Edge-Based Shapley Value in Networks

• Edge-based Shapley value is a framework that assigns value to edges based on their marginal contributions in cooperative games, with applications in GNN sparsification, network allocation, and phylogenetics.
• These methods employ classical Shapley formulas, random-order evaluations, node-induced games, and edge-flow decompositions to capture interaction-centric valuation.
• Practical insights include improved pruning in graph neural networks and fair allocation in economic and biodiversity models, despite computational challenges and varying axiomatic guarantees.

Searching arXiv for papers on edge-based Shapley value and related formulations. Edge-based Shapley value denotes a family of Shapley-style attribution and allocation constructions in which edges, links, hyperedges, or edge-induced interaction structures are the primary locus of value generation or marginal contribution. Across recent literature, the term does not refer to a single universal object, but to several technically distinct frameworks. In graph neural network inference, edges are treated as players in a local cooperative game over a node’s computational graph, and their signed Shapley values are aggregated into global pruning scores for graph sparsification (Akkas et al., 28 Jul 2025). In cooperative game theory on networks, value may be defined on coalitions of edges and then allocated either directly to edges or to nodes through an induced node-side game, yielding an allocation rule termed the edge-based Shapley value (Yamada et al., 16 Jul 2025). In classical network games, a related link-based construction underlies the position value and its weighted generalization, where links are the players of an associated game and their Shapley payoffs are distributed to incident nodes (Kakoty et al., 2023). Other uses include phylogenetic diversity on trees (Wicke et al., 2017), priority-weighted random-order values on directed weighted graphs (Lee et al., 14 May 2026), Hodge-theoretic path-integral allocations on coalition graphs and cooperative networks (Lim, 2022), and edge-responsibility for regular path queries in labeled graphs (Khalil et al., 2022). The unifying theme is a shift from node-only attribution toward interaction-centric valuation.

1. Conceptual scope and principal variants

The most direct meaning of edge-based Shapley value arises when edges themselves are the players of a transferable-utility game. In the GNN sparsification setting, for a target node $v$ and an $l$-layer GNN, the relevant player set is the edge set $E_c(v)$ of the $l$-hop computational graph $G_c(v)$, and each edge receives a Shapley value measuring its marginal effect on the predicted class probability under masked adjacency coalitions (Akkas et al., 28 Jul 2025). In cooperative network games, an edge-based graph game is defined as $(N,E,w)$ with $w:2^E\to\mathbb{R}$, so that coalitions are edge sets rather than node sets (Yamada et al., 16 Jul 2025). In network games and the position-value literature, links are likewise treated as the players of an associated link game $v^E$, after which link-level Shapley payoffs are split among endpoints (Kakoty et al., 2023).

A second variant is node allocation induced by edge-centric valuation. The 2025 cooperative-game formulation defines the paper’s allocation rule as

$$ESh_i(N,E,w)=Sh_i(N,w^N), \qquad w^N(S)=w(\{e\in E\mid e\subseteq S\}),$$

so nodes receive Shapley values computed on a node-side game induced from an underlying edge-side characteristic function (Yamada et al., 16 Jul 2025). This differs from frameworks that first compute Shapley values on edges and then split edge payoff post hoc. The distinction is explicit in that work: “There is no post-hoc split of edge value to incident nodes; instead, node allocations arise by computing the classical Shapley value on $w^N$” (Yamada et al., 16 Jul 2025).

A third variant uses edge structure to deform the random-order distribution underlying marginal contributions. The generalized priority-aware Shapley value introduces a directed weighted priority graph $l$0, where edge weights $l$1 penalize order violations in permutations via a Gibbs factor $l$2. The resulting value remains a random-order expectation over player marginals, but the order law is edge-driven, soft, weighted, and cyclicity-tolerant (Lee et al., 14 May 2026). This suggests a broader interpretation in which “edge-based” need not mean that edges are the final recipients of value; it can also mean that edges are the structural primitives governing marginal-contribution aggregation.

