), what metric works better with your data (e.g., any outliers?), etc.

Edits: Thanks @alex-m for the feedback.

The way you construct these vectors might not be the most helpful to explore the correlations between centrality measures.

Let's say you want to see whether betweenness centrality $C_B$ correlates with eigenvector centrality $C_E$ across a set of nodes $V = (V_1, V_2, ..., V_N)$. Instead of using per-node vector $A$ and $B$ you should build per-metric vector for centralities of interest, i.e., $C_B = (C_B^{(1)}, C_B^{(2)}, ..., C_B^{(N)})$, $C_E = (C_E^{(1)}, C_E^{(2)}, ..., C_E^{(N)})$, where each element $C_B^{(i)}$ or $C_E^{(i)}$ corresponds to the centrality value for node $V_i \in V$.

Next, let's start with linear correlations between $C_B^{(i)}$ and $C_E^{(i)}$. Pearson's $r$ ranging from [-1, 1] will give you a sense of the direction and the strength of correlation, as this paper did when they compared the correlation of centrality measures between ER, SF and real-world networks. A negative R means negative correlation and vice versa; the larger the absolute value, the stronger the correlation.

The paper also proposed another correlation metric, centrality similarities, which is basically a rank correlation value that computes the percentage of overlapping nodes in two ranked lists based on certain centrality measures. Other rank correlation metrics such as Spearman’s $\rho$ and Kendall’s $\tau$ can also serve as supplements if you want to capture non-linear relationships.