# Graph Regression - Is that possible?

DATA

• 30 individuals;
• Each individual has an associated graph;
• Individuals graphs represent the connections between brain regions: nodes are brain regions, while edges' weights are the strengths of the connections. There are 100 brain regions, which are the same across subjects. It implies a total of 4950 edges;
• Each individual has an associated scale (accuracy during a task). So they can be ordered from the worst to the best in that scale.

AIM

I want to build a model starting from individual graphs. The model should generate the graph expected, given an accuracy score. So, it sounds like a "graph regression".

I am aware that this is not simple, so not only straight answers but also suggestions on how to approach the problem are appreciated.

• That doesn't like something that can be solved with 30 data points. You're trying to infer a particular graph structure out of 2^{n choose 2} possible graphs given 30 examples? And you want to do this based on one variable (accuracy score?). – Bar Oct 6 '17 at 14:21
• I agree with the comment above. You could perhaps regress accuracy against a simpler summary statistic like number of edges. – Paul Oct 6 '17 at 14:25
• I thank both of you for your comments. I can use more subjects (up to a maximum of 250) but still the question is the same. Additionally, if you think that this can or cannot be solved in some way, you should justify your answer. – smndpln Oct 6 '17 at 14:31

A simple approach could be to cast the unit of observation as the edge, and the node as a covariate to be treated as a grouping factor. Your outcome would be the accuracy score. The model would thus be

$$\text{accuracy score}_{ij} = f(X_i, X_j)$$

where $i$ indexes outgoing node, $j$ the incoming node, and $X$ characteristics of that node. Perhaps you have other covariates, but at minimum you can calculate various sorts of centrality measures. The $f$ could be most anything, from a mixed effects regression to a random forest.

Or, if you want to predict linkage based on an accuracy score, you could fit a model like

$$pr(\text{linkage})_{ij} = f(X_i, X_j)$$

with accuracy scores among the $X$'s. Fitted values of this model would create a probabilistic graph.

More detailed suggestions are probably possible with more info on your data.

• This is an useful answer. Thanks. There is the possibility to include further covariates (i.e. more psychological measures), and in my opinion a mixed effects model is quite appropriate anyway, even if I was initially thinking at multivariate regression. – smndpln Oct 7 '17 at 15:29