Node-Level Data Valuation on Graphs
Authors: Simone Antonelli
Aleksandar Bojchevski
Venue: Accepted by TMLR ()

Abstract: How much is a node worth? We answer this question using an emerging set of data valuation techniques, where the value of a data point is measured via its marginal contribution when added to the (training) dataset. Data valuation has been primarily studied in the i.i.d. setting, giving rise to methods like influence functions, leave-one-out estimation, data Shapley, and data Banzhaf. We conduct a comprehensive study of data valuation approaches applied to graph-structured models such as graph neural networks in a semi-supervised transductive setting. Since all nodes (labeled and unlabeled) influence both training and inference we construct various scenarios to understand the diverse mechanisms by which nodes can impact learning. We show that the resulting node values can be used to identify (positively and negatively) influential nodes, quantify model brittleness, detect poisoned data, and accurately predict counterfactuals.

Published in Transactions on Machine Learning Research (08/2025)
Node-Level Data Valuation on Graphs
Simone Antonelli
stroons.onnooneel@@lcisps...ddd
CISPA Helmholts Center for Information Security
Aleksandar Bojchevski
a. ..ojcchrrski@@ui--kcch....
University of Cologne
Reviewed on OpenReview: https: opwnrevvie.... net/ forum? id= LNgAApIgDS
Abstract
How much is node worth? We answer this question using an emerging set of data valuation techniques, where the value of data point is measured via its marginal contribution when added to the (training) dataset. Data valuation has been primarily studied in the i.i.d. set- ting, giving rise to methods like influence functions, leave-one-out estimation, data Shapley, and data Banzhaf. We conduct a comprehensive study of data valuation approaches applied to graph-structured models such as graph neural nelworks in semi-supervised transduc- tive setting. Since all nodes (labeled and unlabeled) influence both training and inference construct various scenarios to understand the diverse mechanisms by which nodes can impact learning and final predictions. We show that the resulting node values can be used to identify (positively and negatively) influential nodes, quantify model brittleness, detect poisoned data, and accurately predict counterfactuals
1 Introduction
When we learn from data, a natural question is how each data point influences learning. We can study how given training point affects the learned weights, accuracy, prediction for given test point. Beyond offering insights into our models, this question is also practically important, especially for modern machine learning methods using increasingly larger datasets (Zhou et al. (017). For example, if we gather data from unreliable sources (e.g., the internet) we may want to filter out low-quality instances. When obtaining data from data markels, different providers (e.g., individuals, hospitals) should be equitably compensaled for the data they provide. Data values have been used to quantify train-test leakage, find semantically similar examples Tlaas
al. 2022 detect mislabeled instances (Wang & Jia 2023). and dataset selection (Engstrom et al. 2024
A classical answer to the data valuation question is influence functions (Cook & Weisberg 1982 Koh &
Liang 2017). They approximate the effect of removing data point on the model performance. From linear regression to deep learning, they have been thoroughly studied in various settings, including their limitations [Bauu et al. 2021 Bae et al. 2022). Broadly, data valuation encompasses techniques that try to relate some output of model with the data is trained on. While there is growing research that explores different notions of value, the vast majority is focused on supervised learning for i.i.d. data.
We conduct the first comprehensive data valuation study for graph- based models in the semi-supervised transductive setting. Only two recent works have explored data valuation for graphs. Chen et al. (202)) derive a closed-form estimate for the leave-one-out influence of nodes and edges using a simple graph convolution (SGC) network (Wu et al. 2019 as a surrogate model. Chi et al. (2025a) propose the precedence-const rained winter value where nodes are grouped coalitions based on the graph structure. Our focus is not on proposing new notions, but on rigorously evaluating existing ones. Our results show that both data Banzhaf (Wang &
Jia
2023 and datamodels (Ilyas el. al. 2022 significantly outperform other valuation methods, but neither
TTe code provided at http://hhhw...ceeeetcteeeee///ap

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A v(S (4))
u(S,)
1) (())
S()))
A(S.. = a(Sm. U (()) w(S...
0.2
(1) = -
wckk SCV SS=k-1
E
ASS
k=1
-0.
Figure 1: Data valuation assigns importance to a node based on its marginal contribution to the utility u across various subsets S. When removing the node (and its edges): if model performance decreases, it is con- sidered informative (green); if performance remains unchanged, the node is irrelevant (grey); if performance improves, the node is misleading (red). Weighting schemes w define different data valuation methods.
has been studied for graphs so far. Given the importance and ubiquity of models such as graph neural networks (Ju et al. 2024), we hope our work lays solid foundation for studying data valuation on graphs.
It is clear that not all data are created cqually, e.g. some instances may be noisy or mislabeled. However, even a "high-quality" instance may not be valuable if there are already similar instances in the datasel, indicating that to properly evaluate the worth of an instance we need to carefully consider the appropriate context. Lack of context is why leave-onc-out estimation often fails practice its focus is on the effect of an instance in isolation. In contrast, recent game-theoretic notions such as data Shapley (Ghorbani & Zou, 2019 Jia et al. (019bb and other semi-values (Dubey al. 1981) compute the average marginal contribution of an instance considering all potential subsets of the data. This indeed comes at a significant computational cost, but approximation strategies allow us to compute useful estimates in practice.
The right context is even more important in the graph setting. First, the prediction of a node depends on its neighborhood nodes are not i.i.d. This suggests that looking at nodes in isolation is likely suboptimal and we should consider different subgraphs. The main idea illustrated in Fig. I To estimate the influence of node on some utility function u(-) (e.g. the accuracy), we consider the difference in utility A(S, = u =(S((i))-u(S) with and without node across many subsets S C V of different sizes. Marginal contributions A(S, are merged together using weighted average to obtain the final node value ((i). Different weights recover many of the established data value notions, including data Shapley, beta Shapley, data Banzhaf, and leave-one-out.
Socond, all nodes, cither labeled and unlabcled, inffuence both training and inference. We account for this by constructing different variants (see $ 3) of each data value notion that aim at disentangling these effects. Unsurprisingly, even though the number of training nodes a small percentage of all nodes in the sparsely- labeled scenario (Shchur et al. 2018), the top most influential nodes are mostly training nodes. At the same time, we can find unlabeled nodes which have greater influence than many training nodes. This reflects the complex interplay wetw training and test nodes in graphs.
In addition to influence-based and game-theoretic notions, predictive approaches, like datamodels (llyas et al, 2022) and MLPbV (Wu et al.) 2024), construct surrogate (proxy) models that can predict the utility function for any subset without the need to train the model from scratch. They allow us answer counterfactual questions, i.c. predicting the performance on arbitrary unseen subsets of data. Proxy models are trained on subsets of all possible ((S., u(S))} pairs, where S; C V. As we discuss in 2.1 we can trivially turn any set of data values into predictive (linear) surrogate. Moreover, the resulting data values from predictive and game-theoretic approaches are equivalent for certain configurations (sce 2.2 Thus, this distinction between these types of notions may not be very useful.
Overall, our results show that approaches accounting for subgraphs instead of single-node contribution yield more accurate data values. We demonstrate their usefulness for downstrcam applications: i) finding highly inffuential nodes pruning nodes with high positive (negative) influence causes a drop (rise) in perfor- mance; ii) spotling britlle predictions which depend on small set of support nodes whose removal causes
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misclassification; iii) detecting poisoned (mislabeled) data; iv) estimating counterfactuals such as predicting performance on an arbitrary subset; and v) visualizations that provide further insight about the data.
We thoroughly study the problem of data valuation for graphs, including variants that attribute importance to both labeled and unlabeled nodes. We analyse game-theoretin and predictive notions and show that data- models and data Banzhaf, neither of which was previously considered in the context of graphs, outperform other valuation techniques, including the two proposed graph-specific approaches. Our study was computa- tionally expensive (over 2500 compute hours in total) and storage-intensiv (more than 10TB of raw data). Nonetheless, we show that one obtain accurate estimates even with modest computational resources.
2 Data valuation
Broadly, data valuation methods try to relate the presence or absence an instance in the dataset with the resulting change in some user-defined utility function. Traditionally, this is some measure of model performance such as accuracy, where a high positive (negative) value indicates that including the instance in the data improves (deteriorates) performance. For both game-theoretic and predictive approaches the main idea is to consider different subsets of the training set D and assess the value of missing training samples according to the change in the utility. We denote with o(i) E R the value assigned to instance i.
Utility. Let u : 2| - R be a utility function which maps any subsets S C D to a score indicating the usefulness of the subset. For classification tasks, we commonly have u(S) = ace(A(S)) where A is a learning algorithm that returns a model fs trained on S, and acc computes the accuracy on a held-out set. Sometimes we are not interested in the overall performance but rather how the training data influences specific test point. For example, the utility can be the prediction margin for the test point Tust That is u(S) u(S, Ptest) = margin(Js, Btest) fs(ases) maxy fss fs( where fs(-), returns the logit-score probability for label y, and y" is the ground-truth label. Any other function of (the outputs of) fs is valid.