A fourth line of work interprets Shapley allocation through edge flows on state-transition graphs. In the Hodge-Shapley framework, players remain nodes, but each player’s marginal contribution is represented by an edge flow on a cooperative network, and values at target states are obtained as expected stochastic path integrals or as solutions of a graph Poisson equation (Lim, 2022). Here edge-based structure enters through gradients, divergences, and edgewise marginal flows rather than through an edge coalition game.

2. Formal definitions when edges are the players

When edges are the players in a cooperative game, the Shapley definition is formally classical. For a player set $l$3 and characteristic function $l$4 with $l$5, the edge-level Shapley value of edge $l$6 is

$l$7

This is the canonical edge-level definition given in the general cooperative-game treatment (Yamada et al., 16 Jul 2025). The same paper also gives the equivalent random-order form over permutations of $l$8 (Yamada et al., 16 Jul 2025).

In GNN inference, the local game is instantiated around a target node. Let $l$9 be the edges of the computational graph of $E_c(v)$0. For edge $E_c(v)$1, the local Shapley value is

$E_c(v)$2

where the payoff function is the predicted probability of the target class under masked adjacency: $E_c(v)$3 Here $E_c(v)$4 is the binary mask induced by coalition $E_c(v)$5, and explanations are computed for the predicted class (Akkas et al., 28 Jul 2025). This is a purely local edge-player game whose output is later aggregated across nodes.

In phylogenetic diversity, the player set is instead the leaf set $E_c(v)$6, but the Shapley value admits an explicit edge-based formula because the cooperative game $E_c(v)$7 is additive over tree edges. For an unrooted binary phylogenetic tree with edge lengths $E_c(v)$8, the Shapley value of leaf $E_c(v)$9 can be written as

$l$0

where $l$1 and $l$2 are split counts induced by removing edge $l$3 (Wicke et al., 2017). This is edge-based in a distinct sense: the value is assigned to leaves, but the dependence on the game is fully mediated by edge lengths and split-induced coefficients.

In network games, the position value constructs a link game $l$4 over the edge set $l$5, with

$l$6

where the link coalition $l$7 is interpreted as a subnetwork $l$8. The link Shapley value is then

$l$9

or equivalently via Harsanyi dividends (Kakoty et al., 2023). This provides the edge-level layer from which endpoint allocations are derived.

3. Induced node allocations and relations to classical values

A central issue is how an edge-centric game produces node-level allocations. The 2025 allocation-rule paper adopts induction rather than splitting. Given $G_c(v)$0, it defines

$G_c(v)$1

and then sets

$G_c(v)$2

This rule inherits the classical Shapley axioms on the induced node game, including efficiency, linearity, symmetry, and the null-player property (Yamada et al., 16 Jul 2025). Efficiency takes the form

$G_c(v)$3

The same framework proves a neighborhood restriction: $G_c(v)$4 where $G_c(v)$5 is the out-neighborhood of $G_c(v)$6 in the directed graph (Yamada et al., 16 Jul 2025). Since marginal contributions vanish outside the neighborhood, the induced allocation is local in the graph-theoretic sense.

The relation to the Myerson value is explicit. For a graph game $G_c(v)$7 on nodes, define an edge-side characteristic function

$G_c(v)$8

Then $G_c(v)$9, the Myerson-restricted characteristic function over connected components of the induced subgraph. Consequently,

$(N,E,w)$0

so the Myerson value appears as a special case of the edge-based Shapley construction (Yamada et al., 16 Jul 2025).

The position value and weighted position value follow a different route. The classical position value allocates each edge’s Shapley share equally among its two incident nodes: $(N,E,w)$1 The weighted position value replaces equal splitting with proportional splitting based on exogenous node weights: $(N,E,w)$2 If all node weights are equal, $(N,E,w)$3 reduces to the classical position value (Kakoty et al., 2023). This establishes a second major paradigm: edge-based Shapley at the link level, followed by endpoint distribution.

A distinct node-allocation mechanism appears in Hodge-Shapley theory. There, one solves

$(N,E,w)$4

on a cooperative network with reversible Markov dynamics, where $(N,E,w)$5 is player $(N,E,w)$6’s edge flow and $(N,E,w)$7 is the graph Laplacian (Lim, 2022). The resulting $(N,E,w)$8 is a coalition- or state-wise allocation, not only a grand-coalition number. This allocation satisfies five axioms extending the Shapley axioms and coincides with the classical Shapley value at the grand coalition in the coalition hypercube case (Lim, 2022).

