Substrate · Phase-Transition Modeling · Corrected

Connectomes and the Cost of Sparsification

Why treewidth failed, and what survived.

CorrectedCorrigendum

Raw treewidth and spectral κ failed sparsification robustness; the integration/separation trade-off survived; Eglob + Q appears near-conserved within C. elegans.

Old measure
raw treewidth; spectral-gap κ
New measure
global efficiency E_glob + modularity Q
Status
discovery-stage · measurement, single-organism
Replication
one clean organism sweep; needs replication on ≥3 more connectomes.
Read the corrigendum

The profiles below were written under the treewidth reading; that reading did not survive (see the corrigendum above). They are kept, unchanged in substance, with the superseded measures marked. The integration/separation account — now carried by Eglob and modularity Q — is what held. Every profile reads the same substrate through a different graph invariant and structural regime, and prices it with Ross’s Law.

Read the volume (PDF, 895 pp)

A connectome is a graph the brain wrote: neurons or regions as vertices, synapses or tracts as edges. The quantity that matters for cognition is not how many nodes it has but how integrated it is — how much of the network has to be considered jointly to account for its dynamics. That is precisely treewidth. A low-treewidth connectome is modular: it decomposes into parts you can understand one at a time. A high-treewidth connectome is bound: no part tells the story alone.

This is where the volume’s boldest thread lives — the claim that integrated information, the thing consciousness theories reach for, is the treewidth gap read from inside. We hold that as a hypothesis, not a theorem. But the structural reading below stands on its own: each of five invariants locates a connectome on the axis from shadow (modular, locally reducible) to mirror (globally bound, irreducible), with an equilibrium band — the critical, small-world regime healthy brains seem to favor — between them.

integration cost ≈ exp(Θ(tw(G)))Ross’s Law for connectomes: the cost of accounting for the network as one integrated system scales with its treewidth. Modularity is the brain’s way of staying in the tractable regime while still binding when it must.

Treewidth boundsSuperseded

Treewidth measures how far the connectome is from a set of independent modules — the width of the worst cut across which information must be integrated.

SupersededRaw treewidth failed sparsification robustness for this instantiation; superseded by global efficiency Eglob + modularity Q — see the corrigendum.
Shadow
tw(G) = O(1) — clean functional modules wired mostly to neighbors. Dynamics reduce to local circuits; powerful but decomposable.
Equilibrium
tw(G) = Θ(log N) — the small-world regime: modules plus a few long-range hubs. Enough integration to bind, little enough to stay tractable. This is where healthy cortex appears to sit.
Mirror
tw(G) = Ω(N) — an all-binding network with no reducible parts. Maximal integration, maximal cost — and, on the volume’s reading, where experience would be most irreducible.

Graph degeneracy

Degeneracy is the cheap structural gauge: the densest mutually-connected core of the connectome — its rich club of hubs — computed without solving for integration directly.

Shadow
d(G) = O(1) — no dense hub core; the network peels into modules.
Equilibrium
d(G) = Θ(log N) — a modest rich club, the integrative backbone forming.
Mirror
d(G) = Ω(N) — an irreducible core binding the whole network; large treewidth certified.

Information integration

The entropy-across-a-cut invariant, for a connectome, is integrated information: how much the two halves of the network constrain each other beyond what each holds alone. Area-law integration is shadow; volume-law integration is mirror — the same form the thesis gives consciousness.

Shadow
S ∝ ∂A — integration lives on module boundaries. The system is nearly the sum of its parts.
Equilibrium
S ∝ log N — integration grows slowly; binding is real but bounded.
Mirror
S ∝ |A| — integration fills the network; no partition captures the whole. The system is irreducibly more than its parts.

Fiedler-value connectivitySuperseded

The Fiedler value reads whether the connectome has a natural fault line — a way the network could be split into nearly-independent systems.

SupersededSpectral κ (the Fiedler gap λ₂) failed sparsification robustness for this instantiation; superseded by global efficiency Eglob + modularity Q — see the corrigendum.
Shadow
λ₂ → 0 — nearly disconnected hemispheres or modules. A clean seam exists.
Equilibrium
λ₂ = Θ(1/N) — weakly coupled modules joined by a few tracts. A faint seam.
Mirror
λ₂ = Ω(1) — an expander. No fault line; the network resists every partition and holds as one.

Bipartite treewidth

The bipartite cut is the integration stress test: split the connectome in two and measure the width of the interface the dynamics must cross — the worst case for any modular account.

Shadow
twᵇ(G) = O(1) — the halves interact through a constant-size bottleneck. Cleanly separable.
Equilibrium
twᵇ(G) = Θ(√N) — a boundary-shaped interface, the signature of spatially embedded wiring.
Mirror
twᵇ(G) = Ω(N) — every bisection severs a linear number of connections. The connectome is irreducibly whole.

The same meter, four other substrates

A connectome is one instance of the universal meter. The same five invariants read these too:

Tensor-Network ContractionVector Databases & RAGNeural-Network Quantum StatesSocial Clique Networks
The full matrix