Möbius transforms and Shapley values for vector-valued functions on weighted directed acyclic multigraphs

TL;DR

This work generalizes Möbius inversions and Shapley values to directed acyclic multigraphs with weights, allowing vector-valued (or more generally $R$-module valued) value functions. It introduces projection operators and strengthened axioms (weak elements and flat hierarchy) to obtain a unique Shapley formula, and provides two pathways: (i) predefined projection weights on PDAMGs and (ii) inference of projection weights via path-uniform weighting on DAMGs. The authors establish a robust theory using a path-algebra perspective, define induced projection weights, and derive explicit formulas for Shapley values in terms of total path weights, with a unifying framework that recovers classical Shapley values on Boolean lattices and extends to broader combinatorial structures. This framework opens avenues for vector-valued attribution in applications such as explainable AI, and offers efficient computational schemes via the path-algebra formulation. Overall, the paper delivers a principled, axiomatized method to project higher-order interactions onto root structures across general DAG-based mereologies, with strong theoretical guarantees and broad applicability.

Abstract

We generalize the concept of Möbius inversion and Shapley values to directed acyclic multigraphs and weighted versions thereof. We further allow value functions (games) and thus their Möbius transforms (synergy function) and Shapley values to have values in any abelian group that is a module over a ring that contains the graph weights, e.g. vector-valued functions. To achieve this and overcome the obstruction that the classical axioms (linearity, efficiency, null player, symmetry) are not strong enough to uniquely determine Shapley values in this more general setting, we analyze Shapley values from two novel points of view: 1) We introduce projection operators that allow us to interpret Shapley values as the recursive projection and re-attribution of higher-order synergies to lower-order ones; 2) we propose a strengthening of the null player axiom and a localized symmetry axiom, namely the weak elements and flat hierarchy axioms. The former allows us to remove coalitions with vanishing synergy while preserving the rest of the hierarchical structure. The latter treats player-coalition bonds uniformly in the corner case of hierarchically flat graphs. Together with linearity these axioms already imply a unique explicit formula for the Shapley values, as well as classical properties like efficiency, null player, symmetry, and novel ones like the projection property. This whole framework then specializes to finite inclusion algebras, lattices, partial orders and mereologies, and also recovers certain previously known cases as corner cases, and presents others from a new perspective. The admission of general weighted directed acyclic multigraph structured hierarchies and vector-valued functions and Shapley values opens up the possibility for new analytic tools and application areas, like machine learning, language processing, explainable artificial intelligence, and many more.

Figures (2)

• Figure 1: A directed acyclic graph $G$ with its value function $v$ and synergy function $w$. Below are $G$'s projections$G^{\setminus e}$ and $G^{\setminus\left \{ d,e\right \} }$, where we removed the elements $e$ and $d$, while preserving all of the remaining hierarchical structure. Note that the last graph has turned into a directed acyclic multigraph (DAMG) with multiple parallel directed edges between root $b$ and leaf $g$. By allowing such parallel directed edges, we have achieved that the number of directed paths between root $b$ and leaf $g$ stays the same in each of the graphs. Regarding the Shapley values $\mathop{\mathrm{Sh}}\nolimits^G_r(v)$, one then has to decide how to split and attribute the synergistic contribution $w(g)=8$ of leaf $g$ to the roots $a$, $b$, $c$. We argue, since $d$ and $e$ are considered weak elements, i.e. $w(d)=w(e)=0$ (see \ref{['def:weak-element']}), that the split of $w(g)$ should lead to the same number in each of the graphs $G$, $G^{\setminus e}$ and $G^{\left \{ d,e\right \} }$, where (some of) those elements are projected out, and when the synergy functions are re-computed there. Since $G^{\setminus\left \{ d,e\right \} }$ is hierarchically flat, that is, only consists of roots and leaves, the only uninformative, symmetric way to distribute $w(g)$ is by splitting it uniformly among the 4 directed edges pointing to $g$. By the above arguments, this split should then also hold in the original graph $G$. We reason similarly for the other nodes and get: $\mathop{\mathrm{Sh}}\nolimits^G_b(v) = w(b)+\frac{1}{2}w(f)+\frac{2}{4}w(g)+\frac{1}{2}w(h)=2+1+4+2=9$. This reasoning will generalize to all possible DAMGs and uniquely determine the Shapley values for arbitrary vector-valued functions on them and result in an explicit formula and many desirable properties. This will be shown in \ref{['cor:shapley-main']}.
• Figure 2: The Shapley value of spin $d$ in the higher-order Ising model from Example \ref{['ex:ising']} becomes larger than the Shapley value for spin $a$ whenever the 3-point interaction $J_{bcd}$ exceeds $3$.

Theorems & Definitions (113)

• Definition 2.1: Directed acyclic multigraph
• Example 2.3: Posets $P$ as an DAGs
• Example 2.4: Lattices $L$ as DAGs
• Example 2.5: Power sets as DAGs
• Definition 2.6: Edge and root weighted directed acyclic multigraph ($R$-DAMG)
• Example 2.7: Standard edge and root weight functions of directed acyclic multigraphs
• Remark 2.8: Default choice of edge and root weight function on DAMGs
• Definition 2.9: $R$-DAMG automorphism
• Definition 2.11: Horizontal subset
• Remark 2.12