Weighted Möbius Score Framework

• Weighted Möbius Score Framework is a unified theory that integrates Möbius inversion with Shapley values for attributing contributions across set and graph structures.
• It employs recursive re-attribution through projection operators and weight matrices to evaluate higher-order synergies in acyclic and multi-feature systems.
• The framework supports applications in model explainability, causal inference, fair division, and complex hierarchical scoring using rigorous algebraic and combinatorial foundations.

The Weighted Möbius Score Framework provides a rigorous and unified approach for attributing values or importance across elements of a system governed by partially ordered relationships, encompassing set-theoretic, combinatorial, and graph-theoretic structures. It synthesizes linear algebraic, combinatorial, and game-theoretic ideas to generalize Möbius inversion and Shapley value concepts to encompass general weighted directed acyclic multigraphs (DAGs) and, in parallel, situates all feature and interaction attribution methods on a common mathematical foundation. Central to this framework is the recursive re-attribution of higher-order synergies, uniquely specified via projection operators acting on the Möbius (synergy) transform of a value function, and parameterized by weight matrices encoding different attribution philosophies. The resulting machinery provides both conceptual unification and practical accessibility for applications in explainability, causal inference, fair division, and hierarchical scoring.

1. Algebraic and Combinatorial Foundations

Let $D = \{1, \ldots, d\}$ be a finite set of $d$ elements (e.g., features) or, more generally, let $G = (V, E)$ be a finite directed acyclic multigraph (DAG). For the set-theoretic case:

• The power set $\mathcal{P}(D)$ is ordered by inclusion, forming the Boolean lattice.
• A value function (or game) is $f : \mathcal{P}(D) \to \mathbb{R}$, assigning to each subset a numerical score.

For a general $R$-DAMG (edge- and root-weighted directed acyclic multigraph over commutative ring $R$ and $R$-module $A$), vertices carry values $v : V \to A$, and edge/rroot weights $d$0, $d$1.

The classical Möbius transform $d$2, where $d$3, is given by

$d$4

and in the graph case, a recursive formula along a topological order or via DAG Möbius functions.

The essential insight is that all local attribution maps reside in the $d$5-dimensional vector space $d$6 (or $d$7 in the DAG case), and every linear combination, Möbius inversion, and weighting is an operator on this space (Jiang et al., 2023, Forré et al., 7 Oct 2025).

2. Weighted Möbius Scores: Definitions and Parameterizations

The Weighted Möbius Score Framework introduces a family of attribution methods parameterized by weighting functions $d$8 in the set-theoretic case, or by a path-based kernel $d$9 in the DAG context.

Set Case ($G = (V, E)$0):

• The feature-isolation score is $G = (V, E)$1.
• The Möbius score is $G = (V, E)$2.
• Any weighted attribution is

$G = (V, E)$3

Faithful weights are those for which $G = (V, E)$4 whenever $G = (V, E)$5, ensuring only relevant features contribute to their attributions (Jiang et al., 2023).

Graph Case ($G = (V, E)$6-DAMG):

• The Möbius synergy on vertices is $G = (V, E)$7, recursively as

$G = (V, E)$8

• Attribution proceeds via projection operators $G = (V, E)$9 and a normalized projection-kernel $\mathcal{P}(D)$0.
• The weighted Shapley-style value at root $\mathcal{P}(D)$1 is

$\mathcal{P}(D)$2

where $\mathcal{P}(D)$3 counts root-to-$\mathcal{P}(D)$4 paths, $\mathcal{P}(D)$5 (Forré et al., 7 Oct 2025).

3. Connections to Classical and Modern Attribution Methods

The Weighted Möbius Score framework unifies a broad array of attribution and interaction measures—many of which correspond to specific choices of weighting:

Attribution Name	Weight Function $\mathcal{P}(D)$6	Reference
Möbius Score (MI)	$\mathcal{P}(D)$7 if $\mathcal{P}(D)$8, $\mathcal{P}(D)$9 else	(Jiang et al., 2023)
Shapley Value	$f : \mathcal{P}(D) \to \mathbb{R}$0 if $f : \mathcal{P}(D) \to \mathbb{R}$1	(Jiang et al., 2023)
Shapley Interaction Index	$f : \mathcal{P}(D) \to \mathbb{R}$2 if $f : \mathcal{P}(D) \to \mathbb{R}$3	(Jiang et al., 2023)
Shapley–Taylor Index (order $f : \mathcal{P}(D) \to \mathbb{R}$4)	$f : \mathcal{P}(D) \to \mathbb{R}$5 if $f : \mathcal{P}(D) \to \mathbb{R}$6 ($f : \mathcal{P}(D) \to \mathbb{R}$7), else $f : \mathcal{P}(D) \to \mathbb{R}$8	(Jiang et al., 2023)
ArchAttribute	$f : \mathcal{P}(D) \to \mathbb{R}$9 if $R$0, $R$1 else	(Jiang et al., 2023)

