Shapley Value Graph Sparsification

• The paper proposes a novel post hoc inference method for GNNs that uses signed Shapley values to rank and prune edges, maintaining node classification accuracy with reduced message-passing cost.
• Local Shapley explanations are aggregated globally to form an edge importance ranking, enabling efficient, retraining-free sparsification across variable sparsity budgets.
• Empirical evidence shows that pruning up to 80% of edges on datasets like Cora and PubMed can preserve performance while significantly lowering computational overhead.

Shapley value-based graph sparsification denotes a class of post hoc inference procedures for Graph Neural Networks in which local Shapley-value explanations are converted into a global pruning rule on graph structure. In the formulation studied for node classification, a trained GNN is first explained on the original graph, edge-level or edge-derived local importance scores are aggregated across nodes, and edges are then ranked and removed under a target sparsity budget. The distinctive feature of the Shapley-based approach is that Shapley values are signed: an edge can receive a positive score if it supports a prediction or a negative score if it decreases the target output. This signed attribution is used to retain strongly positive edges, remove weak edges, and prune negative edges earliest, with the aim of preserving predictive performance while reducing message-passing cost and improving interpretability (Akkas et al., 28 Jul 2025).

1. Definition and scope

In this setting, graph sparsification means removing edges from the input graph so that a trained GNN can perform inference on a smaller graph with similar predictive accuracy. The motivation is that, for GNNs, inference cost is often dominated not by parameter matrices but by the graph structure itself, especially the number of edges involved in message passing. If many edges are redundant, weakly useful, or harmful, pruning them can reduce memory and computation (Akkas et al., 28 Jul 2025).

The version emphasized in recent work is post hoc sparsification for inference. The model is trained once on the original graph; explanation scores are then computed; edges are pruned; and inference is run on the sparsified graph. This differs from graph lottery ticket or denoising methods that learn sparsity during training and often require retraining for each sparsity level. The procedure is therefore tied to the behavior of a fixed pretrained GNN rather than to a jointly optimized sparse training objective (Akkas et al., 28 Jul 2025).

Explainability is used as the bridge from prediction to pruning. For node classification, a local explainer assigns importance scores to edges in the node’s computational graph. If those scores are meaningful, then aggregating them across nodes yields a global edge importance score that can be used for pruning. The attraction of this design is threefold: it does not require retraining after sparsification, one can sweep different sparsity levels without recomputing explanations once the scores are available, and the resulting sparse graph is interpretable because it preserves edges deemed useful by the model (Akkas et al., 28 Jul 2025).

A central distinction within explainability-driven sparsification is between non-negative importance and signed contribution. Many explanation methods produce only non-negative or magnitude-like scores. Such methods can reveal that an edge matters, but not whether it matters in a helpful or harmful way. For pruning, this is a substantial limitation: an edge may strongly affect the prediction yet do so by reducing confidence in the correct or predicted class. In that case, magnitude-only ranking can preserve an edge that should instead be removed (Akkas et al., 28 Jul 2025).

2. Signed Shapley attributions as pruning signals

The formal basis of the method is the Shapley value. In the standard cooperative-game formulation, the Shapley value of player $i$ is

$$\phi_i = \sum_{S \subseteq \{1,\dots,n\}\setminus \{i\}} \frac{|S|!(n-|S|-1)!}{n!}\left[f(S\cup\{i\}) - f(S)\right].$$

Equivalently,

$$\phi_i(v)=\sum_{S\subseteq N\setminus\{i\}} \frac{|S|!(|N|-|S|-1)!}{|N|!}\Big(v(S\cup\{i\})-v(S)\Big).$$

Here $n$ is the number of players, $S$ is a coalition, and $f(S\cup\{i\})-f(S)$ is the marginal contribution of player $i$ (Akkas et al., 28 Jul 2025).