4. Axioms, structural properties, and fairness claims

Because many edge-based constructions reduce to ordinary Shapley values on suitably transformed games, standard axioms recur. In the induced-node formulation, efficiency, linearity, symmetry, and null-player hold directly since $(N,E,w)$9 (Yamada et al., 16 Jul 2025). The same work proves a Myerson-style fairness statement for edge removal: for an edge $w:2^E\to\mathbb{R}$0, the differential effect of removing $w:2^E\to\mathbb{R}$1 on the allocations of its endpoints is equal (Yamada et al., 16 Jul 2025). Under additivity of $w:2^E\to\mathbb{R}$2 across disjoint node sets, the rule also satisfies component efficiency: $w:2^E\to\mathbb{R}$3 for any connected component $w:2^E\to\mathbb{R}$4 (Yamada et al., 16 Jul 2025).

In GNN sparsification, the salient structural property is signedness. Shapley values can be positive or negative. Positive $w:2^E\to\mathbb{R}$5 means an edge increases prediction confidence for the target class, while negative $w:2^E\to\mathbb{R}$6 means it decreases confidence and may be misleading or adversarial (Akkas et al., 28 Jul 2025). This signed nature is treated as critical for sparsification, in contrast to many explainers that produce only non-negative importance scores. The same paper highlights standard Shapley properties—efficiency, symmetry, linearly, and dummy—within the edge-player game, emphasizing that signed attributions permit the removal of harmful edges while preserving influential ones (Akkas et al., 28 Jul 2025).

In the phylogenetic setting, efficiency has a particularly transparent per-edge form. If $w:2^E\to\mathbb{R}$7 is the Shapley transformation matrix with entries

$w:2^E\to\mathbb{R}$8

then for each edge $w:2^E\to\mathbb{R}$9, the coefficients sum to $v^E$0 over leaves, which implies

$v^E$1

Thus total allocated leaf value equals total tree length (Wicke et al., 2017).

The priority-aware construction (Lee et al., 14 May 2026) is not an edge-coalition game, but its axiomatization is instructive for the broader category. The value is a random-order value satisfying Weber’s conditions for ROV representations and is uniquely determined by Generalized State-Choice Factorization, Generalized Weight Proportionality, and a Pairwise-Violation Factorization boundary condition (Lee et al., 14 May 2026). This establishes that edge-weighted soft priority graphs can shape Shapley-like values while preserving an axiomatic random-order foundation.

Not all edge-based constructions retain the full classical fairness repertoire. In the hypergraph-based individual Shapley value tied to Forman curvature, efficiency, additivity, and component efficiency hold, but symmetry and null player do not hold in general, and the paper notes that the rule fails Myerson’s fairness axiom in general (Yamada, 2021). This is a useful corrective to any impression that “edge-based Shapley value” automatically preserves the entire Shapley–Myerson axiomatics across all generalizations.

5. Computation and approximation

Exact edge-level Shapley computation is typically combinatorial, and most frameworks rely on structure, decomposition, or sampling.

In GNN inference, exact computation over all edge coalitions of $v^E$2 is infeasible for large $v^E$3. The paper follows GraphSVX and GNNShap in approximating the payoff with a linear surrogate

$v^E$4

estimated by least squares from $v^E$5 sampled coalitions and masked forward passes (Akkas et al., 28 Jul 2025). Per node, the dominant cost is $v^E$6, where $v^E$7 is the cost of a masked forward pass on the computational graph. GNNShap’s GPU batching and kernels reduce wall-clock time substantially; the paper reports that it produced explanations on PubMed and Coauthor-CS where GraphSVX exceeded a 10-hour limit (Akkas et al., 28 Jul 2025).