This table illustrates that classical cooperative game-theoretic solutions (e.g., Harsanyi dividend, Shapley) and recent feature interaction indices are all subcases of the general weighted framework.

In the DAG framework, the attribution generalizes to hierarchies and mereological systems, with the projection kernel and path enumeration specifying the split of synergy across roots, encompassing the classic Shapley value on Boolean lattices as a special case (Forré et al., 7 Oct 2025).

4. Axiomatic and Structural Guarantees

The framework is axiomatically pinned down by several properties:

• Linearity: Both in the ground ring $R$2 and module $R$3, attributions preserve addition and scalar multiplication.
• Efficiency: The sum of all root attributions equals the total synergy (i.e., no surplus loss).
• Null-player/Null-root: Any coalition or root that contributes no synergy under the Möbius transform receives zero attribution.
• Symmetry: Attributions are invariant under automorphisms of the system that preserve equivalence classes.
• Projection property: Weak elements—those contributing no synergy—can be eliminated without affecting the attributions elsewhere.
• Flat hierarchy: For graphs lacking intermediate nodes, the protocol prescribes proportional splits according to path-counting or edge-strengths (Forré et al., 7 Oct 2025).

By enforcing these, the path-uniform kernel (or associated weighting matrix in the set case) is uniquely specified, ensuring a principled split of contributions and guaranteeing interpretability.

5. Interpretations: Game Theory, Causal Mediation, and Beyond

In cooperative game theory, the Möbius score coincides with the Harsanyi dividend and all allocation concepts linear in the dividend (e.g., Shapley, k-order indices) are instantiations of weighted Möbius scores (Jiang et al., 2023). In causal mediation, the mediated interaction effect (MI) assigns an attributable effect to any coalition of mediators/features via inclusion–exclusion, recoverable as a Möbius transform.

Novel attributions such as arbitrary-order MI (for $R$4) become accessible via this framework, leveraging properties of the Möbius inversion to guarantee both efficiency and identifiability.

In the DAG extension, applications to neural network features, general mereological systems, and hierarchical fair division are supported. The recursive projection process encodes the transfer of higher-order synergy to lower-order nodes in the presence of general acyclic dependencies (Forré et al., 7 Oct 2025).

6. Applications and Empirical Insights

The framework has been demonstrated in several key empirical domains:

• Sentiment Analysis (BERT, SST-2): PIE and MI values computed layerwise capture attribution and interaction effects, with MI dominating early in the model and PIE growing in later layers (Jiang et al., 2023).
• Chain-of-Thought Prompting (GPT-3.5-turbo): Möbius score, Shapley value, higher-order indices, and faithfulness-weighted attributions reveal the degree of both individual and synergistic sentence/phrase contributions, distinguishing new insights about prompt efficacy (Jiang et al., 2023).
• Complex hierarchies: Attributions in multiroot and polyedge DAGs can be computed via the recursive synergy-projection formalism, enabling nuanced modeling of multi-path, non-lattice relationships (Forré et al., 7 Oct 2025).

Potential applications extend to feature attribution in machine learning, explainable artificial intelligence for models with structured inputs, fair allocation of value or risk in networked systems, and higher-order effect decomposition in the natural sciences.

7. Summary and Outlook

The Weighted Möbius Score Framework establishes a comprehensive, axiomatically grounded, and computationally accessible theory for weighting and attributing values in systems representable by sets, lattices, or acyclic graphs. By encoding all attribution philosophies as weight matrices or path kernels acting on Möbius transforms, the framework not only unifies existing feature- and interaction-attribution methodologies, but also enables the principled generation of new ones, with algorithmic formulations suitable for high-dimensional and vector-valued applications. Its flexibility and universality render it particularly suitable for future developments in machine learning explainability, causal analysis, and complex system modeling (Jiang et al., 2023, Forré et al., 7 Oct 2025).