For the sparsification method centered on GNNShap, the players are edges, and the task is node classification for the predicted class of each node. For a node $v$ and target class $t$, the explainer assigns an edge-level local score

$\phi_v^t(i,j),$

the contribution of edge $$\phi_i = \sum_{S \subseteq \{1,\dots,n\}\setminus \{i\}} \frac{|S|!(n-|S|-1)!}{n!}\left[f(S\cup\{i\}) - f(S)\right].$$0 to node $$\phi_i = \sum_{S \subseteq \{1,\dots,n\}\setminus \{i\}} \frac{|S|!(n-|S|-1)!}{n!}\left[f(S\cup\{i\}) - f(S)\right].$$1’s prediction for class $$\phi_i = \sum_{S \subseteq \{1,\dots,n\}\setminus \{i\}} \frac{|S|!(n-|S|-1)!}{n!}\left[f(S\cup\{i\}) - f(S)\right].$$2. A positive Shapley value means that the edge tends to increase the target prediction; a negative Shapley value means that the edge tends to decrease it. This is the key reason signed Shapley values are especially suitable for sparsification: they distinguish supportive from harmful edges rather than merely sensitive from insensitive ones (Akkas et al., 28 Jul 2025).

The method interprets a coalition as a subset of graph components retained in an explanatory subgraph, and the value function as the model output for the target class when evaluated on that coalition-masked graph. Exact Shapley computation is intractable because it requires all $$\phi_i = \sum_{S \subseteq \{1,\dots,n\}\setminus \{i\}} \frac{|S|!(n-|S|-1)!}{n!}\left[f(S\cup\{i\}) - f(S)\right].$$3 coalitions, so the paper follows a GraphSVX/GNNShap-style surrogate approximation:

$$\phi_i = \sum_{S \subseteq \{1,\dots,n\}\setminus \{i\}} \frac{|S|!(n-|S|-1)!}{n!}\left[f(S\cup\{i\}) - f(S)\right].$$4

where $$\phi_i = \sum_{S \subseteq \{1,\dots,n\}\setminus \{i\}} \frac{|S|!(n-|S|-1)!}{n!}\left[f(S\cup\{i\}) - f(S)\right].$$5 is a binary coalition mask, $$\phi_i = \sum_{S \subseteq \{1,\dots,n\}\setminus \{i\}} \frac{|S|!(n-|S|-1)!}{n!}\left[f(S\cup\{i\}) - f(S)\right].$$6 means player $$\phi_i = \sum_{S \subseteq \{1,\dots,n\}\setminus \{i\}} \frac{|S|!(n-|S|-1)!}{n!}\left[f(S\cup\{i\}) - f(S)\right].$$7 is included, $$\phi_i = \sum_{S \subseteq \{1,\dots,n\}\setminus \{i\}} \frac{|S|!(n-|S|-1)!}{n!}\left[f(S\cup\{i\}) - f(S)\right].$$8 are approximate Shapley values, and $$\phi_i = \sum_{S \subseteq \{1,\dots,n\}\setminus \{i\}} \frac{|S|!(n-|S|-1)!}{n!}\left[f(S\cup\{i\}) - f(S)\right].$$9 is a bias term. The approximation strategy is to sample coalitions, evaluate model outputs on coalition-masked inputs, fit the surrogate linear model, and use its coefficients as approximate Shapley values (Akkas et al., 28 Jul 2025).

The theoretical appeal is the familiar game-theoretic one: Shapley values provide a fair allocation of the model output across collaborating players by averaging marginal contributions across many subsets. For pruning, the relevant operational consequence is not only fairness but directionality. The edge ranking uses the signed score itself, not the absolute value. A large negative score means the edge is strongly harmful and should be removed, whereas absolute-value pruning would mistakenly preserve it. The reported ablation, in which GNNShap scores are converted to non-negative values by taking absolute values, significantly degrades sparsification quality. This directly supports the thesis that signed attribution, rather than attribution magnitude alone, is the critical ingredient for explainability-driven graph sparsification (Akkas et al., 28 Jul 2025).

