Measuring modularity and information exchange in simple networks
by Esben Kran
As we’ve discussed before, we think a good measure of modularity should be deeply linked to concepts of information exchange and processing, and finding a measure which captures these concepts might be a huge step forwards in this project. Although no such measure is currently in use to our knowledge, there are several that have been suggested in the literature which try and gauge how much different parts of the network interact with each other. Most of them work by finding a “maximally modular partition” and measuring its modularity, with the distinctive part of the algorithm being how the modularity of a particular partition is calculated. For instance:
- Some are derived from tools used to analyse simple unweighted undirected graphs, e.g. the Q-score
Some look at the weights, using e.g. the matrix norms of convolutional kernels. - Some look at derivatives with respect to node input and output, coactivation of neurons, or mutual information of neurons
- We’re also currently working on a candidate measure based on counterfactual mutual information, which we’ll be making a post about soon.
It would be valuable to compare these different measures against each other, and see if some are more successful at capturing intuitive notions of modularity than others.
This isn’t just a theoretical issue either. Right now, it’s looking like e.g. the matrix norm and node derivative measures give very different answers, where one might tell you that a network exhibits statistically significant modularity, whereas the other says there isn’t any.
This suggests the following experiment: taking a very simple system (e.g. the retina task), training it until it finds a solution, and benchmarking and visualising all of these measures against each other on the learned solution.
Some questions you could ask:
- Which modularity measures give rise to similar “maximally modular partitions”? Which ones give partitions that are more similar than others? (this paper suggests a method for comparing the similarity of two different partitions)
- For small networks, you could try visualising the learned solutions and the partitions. Do some partitions look obviously more modular than others?
- Do your results change if you apply them on a solution which hasn’t yet attained perfect performance?
- Try to construct networks that Goodhart a particular measure. How difficult is this? Do the results look like something that a typical training process might select for?