This repository explores whether specific graph-theoretic centrality measures can identify subnetworks resembling modular processing in neural networks.
Specifically, we investigate whether neurons with high in-degree centrality (receiving many strong inputs) but low betweenness centrality (not on many shortest paths) play a special integrative role -potentially acting as modular bottlenecks. The intuition is that there are many inputs that can belong to a particular class, but information is needed upstream only if the module is required.
The experiment:
- Trains a neural network on a subset of MNIST digits (by default 3, 5, 7).
- Constructs a directed acyclic graph of neurons and weighted connections.
- Identifies neurons in the hidden layers with high in-degree and low betweenness.
- Ablates each such neuron and its strongest incoming neurons -input neurons are spared.
- Evaluates how this targeted ablation impacts classification performance per digit.
- Install dependencies:
pip install -r requirements.txt- Train a neural network (example below uses 3 hidden layers of 8 units each):
python train.py --epochs 30 --batch-size 128 --lr 0.001 --hidden-layers 8,8,8 --save-dir ./mlp- Analyze the trained network graph:
python network_analysis.py- Run the experiment:
python are_modules_indegree_nobetween.pyThe ablation experiment runs over multiple candidate neurons that match the centrality profile, disabling the modules individually.
Specifically, for each such neuron:
-The neuron and its strongest incoming connections (based on connection weights) are zeroed out.
-The model is re-evaluated on test images.
-Accuracy is reported per digit class. Class-specific deteroriations have disabled modules processing specific digits.
In short, the experiment allows you to ask:
Do these centrally located “information sinks” support modular class representations?
The figure below presents the classification accuracy following the ablation of neurons identified by their high in-degree centrality and low betweenness centrality. Each triplet of columns corresponds to a different ablation, with bars indicating accuracy for each class.
The results reveal that the method successfully identifies critical output nodes for certain classes. Interestingly, even with significant ablation, the network sometimes maintains or even improves accuracy for other classes.
The graph construction treats neurons as nodes and absolute weight magnitude as edge weights.
Shortest paths are similarly calculated via strongest weights.
Input neurons are excluded from candidate selection to avoid trivial input masking.
If you use this codebase or find it helpful in your research, please cite the following repository:
Panagiotou, F (2025). Neural Network Modularity [Computer software]. GitHub. https://github.com/mknull/neural-network-modularity