Consider a set of features $x_1,x_2,...x_N$ that may be (pearson-) correlated.
I want to group them whenever they have a high correlation between themselves so that I can interpret the coefficients of a linear regression on them.
To this, I constructed an undirected (because correlation is commutative) graph/network where feature $x_i$ is connected to feature $x_j$ when they have a correlation above some threshold.
The task of grouping correlated features amounts to find the components of this graph.
Can I assume that if the edges $(x_1,x_2)$ and $(x_1,x_3)$ exist, then $(x_2,x_3)$ exists too?
I.e. Does $corr(x_1,x_2) \geq z$ and $corr(x_1,x_3) \geq z$ imply that $corr(x_2,x_3) \geq z$?