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LIT-1241: Simon-Ando Decomposability and Fitness Landscapes

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Properties5
StatusActive — read 2026-10-20 (NOTE-977) in full, from arXiv:q-bio/0403030v1; worth reading for the record's only open statement of Simon and Ando's (1961) theorem (Theorems 2.1, 2.2a and 2.2b: block-diagonal A* plus εC, the four time windows T0 < T1 < T2 < T3, and the aggregate Â_IJ = Σ_{i∈I}Σ_{j∈J} A_ij ν*_jJ), for its four-stage account of when a nearly decomposable system is decomposable, aggregable or both, and for its conclusion that quasi-equilibrium, symmetry (equitable partitions) and linkage equilibrium are independent routes to a dynamically sufficient aggregate. Read it knowing that the theorem is quoted without proof and states no rate in ε and no bound on aggregation error; that the landscape "Simon–Ando condition" is an entrywise proxy, met trivially at low mutation rate even on a flat landscape where there is no time-scale separation; that its leading-eigenvector statement fails when block leading eigenvalues coincide, the stochastic case it invokes; and that (6.2b) is wrong for radius κ ≥ 2. Published as Theory in Biosciences 123(2):139–180 (2004), doi:10.1016/j.thbio.2004.04.001.
Version2
Tagscomplex-systems, philosophy-of-biology, mathematics
SupportsCLAIM-386 — For x(t+1) = (A* + εC)x with A* block-diagonal, the Simon–Ando aggregate  = QAV has one-step closure defect exactly QA − ÂQ = QA(I − VQ); the part from the blocks' own dynamics vanishes iff every block has equal column sums, and then the whole defect is ε·QC(I − VQ)
Cited as “paper” byNOTE-977 — Shpak, Stadler, Wagner and Altenberg — Simon–Ando decomposability and fitness landscapes

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Shpak, M., Stadler, P. F., Wagner, G. P. & Altenberg, L. (2004). Simon-Ando Decomposability and Fitness Landscapes. arXiv:q-bio/0403030 (22 March 2004). Published in Theory in Biosciences 123(2): 139–180 (2004). doi:10.1016/j.thbio.2004.04.001.

Read in full in NOTE-977, from the arXiv copy.

Why this is here

The near-decomposable case that its companion, LIT-1242 (read in NOTE-979), only paraphrases: Simon and Ando’s criteria applied, with their timescale separation, which is what THEORY-661 rests on and what QUESTION-221 asks to bound. With Simon & Ando itself Unavailable (LIT-1127), it is the nearest open statement of the theorem’s conditions the record has found. The abstract’s closing remark, that the formalism is unlikely to extend to other kinds of decomposition and aggregation in evolutionary systems, bears on QUESTION-218.

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