Contribution
It puts the existing linear results on aggregability on one footing and names the distinction the record’s individuation line lacked. A system is decomposable when its variables split into non-interacting (or weakly interacting) subsets. It is aggregable when some y = f(x) with fewer variables evolves autonomously. “Aggregativity need not imply decomposability, nor vice-versa.” For linear systems and partitions, it identifies the exact condition for aggregation, column-equitability, and applies it to mutation–selection on fitness landscapes.
Key insight
The question “can these variables be summarized?” is not “do these variables form a module?”. A macrostate can be a function of every microstate, as temperature is of every molecule, and still be dynamically sufficient. For a partition of the variables, the summaries of the blocks close exactly when every member of a block has the same total influence on each other block. That is a symmetry condition (an equitable partition, which the orbits of a symmetry group always give), not a weak-coupling condition. Blocks can interact as strongly as they like.
Assumptions
- Linear, discrete-time dynamics x(t+1) = Ax(t) with A constant, and linear aggregation y = Qx with Q an m × n matrix, m < n, of full row rank. The stochastic case adds row or column stochasticity.
- For the partition results, Q is a matrix of (possibly weighted) characteristic vectors of a partition of the variables.
- For the landscape applications: binary (or, in the appendix, suitably ordered multi-allele) genotypes on a Hamming graph, point mutation only (multiple mutations neglected at low rate), equal rates at all loci, and a linearized mutation–selection operator A = WM on absolute frequencies, with W the diagonal fitness matrix.
- The “best approximation” of Theorem 1.1 is in the Frobenius norm, read as an expected error over state vectors distributed isotropically (equal variances, no cross-covariance) about a uniform point.
Key results
- (1.1)–(1.2). Exact aggregation by Q requires ĀQ = QA. If QQᵀ is invertible, Ā = QAQᵀ(QQᵀ)⁻¹. The converse does not hold: this Ā need not satisfy (1.1) for an arbitrary Q.
- Theorem 1.1. B = QAQᵀ(QQᵀ)⁻¹ minimizes ‖BQ − QA‖ in the Frobenius norm, so Ā is the best approximate aggregate operator for a given Q even when no exact one exists. Holds when: error averaged over an isotropic distribution of states.
- Result 1.1. Every n × n matrix, n ≥ 2, has an aggregation with m < n: take Q to be an eigenvector. Result 1.2. A symmetric matrix has one at every m, from any m of its n real eigenvectors. Such aggregations generally need every microstate to compute each macrostate.
- Lemma 1.0. For a positive stochastic A, the aggregate Ā is stochastic iff the columns of Q sum to one, which every partition satisfies.
- (1.10)–(1.11). For a partition Q of characteristic vectors, a given state x aggregates exactly when it is uniform within each block. The aggregation is exact for every x, and an aggregable state stays aggregable, iff A is column-equitable: for every pair of blocks I, J, Σ_{i∈I} A_ij is the same for every j ∈ J. With weights, a weighted version holds.
- Equitable partitions (Schwenk, McKay, Godsil; Stadler and Tinhofer). The orbits of any subgroup of a graph’s automorphism group form an equitable partition. Theorem 2.1 (cited, not proved here): the spectra of the row and column structure matrices R and S coincide and are contained in Spec(A), and R’s characteristic polynomial divides A’s when A is diagonalizable. So Ā keeps a subset of A’s eigenvalues.
- Simon–Ando as a limiting case. For a nearly block-diagonal system, once each block is within ε of its own equilibrium, the rows of Q are the leading eigenvectors of the blocks and Ā_II is each block’s leading eigenvalue. No error bound is given. Simon and Ando (1961) is cited, not reproduced.
- Fitness landscapes.
- The point-mutation matrix on the hypercube has nested equitable partitions, by allele at one locus, two loci and so on.
- On a multiplicative single-peak landscape, the Hamming-distance classes are equitable. That reduces 2ⁿ variables to n + 1, keeps every eigenvalue with its multiplicity dropped to one (Fig. 1, n = 5: e.g. 0.7366 has multiplicity 5), and reproduces Eigen et al.’s error threshold (Fig. 2).
