Source
Shpak, Stadler, Wagner and Hermisson (2004), Theory in Biosciences 123(1):33–68, LIT-1242, read in full in NOTE-979 from the authors’ manuscript. The condition is the paper’s (1.10)–(1.11). The spectral inclusion is its Theorem 2.1, which it cites from Stadler and Tinhofer (2000). The derivations below are the record’s.
What was actually shown
Let x(t+1) = Ax(t) in ℝⁿ, and let C₁, …, C_m (1 ≤ m < n) partition the variables. Q is the m × n matrix with Q_Ii = 1 if i ∈ C_I and 0 otherwise, so y = Qx is the vector of block sums.
The condition. The block sums evolve autonomously from every initial state iff y(t+1) is a function of y(t) alone. Since the map is linear, that means QA = ĀQ for some m × m matrix Ā.
- If. Suppose that for every pair I, J, the sum s_IJ = Σ_{i∈C_I} A_ij is the same for every j ∈ C_J (column-equitability). Then (QA)Ij = Σ{i∈C_I} A_ij = s_IJ = (ĀQ)_Ij with Ā = (s_IJ). So y(t+1) = QAx(t) = ĀQx(t) = Āy(t).
- Only if. If QA = ĀQ, then for j ∈ C_J, Σ_{i∈C_I} A_ij = (QA)_Ij = (ĀQ)_Ij = Ā_IJ, which does not depend on which j ∈ C_J. That is column-equitability.
QQᵀ is the diagonal matrix of block sizes, so Ā = QAQᵀ(QQᵀ)⁻¹, the paper’s (1.2a).
The spectrum. If wᵀĀ = λwᵀ with w ≠ 0, then (wᵀQ)A = wᵀĀQ = λ(wᵀQ). And wᵀQ ≠ 0, because Q has full row rank. So every eigenvalue of Ā is an eigenvalue of A. The paper’s landscape example shows the multiplicities falling to one (Fig. 1).
Aggregation is not decomposition. Column-equitability constrains block sums, not their size. With A = αI + β(11ᵀ − I), every partition is column-equitable: each cross-block column sum is β|C_I|, and each within-block one is α + β(|C_I| − 1). So every partition aggregates exactly for any β, including β ≥ α, where coupling between blocks is as strong as coupling within them. Re-checked numerically as a VAR(1) with Gaussian noise (NOTE-979): the transfer entropy from the six units into the pair of three-unit block sums was zero to machine precision.
What could have failed: aggregability could have needed more than a condition on column sums, such as conditions on the within-block distribution of the state. The paper’s (1.10) shows that it does for a single fixed state. For all states, it reduces to column sums alone.
What this does not say
- Not that a single block’s sum closes. The vector of all block sums closes. One block’s sum alone generally depends on the others: in the uniform example its transfer entropy was 5.0 × 10⁻² at (α, β) = (0.2, 0.1).
- Not anything approximate. Nearly equitable partitions, and the Simon–Ando case of weak coupling between blocks, have no bound here. The paper treats Simon–Ando as a limiting case without one.
- Not about aggregations that are not partitions. Eigenvector aggregations (the paper’s Results 1.1–1.2) close without being partitions, and THEORY-733 covers every linear map.
- Not about nonlinear systems. The paper’s own Wagner–Laubichler example aggregates genotype classes in relative frequencies that do not aggregate in the linear representation.
Connections
- THEORY-733. Exact aggregation ĀQ = QA is THEORY-733’s VAR(1) criterion L A = B L with L = Q. So in a linear Gaussian VAR(1), a partition’s block sums are jointly exactly dynamically independent iff the partition is column-equitable. This is the partition case of THEORY-733, for the joint vector of block sums.
- CLAIM-388. The uniform-coupling system in CASE-120 is aggregable along every partition, though none is a block. The claim is about the sum of all units and about blocks, and stands.
- QUESTION-218: answers in part. It asks whether a boundary test’s unit should be a subset of members or a function of the whole state. This gives a third kind: a partition of members whose group summaries jointly close while the groups interact strongly. It does not say whether such a partition is one system or several.
- CLAIM-157 and THEORY-661. Equitability is a symmetry condition and near-decomposability a weak-coupling one. They select different partitions, and only the second is a block.
- THEORY-822. For a Markov chain, this condition applied to Pᵀ is Kemeny and Snell’s strong lumpability, and it makes the block label exactly dynamically independent in Barnett and Seth’s sense.
Note of 2026-10-21: the same condition as a causal criterion
Rubenstein et al. (LIT-1334), read in full on 2026-10-21 (NOTE-1037): averaging blocks of linear micro-variables is an exact transformation of structural equation models exactly when the blocks’ column sums are equal, which is this account’s column-equitability for the block that feeds the averaged one (THEORY-847). There it holds only for interventions on whole blocks. So one condition makes block sums jointly closed under the dynamics here and makes their averages a causal model there. The account is unchanged.