Contribution
Makes “is this network modular?” a question with an answer. The definition is borrowed from graph clustering — a module is a set of neurons with strong internal connectivity and weak external connectivity — and measured by running spectral clustering on the weight graph of an MLP and reporting the n-cut of the resulting partition.
The finding: training plus weight pruning produces MLPs that are more modular than randomly initialised ones, and often significantly more modular than random MLPs with the same (sparse) weight distribution.
Key insight
The second control is the whole paper. A pruned network is sparse, and sparse graphs cluster more easily than dense ones — so “pruned networks are modular” is uninformative unless you compare against a random network with the same sparsity pattern statistics. They do, and the effect survives.
That is the methodological point worth carrying: when a structural property appears after an intervention, control for the intervention’s incidental statistical effects, not just against the pre-intervention baseline.
Assumptions
- Graph clusterability of the weight matrix is a meaningful proxy for functional modularity. This is the load-bearing assumption and it is a proxy — the paper measures connectivity structure, not behaviour.
- MLPs on small image datasets; the architecture class is narrow by design, so the graph is well defined.
- Spectral clustering’s partition is representative; results are checked for consistency across cluster counts (2-way reflected in 4-way).
Key results
- Trained and pruned MLPs are more modular than random initialisation and than sparsity-matched random baselines.
- Networks trained with dropout are much more modular — a training choice with a measurable structural consequence.
- Results reported across models trained with and without dropout (10 each in one comparison), with notable spread in n-cut values on CIFAR-10.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Modularity is measurable in a weight graph via spectral clustering | strong | the method |
| C2 | Trained+pruned MLPs are more modular than sparsity-matched random ones | strong | the controlled comparison |
| C3 | Dropout increases modularity | moderate | measured across a small set of models |
| C4 | Graph clusterability corresponds to functional modularity | weak — assumed | not tested here |
Method
Train MLPs on small image datasets, with and without dropout. Prune. Build the weight graph. Spectral-cluster it and compute the n-cut. Compare against random initialisation and against random networks matched on the sparse weight distribution.
Concepts
- Modularity as graph clusterability — a definition that makes the question empirical.
- The sparsity-matched control — the reason the result is not trivial, and the transferable methodological move.
- Dropout as a structural intervention — usually justified as regularisation; here it has a measurable effect on connectivity structure.
Connections
LIT-085 (grokking), read in the same batch, is the other paper here that
inspects weights rather than behaviour — and it goes further, recovering the
algorithm rather than a structural statistic. Read together they mark the two
levels mechanistic work operates at: structure you can measure without knowing
what the network computes, and structure you can only see once you do.
C4 is the gap between them: this paper measures connectivity and hopes it means
function; LIT-085 establishes function directly, for a much smaller problem.
Recommendations
- R1 — When an intervention produces a structural property, control against a random baseline matched on the intervention’s incidental statistics. Topic: analysis and evaluation. Strength: strong, and general.
- R2 — Treat graph clusterability as a proxy and say so; connectivity is not function. Strength: strong.
- R3 — Regularisers may have structural effects worth measuring, not only generalisation effects. Strength: weak — C3 is one observation on small MLPs.
Bearing on the record
Nothing is sourced to this paper and this reading files no practice, and the
document is Superseded — correctly. Interpretability moved to
mechanistic circuit-level work on transformers, of which LIT-085 is a small
example, and a connectivity statistic on pruned MLPs is not where that went.
The reading does not argue with the status. What it records is that R1 is the part worth keeping: the sparsity-matched control is why this paper’s result is not an artefact, and it is the kind of control that structural claims about networks routinely omit.
The document’s takeaways include two that the paper does not support. “Feature disentanglement” is not measured — the paper measures connectivity, not features — and “impact on generalization” is not studied at all; no generalization claim appears. That is two of four bullets asserting results the paper does not contain.
Limitations
- Small MLPs on small images, 2020.
- C4 is assumed throughout and is the claim everything else would need.
- C3 rests on a small number of models with a large spread on CIFAR-10.
- Spectral clustering imposes a partition whether or not one exists; the paper’s controls address this and do not eliminate it.
Open questions
- Does weight-graph modularity predict anything behavioural? The paper’s own framing invites the question and does not test it, and that gap is why the line was superseded rather than extended.