- With two reference peaks, pairs of Hamming distances are equitable, giving (n − D₁₂ + 1)(D₁₂ + 1) classes.
- With three or more peaks, distance classes are not equitable. The counterexample is four loci with peaks 1110, 1101 and 1011. Both 1000 and 0111 lie at distances (2, 2, 2), but only 1000 has a neighbour, 0000, at (3, 3, 3). This was checked here.
- Refining to overlap classes, which split the loci by which subset of peaks shares their allele, restores equitability (a Result, with proof). Fig. 4 compares the two: a squared error of about 10⁻² with distance classes and about 10⁻¹⁶ with overlap classes, over ten steps.
- Nonlinear aggregation. Wagner and Laubichler’s character decomposition aggregates genotypes by allele at a locus, under “pi-additive” fitness and generalized linkage equilibrium. It is dynamically sufficient but not an equitable partition, and it exists only in the relative-frequency (nonlinear) representation.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Aggregation (a closed lower-dimensional description) and decomposition (non-interacting subsets) are independent: each can hold without the other | strong for “aggregation without decomposition”, asserted for the converse | definitions in the introduction; the thermodynamic example; Results 1.1–1.2 give aggregations that mix all variables. No example of decomposition without aggregation is given |
| C2 | A linear aggregation Q is exact iff some Ā satisfies ĀQ = QA, and Ā = QAQᵀ(QQᵀ)⁻¹ is the Frobenius-best approximate operator otherwise | strong | (1.1)–(1.2), Theorem 1.1 with derivation |
| C3 | Every matrix has an aggregation with m < n; a symmetric one at every m | strong over ℂ; Result 1.1 needs a qualification over ℝ | eigenvector argument; see Limitations |
| C4 | For a partition, the block sums aggregate exactly for every state iff A is column-equitable | strong | (1.10)–(1.11); re-derived in THEORY-755 |
| C5 | Hamming-distance classes give exact aggregation for one or two fitness peaks and not for three or more; overlap classes give it for any number | strong | two Results with proofs; the counterexample; Figs. 3–4 |
| C6 | Dynamically sufficient aggregate classes are candidate units of evolution, and equivalence under selection alone does not make one: the classes must also be equivalent under transmission | moderate | argued from C5, citing Lewontin (1970) on dynamical sufficiency |
| C7 | Any aggregation of a linear system is some form of (weighted) equitable partition, and Simon–Ando systems are one | weak | asserted in the discussion; in tension with C3 (see Limitations) |
| C8 | Linear aggregability is “probably a necessary (but not sufficient)” condition for aggregability in the nonlinear representation | weak | conjecture; its own Wagner–Laubichler example appears to contradict it (see Limitations) |
Concepts
- Decomposable: A is block-diagonal under some ordering, so each block evolves alone. “Approximately decomposable” when the off-block entries are small (Simon and Ando).
- Aggregable: some y = f(x) with m < n variables obeys y(t+1) = Φ̄(y(t)), “dynamically sufficient”. Nothing restricts which microvariables each yᵢ depends on.
- Column-equitable partition: within each block of A, every column sums to the same value. Row-equitable is the same for rows; equitable is both.
- Structure matrix: the matrix R (or S) of those block sums, the “quotient graph”. Ā is a normalized structure matrix.
- Overlap class: for k reference genotypes, the loci whose alleles are shared by exactly a given subset of the references.
Connections
It builds on Simon and Ando (1961) and Ando and Fisher (1963) for near-decomposability, on graph-theoretic equitable partitions (Schwenk 1974; McKay; Godsil) and Stadler and Tinhofer (2000) for their spectral properties, and on Eigen, McCaskill and Schuster (1989), whose quasispecies computation it re-derives by aggregation. Its nonlinear contrast is Wagner and Laubichler (2000). A companion preprint by Shpak, Stadler, Wagner and Altenberg applies the Simon–Ando criteria themselves to landscapes (LIT-1241, registered unread).