2.1 Categorization of data valuation approaches
Game-theoretic notions. These methods consider subsets (coalitions) of the training data, treating each data point as a player in a cooperative game. If u(D) represents the outcome of the game, the goal is to fairly distribute to each data point according to its contribution. This concept is at the core of Shapley values (Shapley 1953 Ghorbani & Zou, 2019). well-known framework for assessing data value via the average marginal contribution of adding data point to all possible subsets, weighted by the number of permutations in which that data point appears Let [D]. The Shapley value of data point i is given by
n
ØSHAP (i)
Σ SCDV -S|=k1
[u(S U (i)) u(S)]
(1)
To compute it exactly, we nced train 2| models, i.c. one for each possible subset. Since this is prohibitively expensive there are various approximation techniques such as Monte Carlo sampling. Shapley values are popular since they uniquely satisfy four axioms that were argued to be necessary to ensure fair valuation. These are symmetry, linearity, null player, and efficiency (Shapley 1953).
Kwon & Zou 2022) questions the necessity of the efficiency axiom, which requires all the values to sum up to the utility on the original dataset, i.e. EiED x(1) n(D). Removing this axiom we get the so called semivalues (Dubey et al. 1981] which satisfy the other three axioms. As with any axiomatic approach, these axioms can be debated. For the purposes of this paper, we consider them as given. It turns out that all semivalues can be written in a canonical form
1 @semivalue (i, w) 11
w(k)
Σ [t) - [s]] SCD
"Shapley values were first used machine learning for feature attribution determining which features (e.g. pixels) highly influence the prediction. That is, they served as a post-hoc explainability method. In contrast, Ghorbani & Zou (2019) introduce data Shapley to assoss the influence of training points, which is the setting that we consider in this paper (sce also $
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where w : [n]] + R is weight function such that Σx= (x-1) (i-1) wkkk = n. We can recover Shapley (SHAP) with
w(k) = (x=+)= -1 Leave-one-out (LDO) measures the difference in utility when removing a single data point from the dataset and can be obtained by choosing we(k) m [k: n|. Data Banzhaf (BANZ) assigns uniform
weights w(k) = T Weighted Banzhaf (a-BANZ) has w(k) = na (1 (1-a)" a)" weights (Li & Yu 2023).
Predictive notions. Wang et al. (021)) propose to learn surrogate model on utilities. [lyas et al. (2022) introduce the datamodels framework, where mg {0, 1) + R is trained such that can directly predict the utility for any given subset, i.e. m²(S) Rs u(S). The weights 0 of the surrogate are learned from {(S,, u } where S, is encoded as a binary vector {0, 1) indicating the presence or absence of an instance. They show that even a simple linear model can approximate the mapping from 8 u(S) well. The data values are given by the weights I R (PI of the lincar model, i.c. (bx(t;;) 0) 6,. Wu et al. (2024) propose to instcad use a multi-layer perception as a surrogate which can use additional inpuls.
Influence functions. The number of possible subsets increases exponentially with the size of the dataset. Morcover, for each subset, we need to train the entire model from scratch. If the dataset is large even computing the leave-one-out error which requires "only" n = [D] evaluations can be too expensive. In contrast, influence functions (Cook & Weisberg. 1982) aim to approximately compute the change in the model output (or model weights) when removing a traiming sample. They rely on first and second-order information (gradient, Hessian) to obtain estimates without model retraining. Nevertheless, influence functions perform poorly in non-convex settings as in deep neural networks (Basu et al., 2021 Bae et al. 2022). At best, they perform as well as leave-one-out errors which are suboptimal since they consider instances in isolation. Chen et al. 2023) derive different data values using influence functions for graph neural networks.
From game-theoretic to predictive. Any game-theoretic approach can be made predictive by assuming that the values are the weights of an implicit lincar model (without a bias term). This implics that the predicted value of any unseen subset equals the sum of the data values of the nodes in the subset, i.e. u(S) Eies Φ(i) for any d(i). This is a reasonable assumption given the linearity axiom of semivalues, and it's explicitly made by datamodel values [yyas et al. 2022).
2.2 Properties of data valuation approaches
Connections between different values. To compute both DM and B-BANZ we sample subsets where each node is included with probability a. If a = 0.5 and we use no regularization when fitting me the datamodel values are equivalent to Banzhaf values since the Banzhaf values are the best linear approximation to u in terms of least square loss (Wang& & Jia 2023 Hammer & Holzman 1992). Lin et al. (2022) prove that if we sample a ~ (n, 1) rather than having it fixed, then the optimal weights of a certain regularized linear model converge relatively quickly to the Shapley values as we increase the number of samples.
Limitations of data values. The utility function can be viewed as a vector in R²" which is transformed into a vector of data values & E RR. This mapping is not injective, so there are different utility functions that yield identical values. A natural question is which utility functions are well approximated by data values. Wang et al. (2024) show that a sufficient condition is for the utility to be a monotonically transformed modular lunction. In general, data values may be no better than a random baseline for dataset selection. They also generalize the result from Saunshi et al. (2022) which states that the residual error of the unregularized linear approximation equals to the sum of all u's Fourier coefficients of order 2 or higher. Nonetheless, data values work well in practice. This is remarkable since they approximate u(S) which involves training of a model on S from scratch (often a complicated neural network) and then computing some function of the model out putputs.
Robustness and efficiency. In our context the utility function is stochastic due to randomness in the training process, making the influence of a data point a random variable with an expectation and variance [guyye et al. 2024). Thus, using single sample of u(S) may be unreliable. Wang & Jial (2023) define the safety margin- the largest noise semivalue can tolerate without altering the ranking of any distinguishable pair of data points. They prove that data Banzhaf achieves the largest safety margin among all values. This follows from the uniform weights since (i)xz(i) = Un. (ttSes/////SUUSuu (i)) - -u(S). The standard Monte Carlo estimator samples S from Unif(2 but must be repeated times, once for each i. Wang & Jia (2023) propose an alternative estimator where given samples M = (S,,...,Sm)) i.i.d. from Uni it computes
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A11
Train
Subset
Train
/"
arg arg max f (S,
T V
In Inference
In fnfennce
f (V,8*) Learning
f (S,00) Overall
Figure 2: Venn diagram of node subsets for their Figure 3: Pipeline for the learning vs. overall sig- induced subgraphs. Downward hatches from left right (green) indicate the final subset T.
nals: train on subsct S, then mcasure utility either on V S, respectively. Here, T = (all setting).
[Sell EsEs,, u(S) as the difference of the average utility of the sets Se: C that contain i, minus the sets Sgi S M that do not contain it. Now, maximal sample reuse (MSR) is achieved since all evaluations of u(-) are used in the estimation of all @Banz(t) values. The existence of efficient MSR estimator is unique to the Banzhaf value among all semivalues. Liu et al. (2024) also show that the latter can be approximated with linear regression formulation. Nonetheless, (2023) and use a-BANZ where instances in the subsets are sampled with a fixed probability (rather than uniform). In our experiments, a Q-BANZ outperforms BANZ and shows strong results across all scttings.
BaNZ |Se: EsEse: u(S) 1
apply the same idea of Li & Yu
3 Data valuation on graphs
Semi-supervised setting. Consider a graph g (V,E) where V the set of nodes and & is the set of odges. Let X E R/Vidd be the matrix of d-dimensional node features, A 10, VIx(V| be the adjacency matrix and y E R(V be the vector of labels. Let Ve and Vu be the subsets labeled and unlabeled nodes respectively, and let Ve be further split into training nodes V, and validation nodes V.- Usually, V is small subset of nodes while the majority is in V. (Shchur et al.) 2018). In transductive semi-supervised learning setting, the model is exposed to both V2 and Vu during the training, such that it can propagate label information from the few labeled nodes to the many unlabeled nodes leveraging the underlying graph connectivity and node features.
Graph Neural Networks (GNNs). In each layer k of a GNN, the hidden representation h²²²) of a node v is the result of aggregating information from its neighborhood (v)) and its own representation from the pre- vious layer. Many popular GNNs can be succinetly written in matrix notation as ( = (k-1 where is the non-linearity,
(4) are trainable parameters, S is the spatial graph convolution operator and (0) = X. For example, in GCN (Kipf & Welling 2017) we use the degree normalized adjacency matrix S - D AD- 1, where = A+ In and D E, Aij is the degree matrix. SGC (Wu et al. 2019 uses the same architecture as GCN without the non-linearity resulting in a model that is linear w.r.t. the weights.
