Implementational Mode: Connectomic

Origin	Craver (2007) — organization; neuroscience connectomics (Sporns et al. 2005)
Question	How are the identified components wired to each other?
Licensing evidence	Path patching (edge-level) + specificity of claimed pathways vs. alternatives
Interpretive-validity risk	Confusing temporal ordering (layer sequence) with causal wiring (information flow)
Position in partial order	$I_{\text{con}} > I_{\text{top}}$ — asserts structure (a graph), not just membership (a set)

What this mode claims

A verdict tagged [implementational-connectomic] identifies directed connections between components — which feeds into which, through what pathway. This is stronger than topographic because it asserts structure (a directed graph) rather than just membership (a set). The connectomic claim says: “head $A$ sends information to head $B$ through the residual stream, and $B$‘s computation depends on receiving $A$‘s output.”

The analogy to neuroscience is deliberate: a connectome is a wiring diagram. It tells you the structure of the network without specifying what each neuron computes. In MI, a circuit graph (nodes = heads, edges = information-flow dependencies) is a connectomic claim. It is more than a list of important heads, and less than an algorithm.

Formal characterization

Let $G = (V, E)$ be a directed graph where $V = C$ (the circuit components) and $E \subseteq V \times V$ (directed edges). A connectomic claim asserts:

$\forall (c_i, c_j) \in E: \quad \text{PathEffect}(c_i \to c_j) > \delta \quad \text{AND} \quad \text{PathEffect}(c_i \to c_j) \gg \text{PathEffect}(c_i \to c_k) \text{ for } (c_i, c_k) \notin E$

where $\text{PathEffect}(c_i \to c_j)$ is the causal effect of patching $c_i$‘s output specifically at the path to $c_j$ (holding other paths fixed). This is what path patching measures.

The key distinction from topographic: a topographic claim is invariant to permutation of the component set. A connectomic claim is not — it asserts directed relationships between specific pairs.

What licenses an [implementational-connectomic] tag

1. Path-level causal evidence — path patching (Goldowsky-Dill et al. 2023) or edge attribution demonstrating that the claimed pathway is load-bearing. Activation patching alone establishes node importance (topographic), not edge importance (connectomic).

2. Specificity of claimed paths — patching along the claimed path has substantially more effect than patching along alternative paths of the same length. If every path from $A$ to downstream has similar effect, the claim is not connectomic — it’s just that $A$ matters (topographic).

3. Directionality — the effect must be asymmetric. If corrupting $A$ affects $B$ and corrupting $B$ equally affects $A$ (after controlling for layer order), the “wiring” claim is not established.

4. Convergent structural evidence (optional but strengthening) — weight-space composition scores (QK/OV composition, virtual weights) that agree with the causal graph. When path patching and weight-space analysis agree on the same edges, the connectomic claim is substantially stronger.

What does NOT license a [implementational-connectomic] tag

• Layer ordering alone. In a transformer, every earlier layer’s output is accessible to every later layer via the residual stream. The fact that $A$ is in layer 5 and $B$ is in layer 9 does not mean $A$ connects to $B$ — every layer-5 head “connects” to every layer-9 head in this trivial sense. A connectomic claim must show specific, load-bearing connections above this background.
• Attention pattern inspection. That head $B$ attends to positions where head $A$ has written is suggestive but not causal. The residual stream contains contributions from many components at each position.
• Correlation between head activations. Two heads being co-active is statistical, not structural. They might both respond to the same input feature without being wired to each other.
• ACDC edges without effect validation. ACDC discovers edges via iterative patching, but the discovered graph should be validated with held-out path-patching to confirm edge-level effects.

Worked example: IOI QK composition edges

Claim. In the IOI circuit, the S-inhibition heads (L7H3, L7H10, L8H6, L8H10, L8H11) receive directed input from the duplicate-token heads (L5H1, L5H5) via the residual stream, and this connection is load-bearing for the suppression of the repeated name. [implementational-connectomic]

Evidence:

• Path patching from L5H1/L5H5 output specifically at the path to L7-8 S-inhibition heads shows $\Delta \text{logit diff} = 0.4$-$0.7$
• Alternative paths (L5H1 → name-mover heads directly) show $\Delta < 0.05$ — the information flows through S-inhibition first
• QK composition scores: $\langle W_{OV}^{5.1}, W_{QK}^{7.3} \rangle$ is high relative to random head pairs (top 5% of all pairwise compositions)
• Directionality: corrupting L5H1 degrades L7H3’s attention pattern; corrupting L7H3 does not affect L5H1’s attention pattern

What this is not: This does not specify what the duplicate-token heads compute or what operation S-inhibition performs. It says only that information flows directionally from one group to the other, and that this flow is necessary for the behavior.

Upgrade and downgrade

Direction	What’s required
$I_{\text{top}} \to I_{\text{con}}$ (upgrade from topographic)	Path-level causal evidence that specific edges carry information, not just that nodes matter.
$I_{\text{con}} \to I_{\text{fun}}$ (upgrade to functional)	Specify what each node in the graph does to its input to produce its output. The graph tells you the wiring; the functional claim tells you the components.
$I_{\text{con}} \to A$ (upgrade to algorithmic)	Combine the graph (connectomic) with the node functions (functional) and demonstrate sufficiency of the procedure. An algorithm = wiring + operations + sufficiency.

Instruments that provide connectomic-level evidence

• A02 (Path patching) — direct causal evidence of edge-level effects
• B02 (OV/QK composition) — weight-space evidence for compositional wiring
• B08 (Edge Jaccard) — agreement between methods on the edge set
• A13 (PC algorithm) — observational causal discovery of the graph structure
• C01 (Transfer entropy) — directional information flow between components

Key references

• Goldowsky-Dill, N., et al. (2023). “Localizing Model Behavior with Path Patching.” — Path patching as edge-level causal instrument.
• Conmy, A., et al. (2023). “Towards Automated Circuit Discovery for Mechanistic Interpretability.” NeurIPS 2023. — ACDC; edge-based circuit discovery.
• Elhage, N., et al. (2021). “A Mathematical Framework for Transformer Circuits.” Transformer Circuits Thread. — QK/OV composition as structural wiring evidence.
• Sporns, O., Tononi, G., & Kotter, R. (2005). “The Human Connectome: A Structural Description of the Human Brain.” PLoS Computational Biology. — Connectomics framework.