GNN-Induced Graph Games

• GNN-induced graph games map GNN predictive behavior onto cooperative games, treating nodes or edges as players to quantify contributions.
• They leverage game-theoretic methods, including Shapley and structure-aware values, to overcome limitations in attributing importance in graph data.
• Empirical studies demonstrate enhanced explanation fidelity and sparsity across diverse domains like text, molecules, and board games.

A GNN-induced graph game is a formal framework in which the predictions or structure learned by a Graph Neural Network (GNN) are mapped onto a cooperative game defined over the nodes or edges of a graph. In this paradigm, either nodes or edges are interpreted as "players," and coalitions correspond to subgraphs whose contributions to prediction are quantified through a characteristic function. This enables the use of game-theoretic solution concepts—classically the Shapley value and more recently structure-aware or interaction-based indices—to analyse, interpret, and sometimes optimize GNN behavior. GNN-induced graph games underpin several modern explanation, attribution, and policy-learning algorithms for graph data, particularly in the domain of explainable AI and scalable combinatorial optimization.

1. Formal Structure of GNN-Induced Graph Games

GNN-induced graph games instantiate cooperative games using structural elements of a graph as players and employ a GNN’s predictive behaviour as the valuation mechanism for coalitions. The foundational instance treats a graph $G=(V,E)$ and a trained GNN $f$ mapping $G$ (or its subgraphs) to prediction scores, often class probabilities. Letting either nodes $u_i\in V$ or edges $e \in E$ serve as players, the characteristic function $v(S)$ is defined in terms of the GNN prediction on the subgraph induced by $S$ (nodes or edges). For instance, in the node-player case, as in GStarX, $v(S) = [f([S])]_{c^*} - f^0_{c^*}$, where $c^*$ is the GNN’s predicted class for the full graph and $f^0_{c^*}$ is a baseline (Zhang et al., 2022). In the edge-player model, as in GraphGI, $v(S) = f(A[S], X[S], v)$ quantifies the GNN’s probability for the target class when restricting to edges $S$ within an $L$-hop subgraph (Xian et al., 24 Sep 2024).

This structure allows for the algebraic and algorithmic transfer of game-theoretic concepts—marginal contributions, surplus, coalition synergies—to the analysis of learned GNN functions.

2. Shapley Value and Structural Limitations

The canonical method for quantifying feature importance in cooperative games is the Shapley value, defined for each player $i$ as

$$\varphi_i = \sum_{S\subseteq V\setminus \{i\}} \frac{|S|! (|V|-|S|-1)!}{|V|!} [v(S\cup\{i\})-v(S)]$$

However, when directly applied to graph data, the Shapley aggregation is structure-agnostic: it weights a node’s (or edge’s) marginal contributions solely by the size of coalitions, without regard to connectivity or the actual graph topology. This invariance means that disconnected coalitions—which may never engage in message passing within the GNN—are treated identically to connected ones. In practical terms, this can lead to explanations in which, for example, two nodes or words that never interact in the computation graph are given the same credit as those that do, and nodes whose influence propagates only through their local neighbors are under- or over-scored depending on the sampling of coalitions (Zhang et al., 2022); (Xian et al., 24 Sep 2024).

A plausible implication is that the Shapley value, though equitable on tabular or Euclidean data, may not faithfully attribute importance in cases where model behaviour is fundamentally determined by local graph structure.

3. Structure-Aware and Interaction-Induced Values

To address the limitations of the Shapley value in graph contexts, recent algorithms introduce structure-aware game-theoretic imputations. In GStarX, node importance employs the Hamiache–Navarro (HN) value, which explicitly respects the communication topology. For a coalition $S$ and neighboring node $j\in V\setminus S$, the surplus $p(j,S) = v(S\cup\{j\}) - v(S) - v(\{j\})$ quantifies interaction beyond additivity. An iterative linear system,

$v^{(t+1)} = M \cdot v^{(t)}$

reallocates surplus back into coalition values, converging to a unique limit value $\tilde v$ with the node score $\phi_i = \tilde v(\{i\})$. This computation is reminiscent of message-passing updates:

$$v^{(t+1)}(\{i\}) = v^{(t)}(\{i\}) + \tau \sum_{j\in N(i)} v^{(t)}(\{j\})$$

so the resulting importance blends feature contribution with local topology in direct analogy to GNN processing (Zhang et al., 2022).

Edge-based games, as in GISExplainer/GraphGI, define interaction measures $I(A)$ for all edge coalitions $A$ by

$$I(A) = \varphi([A] | N[A]) - \sum_{e\in A} \varphi(e | N\setminus\{e\})$$

quantifying how much extra value emerges from coalition $A$ over independent edges, and further analyse positive/negative components and net strength $T(A)$ to guide the explanation process (Xian et al., 24 Sep 2024).

