# Graphical LASSO Interpretation

I have a conceptual question about graphical LASSO interpretation. I have been using the huge package in R to estimate an association network for a matrix of node attributes. My understanding of graphical LASSO is that you are performing a penalized regression where each vector of attributes is regressed on all other vectors of attributes. A pair of vectors $i$ and $j$ then have a tie in the resulting network if $\beta_{ij}, \beta_{ji} \neq 0$. In most of the applications I've seen, the resulting network is interpreted in terms of similarity. This paper, for example, identifies a network of senators where senators with similar voting records are in each other's neighborhood. This is intuitive and this is what I find in my own applications.

My question is: why is this the case if a pair of vectors are included in the network when $\beta_{ji} \neq 0$? Vectors $ij$ should then be included if $\beta <0$ as well as when $\beta >0$. The network of association would include ties between highly similar and highly dissimilar vectors, both positive and negative associations. So why is it that the networks this method generates are primarily networks of similarity / positive association? Why don't we also see links between perfectly dissimilar vectors? I would greatly appreciate any help with this question.

• I do not have an answer, as such, but can suggest searching for more examples of graphical LASSO, e.g. see the 'adaptive lasso' network here sciencedirect.com/science/article/pii/S0092656614000701. Connections can be negative. As I understand, graphical lasso methods take a precision matrix, or the inverse of the correlation matrix (i.e. partial correlations), and compute a penalised graph dependent on a value lambda, ranging between [0,1]. Thus, connections certainly can be negative. Commented Feb 24, 2016 at 6:35

What you are describing sounds like the Meinhausen-Buhlmann neighbourhood selection scheme. However, I suppose the graphical lasso function inhuge implements some version of the graphical lasso algorithm - which is related to neighbourhood selection - but different.