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LIT-1242: Aggregation of Variables and System Decomposition: Applications to Fitness Landscape Analysis

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Properties5
StatusActive — read 2026-10-20 (NOTE-979) in full, from the authors' manuscript in Leipzig University's repository; worth reading for the distinction between aggregation (a closed lower-dimensional description) and decomposition (non-interacting subsets), for the exact condition under which a partition's block sums close in a linear system (column-equitability, THEORY-755), and for its application to fitness landscapes, exact for one or two peaks and not for three. Read it knowing that it treats exact aggregation only, paraphrases Simon–Ando without an error bound, states Result 1.1 in a form that fails for some real 2×2 matrices, and overstates in its discussion that every linear aggregation is an equitable partition.
Version2
Tagscomplex-systems, mathematics, philosophy-of-biology
Cited as “paper” byNOTE-979 — Shpak, Stadler, Wagner and Hermisson — Aggregation of variables and system decomposition
Cited as “source” byTHEORY-755 — For a linear system x(t+1) = Ax and a partition of its variables, the vector of block sums evolves autonomously from every initial state iff the partition is column-equitable, every column of each block of A summing to the same value; the aggregate operator then keeps a subset of A's eigenvalues, and the blocks may interact as strongly as they like, so exact aggregation is not decomposition, THEORY-822 — For a Markov chain and a partition of its states, strong lumpability, every state of a block sending the same probability into each block, is the column-equitability of the master-equation operator and makes the block label informationally closed; when the chain is irreducible, an informationally closed label that is itself Markov is strongly lumpable, with no aperiodicity needed; weak lumpability gives a Markov label that need not be closed; and the same holds on the causal-state chain of any stationary process

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Shpak, M., Stadler, P. F., Wagner, G. P. & Hermisson, J. (2004). Aggregation of Variables and System Decomposition: Applications to Fitness Landscape Analysis. Theory in Biosciences 123(1): 33–68. doi:10.1016/j.thbio.2004.02.001. Erratum: doi:10.1016/j.thbio.2005.05.001.

Not read. The summary is drawn from the abstract or from what the work is standardly cited for. A reading should check it first.

Why this is here

A5’s “potentially more disruptive discovery”: aggregation is not decomposition. Bears on QUESTION-218 and CLAIM-157, whose unit is a block, and on THEORY-607. Simon & Ando (1961), which it extends, is held unread as LIT-1127.

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