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Functional Connectomes of Neural Networks
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Proposes building functional connectomes of neural networks from neuron activation correlations and analyzing them with persistent graph homology and closed-form Wasserstein statistics, scaling to large networks.
Problem
Interpreting neural networks is hard; existing persistent homology analyses of network functional connectivity rely on cubic-time computation and fixed thresholds, limiting them to small networks.
Approach
Treat hidden-neuron activation correlations as a complete weighted graph (functional connectome), and use persistent graph homology (MST-based birth/death values) with closed-form Wasserstein distances, barycenters, and variances to compute exact, threshold-free, O(n log n) topological statistics and cluster networks by regularization strategy and input class.
Key results
- In Study 1, the Top method reaches purity 1.00 for batch norm vs. vanilla on all three datasets and 1.00 for dropout vs. vanilla on MNIST and Fashion-MNIST, outperforming baselines in most settings.
- In Study 2 (unsupervised clustering of 200 class-wise connectomes), Top, WD and SWK achieve purity around 0.5-0.6, well above the 0.1 random expectation.
- Top computes exact Wasserstein distances between graphs with thousands of nodes and millions of edges in about one second on an Apple M1 Pro CPU, while conventional persistent homology methods only handle a few hundred nodes.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Top rather than the strongest listed baseline | Clustering functional connectomes across training strategies · MNIST, Study 1, all strategies | purity: 0.85 -> 0.78 ± 0.11 | Hurt |
| Use Top rather than the strongest listed baseline | Clustering functional connectomes across training strategies · Fashion-MNIST, Study 1, all strategies | purity: 0.78 -> 0.87 ± 0.12 | Helped |
| Use Top rather than the strongest listed baseline | Clustering functional connectomes across training strategies · CIFAR-10, Study 1, all strategies | purity: 0.98 -> 0.88 ± 0.11 | Hurt |
| Use Top rather than SWK | Study 2 clustering · MNIST, Dropout | purity: 0.46 ± 0.01 -> 0.57 ± 0.02 | Helped |
| Use Top rather than WD | Study 2 clustering · Fashion-MNIST, Vanilla | purity: 0.62 ± 0.01 -> 0.53 ± 0.02 | Hurt |
| Use Top rather than SWK | Study 2 clustering · CIFAR-10, Batch Norm | purity: 0.47 -> 0.52 ± 0.02 | Helped |
| Use Top rather than SWK | Study 2 clustering · CIFAR-10, L2 | purity: 0.59 ± 0.01 -> 0.54 ± 0.03 | Hurt |
| Apply regularization strategies rather than vanilla training | Neural networks analyzed in Study 1 · MNIST, pairwise regularization-versus-vanilla clustering | purity: Top: 1.00; 1.00; 0.80 ± 0.03 | Mixed |
Surprising findings
- Clustering purity in Study 2 is only about 0.5-0.6 for the best methods, indicating class-wise functional differences are moderate rather than strong.
- Several baselines (e.g., adjacency-matrix clustering) perform near chance in Study 2 (purity ~0.15-0.26).
Limitations
- Persistent graph homology only captures connected components (0th homology) and cycles (1st homology); higher-order features are not captured.
- Analysis is primarily restricted to feedforward architectures; extension to convolutional and recurrent layers is left open.
- Only compact networks trained on MNIST, Fashion-MNIST and CIFAR-10 were analyzed, and only hidden fully-connected neurons were used for connectomes.
Experimental setup
Datasets: MNIST Fashion-MNIST CIFAR-10
Benchmarks: MNIST Fashion-MNIST CIFAR-10
Metrics: purity runtime test accuracy
Baselines: Adj bottleneck distance Wasserstein distance sliced Wasserstein kernel heat kernel persistence image
Builds on: persistent homology persistent graph homology Lloyd's algorithm VGG
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