Bearing on the record
- THEORY-733. For deterministic x(t+1) = Ax, exact aggregation (1.1) is ĀQ = QA. That is THEORY-733’s criterion L A = B L for a VAR(1), with L = Q. So in a linear Gaussian VAR(1), Shpak et al.’s exact aggregation and Barnett and Seth’s exact dynamical independence are the same condition on the coarse map. THEORY-733 supplies the necessity under Gaussian noise. The join is the record’s: neither paper cites the other.
- CLAIM-388. C1 is the claim’s slogan, with temperature as the natural example. C4 sharpens it. Under uniform coupling every partition of the units is column-equitable (each cross-block column sum is β times the block size), so the vector of block sums closes for every partition, though a single block’s sum does not. Re-checked here: with six units split three and three, the transfer entropy into the pair of block sums is 0 to machine precision, and into one block sum 5.0 × 10⁻² at (α, β) = (0.2, 0.1). A non-uniform equitable matrix also closes, while a random one does not (3.1 × 10⁻¹). So grouping the members does find closed objects in that system, and they are not blocks. The claim stands as stated, and a note on it records this.
- QUESTION-218 gains a third candidate unit, between “a subset of members” and “a function of the whole state”: an equitable partition, whose groups interact strongly but whose group summaries jointly close.
- QUESTION-221. Theorem 1.1’s ‖ĀQ − QA‖_F is A5’s closure defect, unweighted. For states with covariance Σ the least-squares operator is QAΣQᵀ(QΣQᵀ)⁻¹, the weighted version A5 asked for (a one-line derivation here, not the paper’s). The paper relates neither defect to coupling strength, so the question stays open.
- THEORY-661, LIT-1127. The paper paraphrases the Simon–Ando result and supplies no error bound. The record still needs the 1961 paper, which is Unavailable.
- THEORY-607. C6 proposes dynamically sufficient classes as units of evolution, which is Bourrat’s question at another grain (classes of genotypes, not collectives of particles). Its exact criterion has no cost term, so the identity partition passes it too, and it does not meet THEORY-607’s promote_when.
- CLAIM-157. An equitable partition groups members whose cross-group coupling may be strong. It is a partition, as CLAIM-157’s grouping is, but it is not a gap-isolated block.
No instruction for machine-learning practice; nothing for the anthology.
Limitations
- Result 1.1 over the reals. The proof takes “at least one eigenvector” of A. A real matrix may have none: a plane rotation has no real eigenvector. With real Q the result holds for n ≥ 3, since a real matrix always has a real invariant subspace of dimension one or two, with m ≤ 2 < n. It fails for some 2 × 2 matrices. This is the same correction THEORY-733 made to Barnett and Seth’s count for VAR(1).
- C7 is overstated. The discussion says “any aggregation of variables in a linear system will be some form of equitable partition”. But Results 1.1–1.2 build aggregations from eigenvectors that mix all variables and correspond to no partition. The statement holds only if “weighted equitable partition” is stretched to any Q, which empties it.
- C8 against its own example. The authors conjecture that linear aggregability is necessary for nonlinear aggregability, then present the Wagner–Laubichler classes. Those aggregate in the relative-frequency representation and, by their own account, not in the linear one. Read literally, the example is a counterexample. They may mean something narrower, and the text does not say what.
- No near-equitable theory. Exactness only. The authors name the approximate case, nearly equitable partitions over stated timescales, as future work.
- Simon–Ando is paraphrased, not derived. There is no error bound and no statement of the timescale.
- Theorem 2.1 is cited from Stadler and Tinhofer, not proved.
- The landscapes are special. Exact aggregation needs fitness functions with symmetries concordant with the mutation graph. The overlap-class fitness function that makes three peaks work is “highly unnatural and contrived”, in the authors’ words.
- Manuscript copy. The published text and the 2005 erratum were not compared.
Open questions
- When is a partition nearly equitable, and over what timescale does its aggregate track the full system? A bound in terms of the departure from column-equitability would answer it. With Simon–Ando’s bound, it would answer QUESTION-221 for partitions.
- Is there a decomposable linear system with no nontrivial partition-based aggregation? C1’s “nor vice-versa” asserts the converse direction and gives no example. A block-diagonal A always lets each block’s own eigenvector aggregations close, so the claim needs a narrower reading.
- Which nonlinear aggregations have no linear counterpart? C8 makes this open.