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Conjectured minimal properties for structural α‑synergy decomposition

Establish whether a function f() on a set X necessarily must satisfy localizability, symmetry, non-negativity, and monotonicity in order to induce an interpretable α‑synergy decomposition defined by minimal loss under removal of subsets of elements.

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Background

The paper proposes extending the α‑synergy decomposition beyond information-theoretic measures to any function on sets that yields a meaningful multi-scale synergy backbone. To ensure interpretability and desirable properties (non-negativity and monotonicity of the decomposition), the author posits a set of minimal conditions on the function f().

Determining whether these conditions are necessary (and possibly sufficient) would ground the structural synergy framework mathematically and clarify which non-information measures (e.g., network communicability, efficiency) can be validly decomposed using α‑synergy.

References

For a function f() on a set X to induce an interpretable α-synergy decomposition, I conjecture that it must satisfy a minimal set of properties: Localizability: If f(X) is an expected value, the f() must also be defined on every local instance of X=x. Symmetry: f(x) is invariant under any permutation of the elements of x. Non-negativity: f(x)) > 0. Monotonicity: if y⊆x, then f(y) ≤ f(x).

— A scalable, synergy-first backbone decomposition of higher-order structures in complex systems (2402.08135 - Varley, 13 Feb 2024) in Section “Extensions beyond information: Structural synergy”