The induced-node allocation rule does not present a specialized approximation algorithm, but its neighborhood restriction reduces the coalition space from all subsets of $v^E$8 to subsets of $v^E$9, giving $$ESh_i(N,E,w)=Sh_i(N,w^N), \qquad w^N(S)=w(\{e\in E\mid e\subseteq S\}),$$0 terms per node in a directed graph (Yamada et al., 16 Jul 2025). The paper notes that standard Shapley Monte Carlo estimators apply directly, and that sparse or low-degree graphs are computationally favorable (Yamada et al., 16 Jul 2025).

GPASV develops a full sampling machinery for edge-shaped random-order values. It uses adjacent-swap Metropolis–Hastings on permutations, with a local acceptance ratio

$$ESh_i(N,E,w)=Sh_i(N,w^N), \qquad w^N(S)=w(\{e\in E\mid e\subseteq S\}),$$1

so only local edge-weight differences and competing prefixes need to be recomputed (Lee et al., 14 May 2026). The framework also provides greedy initialization, direct Monte Carlo estimation of marginal contributions, and self-normalized importance sampling reuse across parameter sweeps (Lee et al., 14 May 2026).

In Hodge-Shapley theory, computation reduces to sparse linear algebra. For each player, one builds the gradient $$ESh_i(N,E,w)=Sh_i(N,w^N), \qquad w^N(S)=w(\{e\in E\mid e\subseteq S\}),$$2, divergence $$ESh_i(N,E,w)=Sh_i(N,w^N), \qquad w^N(S)=w(\{e\in E\mid e\subseteq S\}),$$3, Laplacian $$ESh_i(N,E,w)=Sh_i(N,w^N), \qquad w^N(S)=w(\{e\in E\mid e\subseteq S\}),$$4, and solves a reduced Dirichlet Laplacian system after fixing the reference state (Lim, 2022). This yields all state values simultaneously. The paper explicitly positions the method as a graph-based Laplacian-solver problem (Lim, 2022).

In query responsibility for regular path queries, exact edge-based Shapley computation is often intractable. The exact value is $$ESh_i(N,E,w)=Sh_i(N,w^N), \qquad w^N(S)=w(\{e\in E\mid e\subseteq S\}),$$5-hard whenever a non-redundant conjunct permits a word of length three or more, while for RPQs the problem is polynomial-time computable if and only if all words have length at most two (Khalil et al., 2022). By contrast, additive approximation admits an FPRAS via uniform random permutation sampling of endogenous edges and evaluation of whether the target edge is pivotal (Khalil et al., 2022). Multiplicative approximation is available in polynomial time exactly when all query atoms have finite languages, assuming non-redundancy and conventional complexity limitations (Khalil et al., 2022).

6. Applications and empirically documented behavior

The most detailed application in the recent literature is GNN inference-time sparsification. Local edge Shapley values $$ESh_i(N,E,w)=Sh_i(N,w^N), \qquad w^N(S)=w(\{e\in E\mid e\subseteq S\}),$$6 are aggregated to global scores

$$ESh_i(N,E,w)=Sh_i(N,w^N), \qquad w^N(S)=w(\{e\in E\mid e\subseteq S\}),$$7

where $$ESh_i(N,E,w)=Sh_i(N,w^N), \qquad w^N(S)=w(\{e\in E\mid e\subseteq S\}),$$8 is the set of nodes whose computational graphs contain edge $$ESh_i(N,E,w)=Sh_i(N,w^N), \qquad w^N(S)=w(\{e\in E\mid e\subseteq S\}),$$9 (Akkas et al., 28 Jul 2025). The pruning policy removes edges with the smallest global scores, prioritizing negative $w^N$0, using either a top-$w^N$1 or thresholded selection rule. The reported results show that with mean aggregation, GNNShap can prune $w^N$2 of edges on Cora with less than $w^N$3 accuracy drop for both GCN and GAT; on PubMed-GCN it achieves $w^N$4 pruning with less than $w^N$5 drop, and on PubMed-GAT it matches original accuracy at $w^N$6 pruning; on Coauthor-CS-GAT it matches original accuracy at $w^N$7 pruning (Akkas et al., 28 Jul 2025). The paper also reports substantial MAC reductions, such as Cora-GCN from $w^N$8 to $w^N$9 MACs at $l$00 sparsity and PubMed-GCN from $l$01 to $l$02 (Akkas et al., 28 Jul 2025). An ablation shows that replacing signed $l$03 with $l$04 significantly degrades pruning performance (Akkas et al., 28 Jul 2025).