Training nodes vs. All nodes. In the transductive setting, all nodes influence learning. The training nodes do so directly since the loss is computed using their ground-truth labels supervision signal. The remaining nodes do indirectly since they influence the hidden representation of the training nodes via the graph structure. This mcans that in addition to investigating the value of the training nodes V, as in the standard i..d. setting, we can also compute the value of all nodes V. We refer to these two settings as train and all respectively. Let D be the set of considered nodes, namely either D = V, or = V for the train or all setting respectively. Recall that to compute data values we consider different subsets S C D. We first remove the nodes that are in D but not in S, and then we train GNN on the graph induced by the remaining nodes T V (D) (). In the train setting, T V (V, \. S) and in the all setting T V (V () S.
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Fig. 2 shows Venn diagram representations of the considered subgraphs in the two settings. Since all nodes influence learning we focus on the all setting in the main paper. However, train results are in B C.4
Learning signal vs. Overall signal. After training, we need to evaluate our utility function. We consider the accuracy to measure the influence on the final performance, and the prediction margin to measure influence w.r.t. any individual node (see $ 2). Here, we assume access to ground-truth test labels, since our goal is to understand how nodes influence learning for different models. This is standard in the data valuation literature. In practical scenarios, one can validation labels instead. Importantly, after training the graph induced by T we can compute the utility in two different ways which provide different and complementary insights. In particular, we can compute the utility on the same induced subgraph T, which captures the overall influence of a node since removed nodes are not present during inference. Alternatively we can compute the utility on the whole graph that is we bring back the removed nodes V which only captures their influence during learning. These learning data values mcasure the training signal provided by a node, while the overall data values measure the influence during training and inference. This is in contrast to the i.idd setting where we can only measure the train signal provided by an instance. Fig. 3 depicts the pipelines for computing the values in the two described settings for the all setting.
4 Experimental results and analysis
Setting. As representative game-theoretic notions, we consider three well-known semivalues: leave-one-out (LO0), data Shapley (SHAP), and weighted Banzhaf values (a-BANZ), where a = 0.5 recovers the original Banzhaf value. As a representative predictive notion, we consider datamodels (DM). For approaches based on
MSR we describe the procedure to sample subsets in S B.2 Finally, we consider the precedence-consstraing winter value (PCW) as the only data value custom-designed for graphs since the method by Chen et al. (2023) approximates LOO, which we already cover. We introduce a sampling improvement. to PCW which we call PCWP
(see $ B.2.2 for details). We include two more baselines, namely DEG where a node value corresponds to its degree, and RND which randomly assigns node values.
We evaluate methods on the largest connected component (LCC) of different cilation graphs (Citeseer, CoraML, PubMed, and CoPhysics) and co-purchase graphs (Photo and Computers). We also analyze het- erophilic datasets in $ C.I We use SGC, GCN, and GAT models. Unless specified otherwise, we present results on SGC. See S B for datasets, models, and runtime details. We assume that node's value depends on both its features and its edges, since we completely remove any node v & S. Disentangling features from structure is an interesting direction for future research.
Key findings. We summarize our results in five key findings, each supported by one or more experiments.
I. Nodes exert positive influence within their class and negative influence across classes.
We study how every node influences every other node using prediction margins utilitics. We obtain matrix of values o E R [P)), where a column j of & shows how node j influences other nodes. A positive value [i. means that adding node j to the graph increases the margin of node i, while a negative value means that the margin decreases. On Fig. 4 we show a heatmap of the entire matrix PaN (overall signal, all setting) using GCN to compute utilities on CoraML. We also annotate training nodes and highly influential nodes (outliers) with bigger markers. Since rows and columns are sorted by class, cluster size and degree, the block-diagonal pattern with positive diagonal and negative off-diagonal entries, Icads to the conclusion stated in Finding The results are consistent for different methods, and we observe no qualitative difference between learning and overall values ( A.1)
In the average per-class influence plot Fig. top right), we summarize PDK by averaging node influence for every class pair. Influence is positive within a class and negative across classes, and adding same-class nodes widens this margin. Cluster size strongly correlates with within-class influence, i.e. adding a node to a large community matters less because many others play the same role, and this holds across all datascts (see $ A.I.. Influence is asymmetric, e.g. Theory nodes affect the Rule class more than the reverse, and the smal llstl communities show the strongest negative cross-class influence. These patterns warrant. further study.
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V4
Positive outliers
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Figure (Left) Matrix of Pan on CoraML from GCN utilities, where each cell shows the positive (green) or negative (red) effect of adding node on node i's margin. (Top right) Per-class averages from 4ow shows positive intra-class and negative inter-class effects. (Bottom right) Median positive/negative negative influences (colored by class) from aan reveal that few (training) nodes dominate (vertical stripes in Pun).
II. Influence is concentrated and synchronized within each class.
Concentrated. A small subset of nodes (mostly training nodes) is highly influential. This can be casily seen from the Pιy heatmap on Fig. 4 where few columns have large positive and negative values across many rows, visible as vertical strips. In the class concentration subplots on Fig. (right), show the median (per column) influence of node and split them in positive in the top plot and negative in the bottom plot. We color the nodes by class and mark them as training (crosses) and non-training (circles). The results confirm that most of the influence (cither positive or negative) concentrates among training nodes and few of those have high median value highlighting it is influential for many nodes. Since we sorted Φp by class, cluster size and degree, can actually see how the degree plays an important role for the positive influential nodes while less important for the negative ones. In C.g we study how this behaviour is linked to graph connectivity.
1.0
0.5
139 0
2809
0.0
139
0
2809
Figure 5: Pairwise R2 for CoraML's Pue. Column-wise R2 (left) shows same-class training nodes (V)) affect others similarly; Row-wisc R² (right), shows same- class nodes (V) also share influence.
Synchronized. We additionally see high correlation between many pairs of columns. To study this further, we compute the pairwise R² coefficient of entries in 4xx (either rows or columns) in the train setting learning signal on CoraML. From the block-diagonal structure Fig. 5 (left) that any pair of

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training nodes i and from the same class inffuence all other nodes in the same way since the R score between { i] and $[:,j] is high. Similarly, on Fig. 5 (right)) we also see that any pair of same-class nodes are being influenced in the same way by training nodes as show by high At score between o and & :]- Since nodes within each class both exert and receive inftuence in similar ways, we refer to this effect as synchronization. We observe analogous behavior in the setting and overall signal (see $ A.2
НII. Training nodes are most influential, but a few test nodes stand out.
We look at the most positively and negatively important nodes in the two-hop neighbors (to match the receptive field of a two-layer GNN) of a selected test node according to CoraML's 's Pax encoding the learning signal. Fig. 6 we generate a breadth-first tree from given test node. We color nodes by their classes, size both nodes and edges by the absolute value of their influence w.r.t. the root, and shape nodes as circles if they are training nodes or squares otherwise. Edges are colored groen they positively influence the root node and red otherwise. We that the most. important (positive) nodes are usually training ones (large circles). However, we can find important test nodes (large squares) that are more relevant than some training nodes. These are usually test nodes connected to training nodes from the same class. The class concentration plots in Fig. also confirm this insight (compare crosses to dots). This emphasizes the imporlance of looking at both training and unlabeled nodes within the graph domain. In [$ A.4 we rank nodes according to their values in different settings and show how approaches rank non-training nodes vs. training ones. compare ranks from the learning and overall signals.
A.3 we
Pos. infinence
Size x linfluence]
Neg. influence
Color class label
Train node
Test node
Figure 6: Two-hop breadth-first trees rooted at selected test nodes (center of each radial graph) from CoraML. Nodes are class-colored, and edges are colored by sign (green for positive, red for negative). Both nodes and edges are sized based on Pон influence values. Training nodes (circles) are typically most influential, but some test nodes (squarcs) stand out, often when highly connected to samo-class training nodes.

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IV. Data values are excellent al detecting brittle and poisoned nodes.
Brittle. The support of node is the minimum set of nodes such that if removed, the node is mis- classified. This concept was introduced by Ilyas et al. (2022) for images. Using $[v,;] :] we can estimate nodes that have high influence on v, which turn allows us to estimate the support. We follow Alg.
DM
LOD
PCW
RND
a C--BANZ
SHAP
POWP
DEG
Accuracy
CDF
Ensemble guess
from Tyaas et al. (002), using different approaches to estimate the top-k nodes and the support in the all setting. On Fig. 7 (left) we see that, regard- less of the considered signal (learning at the top of the right-hand column, and overall al the boltom) most nodes are brittle- more than half of the nodes (y-axis) have support size of 15 or less (x-axis), and by just removing them we can misclassify more than 50% of nodes. Computing the exact support is intractable and each data value gives an upper bound. Choosing the best upper bound among all values we arrive at the best estimate marked with a black dash line (ensemble guess). Again we see that DM and a-BANZ are closest to this estimate, and are the only two approaches reaching a reasonable esti- mate in the overall setting where all the other fail. In C.2 we include results for the train setting well, where can draw similar conclusions.
Accuracy
CDF
20
40
200
400
Support size
# of removed nodes
Figure 7: Sampling reuse approaches (DM and a-BANZ) show more consistent performance in predicting sup- ports (left) and estimating values (right). Predicting supports uses CoraML's & with an SGC model, while node removal uses GAT model.