3. Local-to-global sparsification pipeline

The sparsification procedure operates on a single shared graph used at inference, not on node-specific subgraphs. Explanations are local because each node depends only on its $$\phi_i(v)=\sum_{S\subseteq N\setminus\{i\}} \frac{|S|!(|N|-|S|-1)!}{|N|!}\Big(v(S\cup\{i\})-v(S)\Big).$$0-hop computational graph in an $$\phi_i(v)=\sum_{S\subseteq N\setminus\{i\}} \frac{|S|!(|N|-|S|-1)!}{|N|!}\Big(v(S\cup\{i\})-v(S)\Big).$$1-layer GNN, but the final output is a global pruned adjacency obtained by aggregating local explanation scores (Akkas et al., 28 Jul 2025).

The procedure can be stated in four steps. First, for every node $$\phi_i(v)=\sum_{S\subseteq N\setminus\{i\}} \frac{|S|!(|N|-|S|-1)!}{|N|!}\Big(v(S\cup\{i\})-v(S)\Big).$$2, compute local edge importance scores $$\phi_i(v)=\sum_{S\subseteq N\setminus\{i\}} \frac{|S|!(|N|-|S|-1)!}{|N|!}\Big(v(S\cup\{i\})-v(S)\Big).$$3 using a GNN explainer; in the main method this explainer is GNNShap, and explanations are generated for each node and for its predicted class. Second, aggregate local scores into a global edge score. If an edge appears in multiple nodes’ computational graphs, the paper uses mean aggregation:

$$\phi_i(v)=\sum_{S\subseteq N\setminus\{i\}} \frac{|S|!(|N|-|S|-1)!}{|N|!}\Big(v(S\cup\{i\})-v(S)\Big).$$4

where $$\phi_i(v)=\sum_{S\subseteq N\setminus\{i\}} \frac{|S|!(|N|-|S|-1)!}{|N|!}\Big(v(S\cup\{i\})-v(S)\Big).$$5 is an edge, $$\phi_i(v)=\sum_{S\subseteq N\setminus\{i\}} \frac{|S|!(|N|-|S|-1)!}{|N|!}\Big(v(S\cup\{i\})-v(S)\Big).$$6 is the local explanation score of edge $$\phi_i(v)=\sum_{S\subseteq N\setminus\{i\}} \frac{|S|!(|N|-|S|-1)!}{|N|!}\Big(v(S\cup\{i\})-v(S)\Big).$$7 for node $$\phi_i(v)=\sum_{S\subseteq N\setminus\{i\}} \frac{|S|!(|N|-|S|-1)!}{|N|!}\Big(v(S\cup\{i\})-v(S)\Big).$$8, and $$\phi_i(v)=\sum_{S\subseteq N\setminus\{i\}} \frac{|S|!(|N|-|S|-1)!}{|N|!}\Big(v(S\cup\{i\})-v(S)\Big).$$9 is the set of nodes whose computational graphs use edge $n$0. The authors also tried sum aggregation and weighted mean aggregation using model prediction probabilities as weights, but report no significant difference from plain mean aggregation. Third, sort edges by $n$1 in descending order. Fourth, for sparsification threshold $n$2, keep the top $n$3 edges and discard the rest (Akkas et al., 28 Jul 2025).

This ranking rule implies that negative-Shapley edges are removed first. Large positive scores are retained first, near-zero scores are retained later or pruned, and negative scores fall to the bottom of the ranking. The procedure is therefore budget-based: it does not learn a validation threshold, and it does not optimize a connectivity-constrained or symmetry-constrained objective. The paper does not describe an explicit mechanism to preserve graph connectivity, connected components, symmetry constraints for undirected edges, class-balanced structure, or node-specific subgraphs after global pruning (Akkas et al., 28 Jul 2025).

Two clarifications are important. First, the method is inference-only and does not retrain after pruning. Second, the pruning object is an edge in the main method, but not every Shapley-based explainer is edge-based. GraphSVX and FastDnX are node-based explainers, and their node scores must be converted to edge scores by averaging the scores of incident nodes before they can be used in the same global pruning pipeline (Akkas et al., 28 Jul 2025).

