Connectivity Scaling Law

Definition

For a network with connectivity degree d (each node connected to exactly d neighbours), the spectral efficiency is:

c(d) = d · sin(π/d) / (2(d−1))

This measures the fraction of dense-limit information capacity accessible at degree d. As d→∞ (dense limit), c(d)→1. As d→2 (sparsest prime), c(2)→1 by normalisation.

The inverse 1/c(d) is the FLOPs penalty: how many additional inference steps are required to compensate for the missing long-range correlations at connectivity d.

Derivation

The formula emerges from the [[tusk-series]] via the [[sopfr-binding-energy]] function:

1. The Ramanujan coefficient of sopfr at prime p: a_sopfr(p) = p/(p−1)
2. The first-difference (Δsopfr) Fourier factor at frequency 1/p: 2sin(π/p)
3. Combined and normalised to c(2)=1: c(p) = p·sin(π/p) / (2(p−1))

The absolute scale K₂ = π²/12 = ζ(2)/2 is universal: it appears for both the bosonic (sopfr, prime factors with repetition) and fermionic (sopf, distinct prime factors only) Tusk Series.

Key values

d	c(d)	1/c(d) — FLOPs penalty
4	0.471	2.1×
16	0.104	9.6×
32	0.051	19.8×
64	0.025	40.1×
128	0.012	80.9×

At d=16: 1/c(16) = 9.61, consistent with the empirically reported ~10× FLOPs penalty for the Extropic Z1 chip (see [[thermodynamic-computing-membrane]]).

Optimal connectivity

For hardware with energy advantage A over dense compute, the net benefit is B(d) = A × c(d). The minimum-capacity constraint determines the optimal degree:

d* = min { d : c(d) ≥ c_min }

This is the smallest connectivity at which the network achieves sufficient capacity. Any larger d increases quality but reduces net energy benefit.

Fermionic version

The same formula applies whether sopfr (bosonic: prime factors with multiplicity) or sopf (fermionic: distinct prime factors only) is used. K₂ = π²/12 is universal across both; the Möbius exclusion at prime powers is an asymptotically negligible finite-N correction.

Published

• Paper: Spectral Efficiency of Sparse Ising Networks (DOI: 10.5281/zenodo.23529072)
• Addendum to Tusk Series (DOI: 10.5281/zenodo.23501055)