Poisoned.
:[u,:] can also be used to spot corrupted instances. When gathering data from external sources, some instances may be corrupted intentionally (poisoning) or unintentionally (mislabeling). The rank of node values can help detect such instances. Namely, a high data value a labeled node for itself (self-importance) indicates memorization. When a training node has a wrong label and v E S the model must. memorize it achieve low training loss. If & S then the model's prediction does not match the wrong label. This leads to large difference u(S U (v)) u(S), and thus a large value. To test this, we poison 10% of the training data and compute the node values using the margins of the nodes themselves the learning signals in the train setting. In Fig. 9 we show the percentage of poisoned data for CoraML appearing in top-k ranked nodes as A increases, with DM and a-BANZ consistently detecting poisoned data better than the others. We provide details in C.3
the utility. We consider
V. Sample reuse is critical feature for data valuation approaches to achieve good performance.
To evaluate the quality of node values, we can again use their rank. Nodes with high positive value should have positive influence on the model, while nodes with high negative value should have the opposite effect. A stan- dard analysis is to then remove (or add) nodes according to the values' rank and observe the change in model's final performance (test accuracy). In Fig. 7 (right), a GAT model computes learning (lop) and overall (bot- tom) signals, using test margins as utility on CoraML (see S B.1 for details on deriving ranking from margins), and rank nodes for removal based on the values. F Fig. shows analogous removal results across datasets using GCN in the all and learning signal setting, using test accuracy as utility. We expect a steep decline in the performance
remove the first most important nodes and then plateau when removing inffuenntiaa only interested in the initial performance drop. Overall,
nodes. We stop after removing 500 nodes since DM and o-BANZ consistently show strong results and have the steepest decline in performance. PCWP performs worse, but better than PCW and SHAP. We omit. DM on some of the large dalasets due to memory issues. LOO and SHAP converge to the simple RND and DEG baselines on larger graphs. In contrast, PCWP seems to work better on larger graphs (c.g. PubMed and CoPhysics) but still lags behind DM and a a-BANZ, However, the gap between the methods is emphasized in the overall setting. Here the valuation methods have "less information" since if node i S the node is not available during training inference. In C. we report additional experiments in-
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DM
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Figure 8: Most influential node removal according to & computed using GCN: Citeseer, CoraML and PubMed (first row), and Photo, Computers and CoPhysica (secoud row). Pruning high-value nodes leads a perfor- mance (test accuracy) drop, revealing the quality estimated values. and o-BANZ consistently show the steepest declines across datasets.
DM
Q-BANZ
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SHAP
PCW
PCWP
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% poisoned
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a
Figure Ratio of poisoned nodes among top-k ranks CoraML, with DM and a-BANZ identifying most of them.
Figure 10: An increasing number of subsets leads to better predic- tive performance (LDS). Experi- ment on CoraML's +
Figure 11: LDS on counterfactual
subsets. Insets show predicted vs. true utility correlation at 0.5.
Experiment on CoraML's c.
cluding the train setting S C.4 as well as node addition experiment GNNs' sensitivity train/val, test splits Shehur al. (2018), we study value stability across 10 splits in S C.6
C.5
The conclusions are similar. Given
Park et al. )2023) introduce the linear datamodeling score (LDS) defined the Spearman correlation be- tween the predicted utility and the true utility. The idea is that good (predictive) valuation method should accurately approximate the utility on held-out subsets. Fig. 10 shows the improvement in the LDS as increas- ing the number of subsots when the held-out set comes from the same distribution (same = 0.1 for both computing and evaluating the data values) for DM and a-BANZ, Fig. 11 shows instead the LDS performance when the distribution of the held-out dataset changes while keeping a 0.1 fixed for computing the data values. From the former, we can see how from already just 10000 subsets DM and a-BANZ achieve reasonable results. From the latter instead, a-BANZ generalizes better across the counterfactual sets, followed by SHAP and PCWP. On the other hand, DM becomes worse for larger distribution shift (larger a).
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Generally, we notice consistent good performance in approaches taking advantage of the sampling reuse FFig. 7 to Fig. 10). The data value of an instance is updated leveraging information from all the subsets and not only from the ones where the node is included. Indeed, both DM and a-BANZ take advantage of the sampling reuse trick (we study further the limitations of game-theoretic approaches in the context of graphs in S C.8
Discussion and outlook. Finding confirms that nodes tend to reinforce intra-class predictions while dampen inter-class ones. The magnitude varies with community size: smaller groups exert stronger cross-class inffuence, whereas larger ones are redundant, reducing per-node impact. This is because when many nodes carry similar information, their contribution overlap and individual's influence becomes less significant. We find this phenomenon equally compelling for i.i.d. settings and leave its exploration to future work.
shows that a handful of training nodes dominate the valuation and are synchronized. This influence homogeneity suggests opportunities for potential pruming, since we can remove nodes that are simultancously unimportant for many others, but also vulnerabilities, since few nodes have large influence over many test nodes. In the i.i.d. setting, training and test data are presented to the model in separate phases. In the transductive remi-supervised setting, this separation does not occur, and Finding III shows that certain test nodes can exert substantial influence. Understanding what makes these nodes influential, such as by analyzing their neighbor structures, remains an open rescarch direction. With Finding IV we explore two applications for node values that have also been studied in the i.i.d. context. A natural next, step is to identify novel, inherently graph-specific applications for node values.
Interestingly, Finding
I1 Finding
Finally, Finding V reveals that sample reuse is key to accurate data valuation on graphs as well Li & Yu (2024) observed this for the i..dd setting. Indeed, methods like DM and a-BANZ outperform custom graph approaches (c.g.,,PWWP PCWP) by efficiently repurposing subset evaluations, even without neighborhood constraints. This hints at structure-aware MSR techniques benefiting from both adaptive subgraph sampling and sta- tistical sampling reuse, Moreover, with the same subset-evalie -abast-evallttion budget, MSR attains higher valuation accuracy, casing scaling limits on large graphs. In S C.7 we show that values learned with a simpler (faster) graph model transfer well to a more complex one, offering another route around computational bottlenecks.
Together, these findings position node-level data valuation not just as a diagnostic tool but as a lens to rethink GNN design: mitigating brittleness via valuo-aware robust training, exploiting synchronization for efficient learning, and redefining the role of unlabeled data. Bridging game-theoretic principles with graphs, hope to spark research in this direction with the goal of developing more interpretable, efficient, and robust GNNs.
5 Related work
In the literature, data valuation has been adopted mainly in an i.id. setting to study relations between training data and changes in the model's utility or to support prediction's explanations at inference time. This section presents an overview of the works that are closest to ours.
Data valuation in the i.i.d. setting. The seminal works by Ghorbani & Zou 2019); Jia et al. (2019b) extends Shapley value from feature-level to data point granularity to assess training data importance. As this may result in a conallytionally expensive procedure, Jia et al. (0021 2019a) compare the utility of different data attribution methods and propose a fast estimator for Shapley values based on a k-nearest neighbors surrogate for which the exact Shapley values can be efficiently computed. A more recent research direction uses linear surrogates to learn a mapping from a subset of the training set to the model's utility [[yas et al. 2022 Wang et al. 2021). The surrogate then used to estimate the utility from newly sampled subsots and the surrogate weights represent the data values. As the number of possible training subsets is exponential w.r.t. the number of training data points, the learning of the surrogate may be time-consuming. In follow-up work, Park et al. (202)) employ Taylor approximation to linearize considered deep neural network such that the onc-step Newton approximation can be applied in non-lincar setting. They show that in such a way, just a few training subsets are needed to attribute imporlance to training data accurately, Wang & Jia (2023) introduce the concept of data Banzhaf for assessing the values of training data and showing its robustness in differentiating data. Except for data Shapley, these approaches have not yet been applied to the graph domain. In our work, we incorporate and compare all these approaches to investigate their behaviour when structural information comes into play. Further, (Sundararajan et. al. (2020) introduce the Shapley-Taylor
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interaction index, generalization of the Shapley value to feature interactions of bounded order. Following this work, Tsai et al. (023)) propose Faith-Shap, variant satisfying the standard Shapley axioms via a least-squares formulation, while Fumagalli et al. (2023) develop SHAP-1Q, a unified estimator applicable to any cardinal interaction index. Also, Li & Yu (0024) propose an estimator based on maximum sample reuse to approximate any probabilistic value, notably improving convergence for Shapley-like attributions. Although they address the computational bottlenecks of subset sampling, their evaluation is limited to feature attribution. We leave the investigation of interactive indices in the graph-data valuation setting as future work. Most recently, Liu et al. (2024) introduced Kernel Banzhaf, which frames Banzhaf value estimation as linear least-squares regression problem with pair subset sampling procedure that considers the complement of each subset, improving the value estimation.