4. Algorithms and Solution Procedures

GNN-induced graph games support a variety of algorithmic procedures. The GStarX algorithm for node-based explanations (Editor’s term: "structure-aware node attribution") includes:

1. For graphs of order $\le m$, enumerate all coalitions, compute $v(S)$, form the surplus matrix, and iterate to convergence to $\tilde v$.
2. For larger graphs, subsample coalitions or subgraphs, combine Monte Carlo estimates.
3. Rank nodes by their HN value; return the induced subgraph on the top $k$ nodes for explanation (Zhang et al., 2022).

GraphGI, for edge-based games, applies a sequential decision process:

1. Initialize selected edge set $C_0 = \varnothing$.
2. At each step $k$, consider candidate edges incident to $C_{k-1}$ or the target node, and choose the one maximizing the gain in interaction strength $\Delta(e) = T(C_{k-1}\cup\{e\}) - T(C_{k-1})$.
3. Continue until marginal gain is nonpositive or a step cap is reached; return the resulting subgraph.
4. All Shapley and interaction terms are estimated by Monte Carlo sampling for tractability (Xian et al., 24 Sep 2024).

In aggregated policy learning or combinatorial optimization, such as ScalableAlphaZero, a GNN induces a graph game wherein board positions are encoded as graphs, and the GNN’s message passing enables scale-invariant policy and value estimation. Self-play experiences, potentially combined with subgraph sampling, yield model update targets. The algorithm’s curriculum moves from small to larger graphs without architecture change, leveraging the size-agnostic property of GNNs (Ben-Assayag et al., 2021).

5. Empirical Evaluation and Benchmarks

Empirical validation across text, molecular, and synthetic benchmarks demonstrates the practical value of structure-aware graph games for GNN explanation. In GStarX, quantitative fidelity scores (harmonic F1-style) on six datasets show GStarX achieving an overall average of 0.5732 compared to a best-baseline average of 0.5569. On GraphSST2, GStarX correctly isolates sentiment-bearing words and their dependency edges in explanatory subgraphs, outperforming Shapley-based methods that may select disconnected components. On MUTAG, GStarX highlights chemically functional groups (e.g., oxygen atoms in NO$_2$ moieties) with higher fidelity than baselines that dilute importance across the graph (Zhang et al., 2022).

GraphGI achieves highest fidelity (84–99%) and greatest sparsity (70–90%) on synthetic BA-Shapes, BA-Community, and Tree motifs compared to baselines including GNNExplainer, PGExplainer, and SubgraphX. On real-world citation networks Cora, CiteSeer, and PubMed, GraphGI yields 15–20% fidelity at sparsity above 80%, compared to 5–13% for baselines, with efficient scalability up to $\sim$20,000 edges (Xian et al., 24 Sep 2024).

In board games, ScalableAlphaZero, which employs a GNN's induced graph game structure to achieve scale-invariant play, matches or outperforms standard AlphaZero trained on large boards, despite only being trained on smaller instances. Training time is drastically reduced, e.g., Othello 16×16 reaches parity with greedy baselines in about 3 hours for the GNN versus 4 days for a CNN. The same model generalizes from $9\times9$ boards to $20\times20$ with no architecture modification and minimal retraining (Ben-Assayag et al., 2021).

6. Theoretical Foundations and Properties

GNN-induced graph games inherit the axioms (linearity, symmetry, dummy, efficiency) of classical cooperative game theory. In both node- and edge-based games, Shapley and interaction-based solution concepts preserve strict sum scores (the sum of individual or coalition attributions equals the total gain in GNN prediction due to all included components). For Monte Carlo-based algorithms, error bounds are guaranteed by concentration inequalities (e.g., Hoeffding’s), yielding accurate approximations. Properties such as monotonicity of interaction gain at each selection step, and fidelity-sparsity trade-offs are theoretically established under local Lipschitz conditions on the GNN aggregator (Xian et al., 24 Sep 2024). The structure-aware surplus allocation in HN values guarantees that only connected components contribute to node or edge scores, ensuring that the explanation process adheres to the topology and message-passing semantics of the underlying GNN (Zhang et al., 2022).

A plausible implication is the capacity for GNN-induced graph games to both faithfully explain GNN predictions and to provide a principled foundation for scale-agnostic reinforcement learning, strategy derivation, or attribution, even in the presence of combinatorially large input spaces.

7. Broader Impacts and Future Directions

By formalizing the interaction between GNN architectures and cooperative game theory, GNN-induced graph games underpin explainability, robustness, and transfer learning for models tasked with non-Euclidean data. They enable structure-aware, theoretically principled attributions that respect underlying graph topology—addressing a core deficiency of prior feature-attribution methods. Ongoing and prospective research directions include:

• Extension to edge- and motif-level attributions and fully joint subgraph-level games,
• Enhanced scalability via advanced sampling or approximation,
• Application to new domains (e.g., chemistry, natural language, network science),
• Unified theoretical frameworks for the convergence and fairness properties of structure-induced values.

Recent evidence suggests that such frameworks may unlock both improved interpretability and efficiency in combinatorial AI beyond mere model explanation, supporting control, search, or optimization on large structured domains (Zhang et al., 2022); (Xian et al., 24 Sep 2024); (Ben-Assayag et al., 2021).