Is there a meaningful way to 'quantify' a group representative partition of its network when subjects within group have their own unique partitions?

Question

I currently have a dataset which can be split into two groups: disease vs control. Each group consists of $n_{disease}$ and $n_{control}$ subjects respectively. The dataset itself is a correlation matrix computed from fMRI timecourses, thus every subject has its own correlation matrix pertaining to their brain activity during acquisition.

What I'm interested in is how the correlation matrix can be partitioned into separate, and non-overlapping modules. The reason for this is because we believe the brain activity given the condition would exhibit activity profiles reflected in the correlation matrix that may result in unique modules not observed in the control group. I've currently implemented the weighted version of the Louvain modularity algorithm from the Brain Connectivity Toolbox (a matlab toolbox that the field uses), and so what I've done is quantified every subject's Louvain community.

I've tried to address concerns regarding the issues of randomness when the Louvain algorithm initializes by runnning 250 iterations of the algorithm on a subject's correlation matrix before coming to a consensus partition by creating an agreement matrix that is iterated by another implementation of the Louvain algorithm with the threshold $\tau = 0.4 $

So whilst I have $n_{disease}$ individual Louvain communitiy/partition vectors and $n_{control}$ partition vectors, I was wondering if there was a way to get a group partition (one for disease, and one for control), before quantifying local nodal metrics such as the Participation Coefficient and within degree modules z-score. The reason for this is because whilst I can calculate the Particpation coefficient and WMD Z-score for every subject based on their own individual Louvain module vector, I would still like to have a consensus on what the group partition is of the correlation matrix given the experimental condition. Additionally, it may be more prudent for me to quantify the wmd z-score and participation coefficient based on a group's representative modularity structure, instead of individual structures as the end objective is to compare between groups, not between individual subjects.

Answer 1

I would start with modelling the stochastic processes that generate your time course data. This is partly to account for uncertainty in the original data generating process, but it also allows you to include mixed effects for the two groups as well as mixed effects for the individuals.

Once you have sampled from the posterior distribution, you can compute correlation matrices on each posterior sample. You can then split the posterior correlation matrices by the two groups and partition the nodes of your graph based on w/e algorithm.

From here you do not have a "group partition" per se because there isn't a unique notion of such a thing. But what you can do for each set $S$ across the partitions is compute an estimate of the independence gaps

$$\Pr[S, \text{Disease}] - \Pr[S]\Pr[\text{Disease}]$$ $$\Pr[S, \text{Control}] - \Pr[S]\Pr[\text{Control}]$$

which quantify statistical dependence in a signed way. A normalization is available.