Data valuation for graph-structured data. Graph data valuation relating data to model predictions is still under investigation. To our knowledge, only two works try tackle this direction. Chen et al. (2023) adapt influence functions to a graph-structured model to approximate the lcave-ono-out training. However, they consider only training nodes in transductive setting, without capturing the information coming from explicitly removing unlabeled nodes. We instead consider more comprehensive picture by studying the transductive scenario, looking at the influence of all nodes, whether labeled or not. Instead, Chi et al.] (2025a) propose new coalition sampling based on the Winter value that considers the graph's structure when creating a permutation of nodes to process and estimate each node's contribution. However, this work focuses on the inductive scenario where unlabeled nodes do not play a role during the model training.
Graph data valuation inference time. When applied at inference time, a data valuation approach assesses the most important input patterns that mostly influence the final model's prediction (feature code/edde//ubbgaapg
attribution). Particularly for graphs, this translates into highlighting the graph struc- ture or node features that cause the prediction for the input. For instance, Duval & Malliaros
(2021) compute
graph structure and node feature explanations for a single example by constructing a surrogate model on a perturbed graph and computing Shapley values as explanations. [Akkas & Azad (2024) explain predictions by computing Shapley values edge-wise and outpulting the subgraph with edges from the lop-k values. Chh- ablani et al. (2024) use Banzhaf value in combination with thresholded utility functions on edges to provide counterfactual explanations. Bui et al. (2024) introduce the Elyerson-Tayle index which includes structure information inside the Shapley value to identify important motifs for the prediction. Finally, Chi et al. (2025b) introduce a structure-aware Shapley value utility by leveraging the precelence-coostrrannee sampling from PCW (Chi et al. 2025a), adapted to the inference setting, and combines it with transferable feature extraction and a Shapley-guides optimization procodure tailored to structure-aware utility learning. Differently from the mentioned works, we employ data valuation to discover relations between nodes involved in the training and the final output of the model. Namely, we do not treat the GNN as a fixed black-box function.
6 Conclusion and further research
We present the first extensive study of data valuation methods for graph-structured data. We introduce different data valuation scenarios, and apply state-of-the-art data valuation approaches that have not yet been investigated in the context of graphs. Our results demonstrate that these methods significantly outperform the latest efforts in graph data valuation across multiple applications. We also show how different utility functions can enable several applications. We summarize our results in five key findings that emphasize properties of different methods. Due to the need to train numerous models on different subsets, our study was computationally intensive, and more effort is needed to develop more efficient data valuation methods for node values (both in terms of resources and time efficiency) in particular for large graphs. Future directions also include exploring alternative proxies for datamodels that account for graph structures and a deeper investigation into settings where nodes are simply unlabeled rather than removed.
Acknowledgments
This work used the RAMSES herf-peeformance computing cluster the University of Cologne's ITCC, whose support is gratefully acknowledged.
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Training nodes
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Figure 12: Heatmap of the learning signal values 4 CoraML for different approaches. Green and red dots represent positive and negative importance respectively. Crosses represent highly impactful nodes (with values larger than the ones indicated in the bars). Black crosses mark the training nodes.
A Analysis of node values
Focusing the values computed with SGC the model on CoraML, more carefully analyse the computed values in different settings.
A.1 Node- and class-level influence
Node influence. To provide more detailed comparison in support of Finding 1 study the values + under both learning and overall signals. Fig. 12 and Fig. 13 show the heatmaps of these values, sorted by class size and degree. Positive (green) entries indicate that a node's removal has beneficial effect on performance, while negative (red) entries show a detrimental effect. Black crosses mark training nodes; their corresponding vertical stripes illustrate how training nodes often exert influence on multiple other nodes supporting also Finding II]
Some methods (e.g., PCW and PCWP) assign values close to zero under the overall signal, mirroring their less reliable performance in the removal experiment. As result, their final rankings can approximate random noise (see lower plots in Fig. 7)
Cluster influence. Fig. 14 and Fig. 15 show the average & values on a per-class basis for the learning and overall signals. We overlay the confusion matrix of the model's predictions, allowing us to interpret how higher positive influence (darker green) correlates with a lower rate of false positives. Notably, PCW and PCWP have difficulty identifying highly negative influence in the overall setting, causing negative values diminish in magnitude.
Fig. 16 shows the average class influence across all the datasets using $ Pa-BANz (learning signal). As pointed out in Finding I smaller communtices get the highest value than larger ones. We see how this effect is consistent across all the considered datasets.
Node influence visualization. Finally, to support Finding III we extend the breadth-first search influ- ence visualization to the overall in Fig. 17 As before, node size encodes importance for a given test node, and the color of the incoming edge (green or red) signifies whether the influence is positive or negative. We again observe that training nodes typically the most influential, although certain test nodes within the
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Training nodes
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DM
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Figure 13: Heatmap of the overall signal values + on CoraML for different approaches. Green and red dots represent positive and negative importance respectively. Crosses represent highly impactful nodes (with values larger than the ones indicated in the bars). Black crosses mark the training nodes.
same class (or those tightly connected to highly impactful training nodes) can also assume significant roles. In the overall signal, more outliers are apparent in terms of node size, indicating that a few nodes can exert an especially large effect on model outcomes.
A.2 Synchronized class influence
Analogously to Fig. 5] we investigate how nodes in the same class tend to exhibit similar influence patterns, both in how they influence others and in how they are influenced themselves (Fig. 18). To quantify these effects, calculate the pairwise R² coefficient of the rows (bottom) and columns (top) of 4 for different experimental settings.
On the left-most. column of Fig. ig. 18 we focus on the train setting, overall signal. As in Finding II we observe that nodes within class tend to influence other nodes in similar manner (top), resulting in a pronounced clustering effect. Similarly, the row-wise 12 shows that nodes from the same class are also influenced similarly by other nodes (bottom), aligning with our earlier findings that training nodes can exert broad impact across the graph.
we focus on the all setting and examine the two different signals. The middle column corresponds to the learning signal, while the right column corresponds to the overall signal. Similar class-wisc clustering patterns are visible in the learning plots (particularly for the row-wise 1P),, but they become weaker under the overall signal. We leave this as an interesting direction to investigate more in the future.
On the right-most two columns of Fig. 18
Overall, our results highlight a similar influence pattern how nodes from the same class exert and receive influence to and by others.
A.3 Learning vs. overall rank
Bocause learning and overall signals can provide complementary insights, we investigate how these two perspectives differ in the rankings they produce. Specifically, for each approach, we compute the overlap in the top k% of node rankings derived from the corresponding & values for CoraML. Fig. 20 summarizes these
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Q BANZ
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079
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579
Figure 14: Average class-wise influence across different approaches according to 4 capturing the learning signal. Classes are sorted by their size. Cell annotations are the confusion matrix values for model predictions.
DM
BANZ
LOO
Rule
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Figure 15: Average class-wise influence across different approaches according to + capturing the overall signal. Classes are sorted by their size. Cell annotations are the confusion matrix values for model predictions.
results, illustrating that DM and a-BANZ yield consistently high overlap between the learning and overall rankings. This stability suggests that, even when node is removed for both training and inference, these methods capture node importance in way that remains robust across varying levels of information.
By contrast, PCW and PCWP show notable variations in their top 1% rankings under different removal con- ditions. These fluctuations underscore the challenge of maintaining stable value estimates for nodes whose significance may shift when the graph and model context change between training and inference.
A.4 Rankings of approaches
Because DM and a-BANZ demonstrate similar performance, we also examine how their node-level rankings report Kendall's correlations computed from
compare to those generated by other methods. In Fig. 21
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CoraML
teseer
PubMed
Node class
0.010
0.04
0.00050
0.02
0.00025
0.005
0.00
0.00000
0.000
Photo
Computers
CoPhysics
0.02
0.0010
Node class
0.010
0.01
0.0005
0.005
0.0000
0.000
0.00
Figure 16: Average class-wise influence from a across datasels. Classes sorted by their size. Adding nodes from smaller communities highly impact the margins positively for nodes from the same class and negatively for nodes from different classes. This effect weakens the larger the community is.
the margin-based & values for CoraML, showing that DM and a-BANZ also exhibit a strong aligument. As expected, PCW and PCWP show notable correlation too.
Fig. 22 and Fig. 23 illustrate how each method ranks nodes according to the learning signal for CoraML, using Lest accuracy and lest. margins, respectively, as the utility (see) S B.1 for an explanation on how to get influence scores from the per-node margins matrix). We highlight training nodes in gray, validation nodes in orange, and test nodes in blue. Although training and inference), the majority of top-ranked nodes remain in the training set. Nevertheless, certain test nodes appear prominently in these rankings, suggesting that their removal adversely impacts model performance. For instance, DM often assigns high importance to particular test nodes which is consistent with Finding III Overall, these results underscore that while most training nodes drive performance, there can select test. nodes that substantially influence model outcomes as well.
the all setting (where nodes can be removed at both
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Pos. influence
Neg. influence
Size o linfluencel
Color = class label
Train node
Test node
II
Figure 17: Two-hop breadth-first trees for overall signal 4px. Nodes are class-colored, and edges are colored by sign (green for positive, red for negative). Both nodes and edges are sized based on influence values on the root.