4. Relation to GraphSVX and cooperative-network generalizations

GraphSVX is a closely related antecedent because it provides a local, post hoc, model-agnostic Shapley-value explanation method for GNNs. Its players in node classification are not edges directly, but the features of the explained node and the graph nodes other than the explained node that may affect its prediction. Coalitions are encoded by binary masks, mapped to perturbed graph inputs, and explained with a weighted linear regression surrogate. The surrogate objective is

$n$4

with SHAP-style kernel weights, and the weighted least-squares estimator is

$n$5

GraphSVX thus already provides a principled way to rank graph components by their marginal effect on a prediction under coalitional perturbations, but it is not a graph sparsification method per se and does not directly score edges or optimize a global sparsified graph objective (Duval et al., 2021).

This distinction matters for sparsification. GraphSVX is explicitly local, whereas the sparsification method in (Akkas et al., 28 Jul 2025) requires a global edge ranking. GraphSVX also defines structural importance through node isolation rather than direct edge masking. A plausible implication is that GraphSVX naturally supports local node-induced subgraph sparsification more directly than global edge pruning, unless an additional conversion step is introduced from node scores to edge scores. The comparison reported in (Akkas et al., 28 Jul 2025) is consistent with this: GraphSVX often provides good explanations but weaker sparsification, in part because it treats nodes as players and requires conversion to edge scores (Akkas et al., 28 Jul 2025, Duval et al., 2021).

A broader theoretical extension appears in work on cooperative networks and Hodge-Shapley value. That framework does not propose an edge-pruning algorithm, a spectral sparsifier, a cut sparsifier, or a backbone extraction method. Its contribution is instead to reinterpret the classical Shapley value as an expected stochastic path integral on a coalition graph, generalize the valuation to all partial coalition states, extend from coalition marginal differentials to arbitrary antisymmetric edge flows on general graphs, and compute the resulting allocations via a graph Poisson equation and Hodge-theoretic operators. In particular, for reversible Markov chains the statewise value allocation satisfies

$n$6

This is a theoretical foundation for edge-centric and path-centric attribution, but it is not itself a sparsification method (Lim, 2022).

Taken together, these works locate Shapley value-based graph sparsification at the intersection of three ideas: local GNN explanation by coalitional perturbation, aggregation of local attributions into a global pruning score, and a more general view of signed contribution on graph elements. This suggests that current sparsification methods use only part of the available Shapley machinery, namely the edge-ranking function derived from local prediction explanations.

5. Empirical evidence and comparative performance

The main empirical study evaluates semi-supervised node classification on four datasets—Cora, CiteSeer, PubMed, and Coauthor-CS—with 2-layer GCN and 2-layer GAT models. The settings include hidden size 16 for Cora, CiteSeer, and PubMed; hidden size 64 for Coauthor-CS; 8 attention heads for GAT; dropout 0.5; ReLU; 200 epochs; and learning rate 0.01. Experiments are repeated 5 times and averaged. The main evaluation metric is test accuracy after sparsifying the graph; the efficiency metric is MACs for message passing during inference (Akkas et al., 28 Jul 2025).

Dataset	Nodes / Edges	Features / Classes
Cora	2708 / 10556	1433 / 7
CiteSeer	3327 / 9104	3703 / 6
PubMed	19717 / 88648	500 / 3
Coauthor-CS	18333 / 163788	6805 / 15

Against explanation baselines—Saliency, Guided Backpropagation, Integrated Gradients, GNNExplainer, PGExplainer, FastDnX, GraphSVX, and GNNShap—the reported finding is that GNNShap-based sparsification consistently gives the best or among the best accuracy at high sparsity. Explicit examples include the following: on Cora with both GCN and GAT, GNNShap prunes 80% of edges with less than 2% accuracy drop; on PubMed with GCN, it also prunes 80% of edges with less than 2% drop; on PubMed with GAT, it matches original accuracy even after 80% pruning; and on Coauthor-CS with GAT, it matches original accuracy even after 55% pruning. The paper further states that Shapley-value sparsification outperforms existing explainers on three of four datasets across two models, with CiteSeer/GCN the notable case where methods look similar (Akkas et al., 28 Jul 2025).

