While my question is related to biology and is quite complex, I am simplifying this problem using an example scenario. Let's say we have 4 groups of people, with same number of individuals in each group. We know that each group is different in terms of the extrovertness of people in that group, but we don't know the degree of extrovertness of these groups. We put these four groups of people in a large room and let them interact with each other - one on one. The people of highly extrovert group will interact with more people than the people of other groups and so on.

At the end of this experiment, we can expect 10 different combinations of interactions as shown below, where 6 combinations are between individuals of different groups (inter-group interactions - orange boxes) and four combinations are between individuals within each group (intra-group interactions - gray boxes).

At the end of this experiment, we know the number of inter-group interactions (i.e. number of interactions for each orange box), but the number of intra-group interactions (i.e. number of interactions for each of the gray boxes) is missing. Based on the numbers for inter-group interactions, we see that some groups have overall high number of interactions with people of other groups, resembling groups with highly extrovert people, and some groups have least interactions overall, indicating groups with introvert people. Therefore, the expectation is that the groups with extrovert people will have high number of intra-group interactions and low numbers for introvert groups.

So the questions is, would it be possible to determine rough estimates of number of each intra-group interaction (gray boxes) based on the information that we know? Since, I have limited information here, I can't think of a way to answer this question using any deterministic approach. I am wondering if any probabilistic approach can be used to solve this problem.

Thank you so much in anticipation of any help. I would highly appreciate any help here.

Disclaimer: I am a novice in probability and therefore, I am not entirely sure what to google for to find a solution to my problem. I am hoping that the experts in probability and statistics can direct me in the right direction.