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Train
A11
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1.0
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A
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Pairwise R2 coelficients of 4 under different settings. The first row shows the column-wise nea measures, and the second row shows the row-wise R² measures. The left-most column corresponds to the Train setting, overall signal. The right-most two columns correspond to the All setting, where the middle column represents the learning signal and the right column represents the overall signal.
Figure 18:
DM
BANZ
SHAP
LO0
PCW
PCWP
1.0
1.00
Accuracy
0.75
SHAP
Overlap
DM
BANZ 0.60
0.75
0.50
0.5
LOO
0.50
0.25
SHAP
0.0
PCW -0.140.130.000.2
0.25
200
400
5
10
50
PCWP 0.176.150.080.211
# of removed nodes
Top-k (%)
0.00
Figure 19: SHAP be improved by adding 10% more nodes as con- text in its evaluated subsets.
Figure 20: DM and Q-BANZ show consistent high overlap between learning and overall rankings.
Figure 21: Kendall's among ap- proaches indicates DM and a-BANZ share a high rank correlation.
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Train nodes
Val nodes
Test nodes
Infiuence value
DM
(-BANZZ
LOO
Inffuecce value
SHAP
PCW
PCWP
Node index
Node index
Node index
Figure 22: Node rankings from learning signal + for CoraML using test accuracy utility.
Train nodes
Val nodes
Test nodes
Inftuence value
DM
a-BANZ
LOO
Influence value
SHAP
PCW
PCWP
Node index
Node index
Node index
Figure 23: Node rankings from learning signal O for CoraML using test margins utility.
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B Further experiment details
Datasets and models. Tab. summarizes the datasets used in our experiments. Following Shehur et al. (018)) we employ stratified sampling to select the training nodes (20 per class) along with equal number of validation nodes. The remaining nodes serve the test set. We choose SGC, GCN, and GAT architectures, cach configured with 2 layers. For GCN and GAT, we use 16 hidden channels, dropout rate of 0.5, and a ReLU activation. Specifically for GAT, we adopt the GATv2 variant from Brody et al. (0022 using 8 attention heads.
Table 1: Summary of datasets. Homophily is computed the LCC.
Dataset
Nodes
Edges
Features
Classes
Nodes (LCC)
Edges (LCC)
Homophily
# Train /al/ /est
CoraML
2995
16316
2879
7
2810
15992
0.784
140 114
2715
Citeseer
4230
10674
602
6
1 1881
3804
0,928
120 / 120
3990
PubMed
19777
88 648
500
197
648
0.802
60 60
19597
Photo
7 7650
238 162
745
74887
238
0.827
160 160/160/ 7330
Computers
13 752
491 722
767
10
13381
491 556
0.777
200 200/200/ 200
13352
CoPbysica
34 493
495 924
8.455
5
34 493
495
0.931
100 / 100 /44293
Texas
183
325
1 1703
5
183
325
0.108
87
59)
37
Wisconain
251
515
1 703
5
251
515
0.196
120
80
51
Training setting. We train GCN and GAT using the Adam optimizer with learning rate of 0.01 for 3000 epochs. Early stopping is based the validation loss, with 50 epochs of patience. Results are averaged over 10 runs for each model, except for larger datasets where we use runs. In contrast, SGC leverages closed- form solution by relaxing the classification objective into a regression problem. Due to the absence of non- linearities, the model weights can be derived as W X,H, where X (X x + and 5 For all models, we conduct experiments using 5 different train/ val/Lest, splits and report average performance. We train each node's datamodel with the ridge regression (with vals-vvaliation) implementation from scikit- learn (eedregosa al. 2011) using its default hyperparameters.
Running these experiments CPUs generally proves faster than on GPUs, given the shallow architectures. We parallelize the runs of different models with the joblib library (Joblib Development Team 2020), leverag- ing the larger RAM capacity (compared to GPU memory) to exccute as many jobs as possible simultancously. Experiments are performed on a cluster of 136 nodes, each equipped with 2x AMD Epyc 9654 processors (96 cores, 2.4-3.7 GHz) and 768 GB of RAM. For larger datasets, such as CoPhysics, we switch to a bigger parti- tion with 8 nodes (same CPU specifications) but 3 TB of RAM to accommodate higher memory requirements.
Running times. We split the computational cost of each method into two components: gathering utility across the selected subsets and computing the final data values. Tab. 2 Tab. 3 and Tab.. 4 show the total cost for each approach under 50 000 subsets with a = 0.1 for different GNNs. Among methods with comparable performance, Q-BANZ exhibits the best overall runtime. A a_" in the tables indicates the approach ran out of memory.
Table 2: SGC computation times.
Daraset
INM
120
SHAP
Train
Thits
T
S1.0
87
Cocpasers Co
18
9
Datamodels training. By default, DM excludes any subset that contains a target sample in the training set to prevent information Icakage (myys et al., 2022). particularly when evaluating predictive performance. However, this procodure not always feasible across different graph settings. Specifically, when computing a node's learning signal value, we exclude subsets containing that node from the sampled distribution used to
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Table GCN computation times.
Dataset
De
B-B2
1.00
SHAP
PCKP
Train
Tron
Citeseer
52 d0l
16 m dit
31
a
mm J6
30 s
2m 22
53 35
28 2
10 m d8
10 68
Coralt
h 53
11
56 u
3
54
16
24 12
1 m 7
14 54
3 12
39 30
17 u 39
15 1
3.
57
h
44
Photo
10 h
58
24 m 27
10
17
m
34
50
m 30
22 32
11 1.133mm 58
26 26
24
21 25
Coopatera
d 1.d11k
47
17
dd1
20
49
1 03 mm 28
19 m 12
18
5 22
27 m 88
27
4.3. 57
1
Table 4: GAT computation times.
Datuwet
BACZ
LO0
PCWP
Tratn
Train
All
Train
Train
Trem
Oscesene
48 h 17 m.
h i
20x
Gorake
18 15 20
52
18
m 56
h 13 2
16 2..00 50
1 5
44
Publied
17 00
17.h 46 60
B 22
10
m 82
10th 56 15a
2 12
9 24
92 92
47 m 58
Photo
12 30 27
1 17
m 51
44 m 25
14 18
20
dd 1 02 0.
7..9 39
d1
41 m 53
Competure
60
26
2 80x
mm
1 1: m0
1 14 20
21
CoPhystic
d 12
30
13
train the datamodel Conversely, when computing the overall value, excluding those subsets would leave no data available for training the datamodel, making this approach impractical.
Banzhaf margins utility. In our work, we adopt the MSR implementation to approximate the Banzhaf values:
ÓRANZ
u(()
Σ u(S).
SEal
SESE:
SESei
While this straightforward when each node's utility is defined for every other node, when computing the overall signal in the all setting, the utilities may be undefined for some nodes. Consequently, the sum of utilities has to be averaged according to the computed utilities. Namely, the aggregation depends on whether nodes are inside or outside the subsets considered for node i. Concretely, given arrays of margins (s for a node i across different subsets S, utilities are undefined for node j that is not in S. Thus, dividing by [Seil in cases where not all those subsets actually define a margin for j, artificially reduces the measured impact of é on j. For this reason, we instead average the inffuence accounting for the the number of times node j appears in the considered subsets Sgi, and analogously for S4z.
B.1 Nodes ranking from margins to influence scores
In the node influence experiment, each node needs single influence score. However, when margins serve the utility, we obtain influence value for every other node's prediction, $ R To derive a final score and subsequently ranking, average each node's contribution across all other nodes' predictions. Concretely, we form a matrix of influence scores where row i indicates how other nodes influence node i's prediction. We then compute the mean of each column in this matrix to obtain a final score for each node, which we use to produce the ranking.
B.2 Subsets sampling
Because it is infeasible to evaluate utility for all 2 subsets, a common approximation strategy to sample subsets. A straightforward but effective approach is to include each instance independently with probability a, i.e., Pr(i S]
for every E DD As a result, the subset. size follows Binomial distribution with expected value [D), implicitly capping the average subset size. We adopt this method for both BANZ and DM values. As shown in s B.2.1. is an important hyperparameter that we tune. To approximate the SHAP values, we randomly sample a permutation of the nodes, then scan them in order until reaching user-defined truncation threshold. For a fair comparison, each method uses the same total number of subset evaluations.
In the all setting, this exclusion applies V. well.
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a Q-BANZ 0.01
a-BANZ 0.05
a-BANZ 0.08
a- D-BANN 0.25
BANZ 0.75
Q-BANZ 0.03
a-BANZ 0.06
cx-BANZ 0.1
a-BANZ 0.5
BANZ 0.9
Accuracy
0.75
0.50
0.25
200
400
0
200
400
0
200
400
0
200
400
# of removed nodes
# of removed nodes
# of removed nodes
# of removed nodes
Hyperparameter search for the best for subsets' sampling CoraML in the all setting. From left to right, subplots report the results for learning & test margins utility, learning & test accuracy utility, overall & test margins utility and overall test accuracy utility.