The reduction in graph complexity is substantial. Examples reported include Cora + GCN, from 305,000 to 110,000 MACs at 80% pruning, a 64% reduction; Cora + GAT, from 2,865,000 to 1,040,000 MACs at 80% pruning, also a 64% reduction; PubMed + GCN, from 2,058,000 to 711,000 MACs at 80% pruning; PubMed + GAT, from 13,003 to 4,493 in the paper’s text, with the stated reduction about 65%; and Coauthor-CS + GAT, from 100,530 to 50,804 at 55% sparsity, a 49% reduction. These numbers support the central claim that message-passing complexity can be cut sharply while preserving predictive performance (Akkas et al., 28 Jul 2025).

The comparison with graph lottery ticket baselines—UGS, WD-GLT, and FastGLT—places the approach in a different part of the design space. GNNShap achieves higher sparsity with less accuracy loss, while explanation-based pruning offers two practical advantages emphasized in the paper: no retraining for each pruning percentage, and no dependence on labels for all edges because explanations rely on predicted classes at inference. FastGLT is reported as competitive, especially on PubMed, but it loses more accuracy at high sparsity (Akkas et al., 28 Jul 2025).

Scalability observations are also part of the empirical picture. GNNShap is described as much faster than GraphSVX because it explains edges directly and uses GPU-based coalition sampling and prediction. GraphSVX could not finish on PubMed and Coauthor-CS within a 10-hour time limit. This suggests that direct edge-level players and GPU-accelerated coalition evaluation are not incidental implementation details but materially affect the practicality of Shapley-based sparsification on larger graphs (Akkas et al., 28 Jul 2025).

6. Limitations, misconceptions, and open questions

The most immediate limitation is computational overhead. Even with approximation, Shapley-based explanations require many coalition evaluations and remain much more expensive than simple gradient-based scores. The method is therefore practical as a post hoc preprocessing step for repeated inference, but may be costly when the graph is huge, when explanations are needed for every node, or when the model changes frequently (Akkas et al., 28 Jul 2025).

A second limitation is dependence on explainer quality and on model quality. The explanations are computed for the predicted class, not necessarily the true class. If the underlying GNN is inaccurate, the explanations may also be misleading because they explain incorrect predictions. This is especially relevant for any global sparsifier constructed by aggregating local explanations: local explanation quality does not guarantee global utility (Akkas et al., 28 Jul 2025).

Several common misconceptions are clarified by the current literature. One misconception is that any high-fidelity explainer is automatically suitable for pruning. The evidence in (Akkas et al., 28 Jul 2025) argues otherwise: pruning requires directional contribution, not merely sensitivity magnitude, and the non-negative GNNShap ablation shows that discarding sign information significantly hurts sparsification quality. Another misconception is that all Shapley explainers naturally provide edge scores. GraphSVX does not; it attributes importance to nodes and features, so edge sparsification requires an additional conversion step that is not part of the original method (Akkas et al., 28 Jul 2025, Duval et al., 2021).

There are also unresolved design questions. The paper does not provide the exact GNNShap coalition sampling settings, the number of coalition samples $n$7, the exact value function implementation under graph masking, or the exact regression loss for the surrogate model within that paper itself. It also does not provide asymptotic end-to-end sparsification complexity beyond the general statement that exact Shapley is $n$8 and approximation uses $n$9 coalitions. Those details would need to be recovered from the cited GNNShap source rather than from the sparsification paper alone (Akkas et al., 28 Jul 2025).

Finally, the method sparsifies only the graph. It does not sparsify model weights, unlike some graph lottery ticket approaches. Theoretical guarantees are likewise limited: the cited works motivate Shapley values through fairness axioms, local coalition reasoning, and, in the more general cooperative-network setting, Hodge/Poisson structure, but they do not prove that removing all low-value components preserves prediction under simultaneous pruning. This suggests that Shapley value-based graph sparsification should be understood as a principled and empirically effective ranking-and-budgeting framework, rather than as a method with formal preservation guarantees for graph connectivity, spectra, or robustness (Akkas et al., 28 Jul 2025, Lim, 2022).