In economic and logistical networks, the induced-node edge-based Shapley value is illustrated through content platform networks and supply chain logistics. In a one-platform, three-content case, the computed allocation is

$l$05

while a two-platform, five-content case yields

$l$06

illustrating how exclusivity can produce higher allocation than raw route count alone (Yamada et al., 16 Jul 2025). In supply-chain logistics, route valuation is defined as

$l$07

with $l$08 in experiments (Yamada et al., 16 Jul 2025). The reported allocations identify redundancy and cost sensitivity in several five-node scenarios (Yamada et al., 16 Jul 2025).

In phylogenetic diversity, the edge-based formula is used both analytically and critically. The paper shows that non-isomorphic unrooted trees can have permutation-equivalent, even identical, Shapley transformation matrices and identical null spaces, so split counts or leaf Shapley values do not identify tree topology (Wicke et al., 2017). It also proves a negative result for biodiversity prioritization: for a constructed family $l$09, the top $l$10 species ranked by Shapley value can all lie in a subtree whose total phylogenetic diversity tends to $l$11 as $l$12, while the total diversity of the full tree tends to $l$13 (Wicke et al., 2017). This is one of the clearest demonstrations that an edge-based Shapley decomposition need not be an optimal selection heuristic for a constrained combinatorial objective.

In hypergraphs, the individual Shapley value linked to Forman curvature defines local edge-characteristic functions $l$14 and $l$15 such that

$l$16

with an analogous identity for $l$17 and $l$18 (Yamada, 2021). For the modular curvature-based choice, the paper derives the closed form

$l$19

and reports the vector $l$20 on its worked hypergraph example (Yamada, 2021).

7. Limitations, misconceptions, and open directions

A recurrent misconception is that edge-based Shapley value is a single standardized definition. The literature shows otherwise. At least four non-equivalent meanings coexist: Shapley on edge coalitions; node Shapley on an induced edge-centric game; edge-informed random-order values; and edge-flow-based coalition-state allocations. Formulas, axioms, and computational burdens depend heavily on which construction is meant.

Another misconception is that edge-based formulations always improve fairness or identifiability. The phylogenetic case shows that Shapley values and split counts do not reconstruct topology and can fail as a prioritization criterion for maximizing diversity under a budget (Wicke et al., 2017). The hypergraph formulation shows that symmetry, null-player, and Myerson fairness can fail in general (Yamada, 2021). In query responsibility, exact edge-based Shapley computation is almost always hard once path languages become expressive enough (Khalil et al., 2022).

In GNNs, the principal limitations are computational and model-dependent. Even with GPU acceleration, per-node Shapley estimation on very large graphs remains costly; explanations depend on the trained model, so inaccurate models can induce misleading attributions; higher-order edge interactions remain challenging for linear surrogates; and the reported pruning is unconstrained with respect to connectivity preservation (Akkas et al., 28 Jul 2025). The paper identifies dynamic graphs, better sampling strategies, caching, distributed implementations, and connectivity-constrained objectives as natural extensions (Akkas et al., 28 Jul 2025).

In induced-node economic networks, the main limitation is dependence on the specification of $l$21. Mis-specified route or contract data directly bias allocations, and large-degree nodes remain expensive despite locality reductions (Yamada et al., 16 Jul 2025). The paper notes future work on approximation algorithms, dynamic or stochastic networks, and hypergraph extensions (Yamada et al., 16 Jul 2025).

In priority-aware valuation, the edge-priority graph itself may be noisy or cyclic due to finite data, and high $l$22 can overconcentrate attribution on hub-like players (Lee et al., 14 May 2026). The paper’s main empirical conclusion is that priority-aware valuation is not a one-button procedure: different balances of pairwise edge priority and node-level soft priority produce substantively different outcomes (Lee et al., 14 May 2026). This suggests a broader methodological caution for edge-based Shapley models: once interactions rather than individual players structure the valuation problem, modeling choices about how edge information enters the game often dominate the formal choice of Shapley solution concept itself.