Figure 24:
We sample 50000 subsets per method. For a-BANZ and DM, we tested various a values and identified 0.1 optimal for CoraML (details in $ B.2.1: we then apply this choice to the remaining graphs. For SHAP, we adopt the implementation of Ghorbani & Zou (2019), which stops processing permutations after 25% of the nodes have been scanned. Consequently, we set the number of permutations to 50,000 [w-te] with tr = 0.25. When computing node values under the all setting, permulations may start with nodes outside the training set, causing random predictions to introduce noise. Therefore, we begin updating node values only once we reach the first training node in cach permutation and truncate 25% from that point. Similarly, for PCW, we choose the number of permutations so that the total number of subsets remains 50 000. In Chi el al. (2025a), the default truncation ratios tr for 1-hop and 2-hop neighbors are 0.5 and 0.7, respectively; we optimize these ratios to obtain PCWP, which yields slightly better performance (see B.2.2
B.2.1 Choice of the hyperparameter
a is the probability of including each node from D in a subset, effectively determining the subset size. For example, in the train sctting, where D Vt: choosing a = 0.1 yiclds subsets comprising 10% of V. plus all the other nodes. This parameter serves as hyperparameter for DM and a-BANZ and can be selected via a grid or random search. Due to the high compulational cost of performing such search on every dataset, determined the optimal for CoraML and reused it across all other datasets, acknowledging it may be suboptimal in certain cases. Nonethcless, as demonstrated in 5 4 this choice of a outperforms other methods on most datasets.
Fig. shows how a-BANZ performs in the most influential pruning experiment under various a values used to sample subsets for node-value prediction. We observe that a = 0.1 and a = 0.25 yield similar estimates. How- ever, because smaller subsets reduce training time for the models, we chose a 0.1 as the best overall setting.
B.2.2 Optimized PCW
In PCW, we compute marginal contributions for nodes only within the first (1 11-th tr) portion cach node's child sub-trees. We approximate the contributions the remaining sub-trees as 0, stopping once we reach exactly 000 subsets. Because this approach was originally developed for an inductive setting, its default trunca- tion ratios appeared overly tailored to that context. We discovered that optimizing these truncation ratios to maximize the number of considered permutations improved performance in the transductive setting. Conse- quently, we set both the 1-hop and 2-hop neighbor truncation ratios to 0.99, referring to this variant as PCWP.
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C-BANZ
LO0
SHAP
POWP
Accuracy
Accuracy
1
100
100
100
200
100
200
Accuracy
1
Accuracy
50
50
50
100
0
50
100
of removed nodes
# of removed nodes
of removed nodes
# of removed nodes
Figure 25: Node drop for Texas dataset. Rows corre- spond to the two valuation settings (all at the top, train at the bottom); columns to the two utility sig- nals (left: learning, right: overall). Steeper initial declines indicate more faithful value estimates.
Node drop for Wisconsin dataset. Rows correspond to the valuation settings (all at the top, train at the bottom); columns to the two utility signals (left: learning, right: overall). Steeper ini- tial declines indicate more faithful value estimates.
Figure 26:
x V.
Positive outliers
Negative outliers
Avg. per-olass influence
Class index
30
couceutration
Median influence
index
Node
10
Mediaa inllucuce
-2
x
Node index
Figure 27: Ppn's 's heatmap on Texas graph (all-learning scenario). Rows and columns are sorted by class, cluster size and degroe. Green (red) values mean that adding column-node, j increases (decreases) the margin of row-node i. We can see positive intra-class influence, but weaker mixed off-diagonal colors, confirmed also by the average per-class influence heatmap. Influence is concentrated few (training) nodes.
C Additional experiments
C.1 Heterophilic datasets
To check whether our findings still hold in the heterophilic settings, we repeated our analysis on the Texas and Wisconsin (Pei et. al.] 2020) hyperlink-web graphs, where the homophily ratio is less than 20% (see
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Positive outliers
Negative outliers
Avg. per-class influence
Class index
30
Class concentration
Median inffeence
20
Node index
10
Median inflnence
V
V
-10
Node index
Figure 28: Pon's 's heatmap on Wisconsin graph (all-learning scenario). Rows and columns are sorted by class, cluster size and degree. Green (red) values mean that adding column-node: increases (decreases) the margin of row-node i. We can see positive intra-class influence, but weaker mixed ofl-diagonal colors, confirmed also by the average per-class influence heatmap. Influence is concentrated in few (training) nodes.
Tab. 10. The experimental protocol, arper-parametent and number of sampled subsets (50000) with = 0.1 for sampling-ren methods) are exactly the same as in the homophilic benchmarks. We used the 10 different node splits that are provided with the datasets. Finally, we figure out that in the Texas dataset, there is only one node belonging to class 1. We then omit this class when reporting the per-class average influence.
Fig. 25 and Fig. 26 show the loss in accuracy we observe when cumulatively removing the top-valued nodes. a-BANZ and DM again show the largest accu- racy drop across different settings (all setting on top and train setting at bottom vs learning signal on the left and overall signal on the right). The shape of the drop curves mirrors what we saw on homophilic graphs: an initial steep fall followed by a plateau once low-value nodes start to dominate. The plateau is reached sooner here because both datasets are an order of magnitude smaller.
DM
SHAP
PCW
Ensemble guess
a- BANZ
LO0
PCWP
CDF
1
CDF
In Fig. 27 and Fig. 28 we repeat the same analysis performed in Fig. 4 We show heatmaps of the val- ues Ppx the all setting using the learning signal, and still observe clear positive blocks along the di- agonal (nodes help peers of the same class), but the off-diagonal entries
0
0
20
40
0
20
40
Support size
Support size
longer strongly negative. In some small classes they even turn slightly positive, reflecting the fact That cross-class neighbors are the norm rather than the exception on these graphs. Con- sistent with the homophilic case, influence highly concentrated: in both datasets fewer nodes have the most of the influence value, and most of these are still training nodes.
Figure 29: Brittle prediction in the train setting for SGC (left) and GCN (right). DM and a-BANZ consis- tently approach the best guess regardless of the con- sidered learning (top) or overall (bottom) signals.
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Influence
2-
DM
Q-BANZ
LOD
SHAP
PCWP
Node index
Node index
Node index
Node index
Node index
Figure 30: Nodes rankings according values computed on a graph with 10% of poisoned (red) nodes.
C.2 Brittleness in the train setting
We replicate the same experiment used in Finding IV but now in the train setting, to assess whether each approach maintains strong predictive performance. In Fig. 29 we evaluated both the learning (top) and overall (bottom) signals computed with SGC (left) and GCN (right), and in each scenario, DM and a-BANZ demonstrated performance closest to the best guess among all methods.
C.3 Memorization ranking
Fig. 30 shows the rankings the approaches as a result of the memorization experiment. As explained in Finding IV] we poison 10% of the training data and expect a good data validation approach to rank these nodes as the most important for their prediction seeff-importance. We show the results for CoraML, where the number of training nodes is 140 resulting in 14 poisoned nodes. We then show only the top 14 nodes ranked by the approaches. The results show that DM and a-BANZ rank first the poisoned nodes establishing as robust approaches for detecting poisoned or mislabeled data.
C.4 Influential nodes in the train setting
Here present the results for the node influence experiment. in the train setting, namely attribute importance (and then remove) only training nodes. In Fig. 31 show how the approaches capture the real value considering both the learning (top) and overall (bottom) signals, across different models SGC (first column), GCN (second column) and GAT (third column). Notably, we observe the approaches struggle more in attributing the right values when non-linearities are introduced in the models (compare SGC w.r.t. GCN and GAT). Consistently with previous results, DM and a-BANZ perform better than the other approaches.
In this experiment, we train DM by considering all possible node subsets, regardless of whether particular node is included. Unlike the procedure described in $ B3 this setup is feasible here because are no longer focusing on maximizing predictive performance (where information leakage would pose a risk). Instead, our primary goal is to evaluate each method's ability to identify problematic data.
C.5 Most influential addition
In Fig. 32 we show the results for the most influential node addition experiment. In this experiment, we start with an empty graph and then incrementally add nodes in order of their importance, as determined by each method's computed node values +. We evaluate the predictive performance (test accuracy) of the model after each batch of added nodes. Contrary to the most influential pruming, because we are adding most influential nodes first, we anticipate a rapid initial increase accuracy, followed by a plateau once most of the crucial nodes have been included, We present results computed according both learning (top) and overall (bottom) values. Similarly to the removal experiment, DM and o-BANZ better assess node values.
C.6 Stability
A crucial aspect of reliable data valuation is consistency producing similar ranking results across multiple runs of the same experiment. Fig. 03..33111 illustrass the stability of each approach, measured over 10 random seeds for each of the crain/val/test splits in CoraML. Fach seed defines different puumutiopermutttionn of nodes
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DM
BANZ
SHAP
LOD
RND
0.75
Accuracy
0.50
0.25
0.00
0.75
Accuracy
0.50
0.25
0.00
100
50
100
# of removed nodes
# of removed nodes
# of removed nodes
Figure 31: Most influential node removal for CoraML across models in the train setting: SGC (first column), GCN (second column) and GAT (third columm). learning signal (top) and overall signal (bottom) 4.
DM
BANZZ
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Figure 32: Most influential node addition for CoraML across models: SGC (first column), GCN (second column) and GAT (third column). learning (top) and overall (bottom) signal of $. Adding high-value nodes leads to a performance (test accuracy) raise until reaching plateau given by keep adding uninfluential nodes.
for valuation. We report the average performance of each method (left) and observe that DM and a-BANZ exhibit. the lowest. variance across runs, making them the most reliable choices for data valuation on graphs.
We also evaluate ranking stability (right panel) by mcasuring the percentage of top-k nodes that overlap across different seeds CoraML. Once again, DM and BANZ demonstrate near-identical rankings across all seeds for given train/we aaii/val/tes test split, underscoring their reliability and consistency.
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Published in Transactions on Machine Learning Research (08/2025)
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Figure 33: Approaches stability: (left) in predicting node values; (right) in predicting the ranking for given split across 10 seeds.
Figure 34: Node removal using data values computed and evaluated by each model (solid line) or trans- ferred from SGC (dashed linc).
C.7 Transferability
we repcat the node removal experiment (similar to Fig. 8 with two variants: i) Computing data values with a GCN model and evaluating performance on the same GCN (solid line, no transfer). ii) Computing data values with SGC and seeing whether these values transfer (dashed line) to GCN and GAT.
In Fig. 34
Using B-BANZ and margins as the utility metric CoraML, observe nearly identical performance among these variants, indicating that data values capture intrinsic properties of the data rather than the specifics of each model. This finding further suggests that cheaper models (e.g. SGC) suffice to compute data values that effectively transfer to more complex architectures.
C.8 Empirical limitation of game-theoretic approaches
We investigated why game-theoretic methods underperform compared to MSR-based approaches. Our anal- ysis suggests that those methods commonly evaluate subsets containing only a small number of nodes when building up from empty sets via permutations. This yields many degenerate subgraphs and introduces noise in the final node value estimates. In contrast, randomly sampling larger subsets seems to better capture the underlying graph structure.
To address this, we propose small modification to SHAP that ensures every evaluated subset includes bit more context. Specifically, whenever node is added to the subset during the permutation scan, also include an extra 10% of the total nodes (in line with a = 0.1 in MSR-based approaches). As in the standard data Shapley procedure, only the value of the newly added node is updated; the difference is that the node's influence is now assessed in a richer structural context.
Fig. illustrates the impact of this modification: the dashed line shows that even a modest increase contextual nodes yields more realistic Shapley values, thereby boosting performance. Determining the optimal fraction of additional context nodes remains open question and is left future work.
C.9 Unstructured data values
We assess the contribution of graph connectivity by comparing standard two-layer GCN with an "un- structured" variant whose adjacency matrix is replacod by the identity, so that every node only sees itself. We keep the same optimizer, initialization, and train/val/ten splits for both the two models. Because the identity graph leaves each node isolated, any difference we observe must come from information that the CoraML graph propagates into the node representations.
After training, we compute the learning influence values + (real graph) and Pia (identity) in the all setting, take the median influence value that a node v exerts on others, and plot the difference A(v) = median( (fl,, :l) modian :l)) Fig. 35 The left panel shows that message passing amplifics positive influence, especially for training nodes, whereas most non-training nodes remain close to zero. The right panel indicates that already negative influences become slightly less negative, with only few outliers moving in
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Published in Transactions on Machine Learning Research (08/2025)
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Figure 35: A median influence (GCN GCNia) for positive (left) and negative (right) learning influence values in the all setting shows that real connectivity of CoraNL mainly amplifies training node influence, leaving most non-training nodes unchanged.
the opposite direction. Together these observations confirm that the specific connectivity pattern of CoraML amplifics node influence.
32


Reviews:
[{"review_id": "EY4nyiFAOZ", "reviewer": "Reviewer_ZMfm", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n(1) The study covers a wide range of data valuation methods and provides thorough experimental evaluation across multiple citation and co-purchase graph datasets with different GNN architectures. The experimental scope includes appropriate baselines with multiple evaluation criteria.\n\n(2) The distinction between learning and overall signals is valuable for understanding how nodes influence both training and inference in transductive settings. The five key findings provide actionable insights about graph data valuation that extend beyond the i.i.d. setting.\n\n(3) The paper includes solid experimental methodology with proper consideration of different signals, various utility functions, and extensive ablation studies.\n\nWeaknesses:\n\n(1) The dataset diversity is constrained to citation and co-purchase graphs. Evaluation on other graph types could give readers more comprehensive understanding and insights.\n\n(2) The contribution is primarily empirical with limited technical novelty. The paper does not propose new data valuation methods specifically designed for graphs.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper presents the first comprehensive empirical study of data valuation methods for graph-structured data in semi-supervised transductive settings. The authors systematically evaluate existing data valuation approaches including game-theoretic methods like data Shapley and data banzhaf, predictive methods like datamodels, and the recently proposed graph-specific precedence-constrained winter value on node classification tasks. Key contributions include (1) adaptation of i.i.d. data valuation methods to graph settings with careful consideration of labeled/unlabeled node interactions, (2) introduction of learning vs. overall signal distinction for graph data valuation, (3) extensive empirical evaluation across multiple datasets and models, and (4) five key findings about node influence patterns, brittleness detection, and the importance of sample reuse in graph data valuation.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "(1) It would be good to add more detailed discussion of computational bottlenecks and potential solutions for scaling to larger graphs. \n\n(2) It will be also great if authors could consider evaluation on additional graph types beyond citation and co-purchase networks, e.g. biological networks.", "broader_impact_concerns": "The work focuses on methodological evaluation of data valuation techniques and does not raise significant ethical concerns. The applications to detecting poisoned and mislabeled data could have positive security implications.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "FL8kLb1goA", "reviewer": "Reviewer_GAFF", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n\n- This work provides a comprehensive study of data valuation for graph-based models in semi-supervised settings, rigorously evaluating existing methods not previously applied to graphs.\n- The paper evaluates a broad range of methods, including game-theoretic and predictive (DM) approaches, alongside graph-specific methods.\n- The analysis yields several key findings on node influence dynamics, covering detection of brittle/poisoned nodes and the importance of sample reuse in graph data valuation.\n\n**Weaknesses**\n\n- The current approach to node removal for data valuation makes it challenging to disentangle the distinct influences of node features versus graph structure (edges), especially concerning noise sources.\n- While providing valuable insights, the paper could offer more explicit guidance on how these observations can be leveraged to design improved graph datasets or more robust and efficient GNN algorithms.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This work provides an empirical study on existing data valuation techniques. The study constructs various scenarios to understand how both labeled and unlabeled nodes influence GNN training and inference. The resulting node values effectively identify influential nodes, quantify model brittleness, detect poisoned data, and predict counterfactuals. The results further highlight that data Banzhaf and datamodels outperform graph-specific methods.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- As pointed out by the weaknesses, maybe add (1) some discussions on the limitations of the evaluation strategies and (2) some insights about dataset construction and algorithm design.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "XGxiT6ssU8", "reviewer": "Reviewer_BmWp", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strength**:\n\n1. Overall, this paper is well written and easy to follow.\n\n2. Node valuation is an interesting topic and is significant to the community.\n\n3. I admire the comprehensive experiments conducted by the authors, which I believe can well support the key claims in this paper.\n\n**Weakness**:\n\n1. What is the homophily ratio of the used datasets? Datasets with low homophily ratios, like Texas and Wisconsin [1], should also be considered to enhance the claim, especially Key Findings 1.\n\n2. Dataset information is missing in most Figures' captions and the corresponding textual content, making it hard for readers to follow the experimental settings.\n\n3. I notice that the Appendix includes a wide range of experiments, including analyses of detecting influential nodes or abnormal nodes. I suggest that authors should add corresponding notes in the main paper to facilitate the readers, especially considering the considerable \nlength of this paper.\n\n4. I agree that the method is computationally intensive, as the authors claim in Section 6, which I believe is a bottleneck for further application, especially for large graphs with millions of nodes.\n\n[1] GBK-GNN: Gated Bi-Kernel Graph Neural Networks for Modeling Both Homophily and Heterophily. WWW 2022.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors present an extensive study of data valuation methods for graph-structured data. Comprehensive experiments are conducted on a wide range of datasets, and several interesting findings are summarized. The authors also demonstrate the benefits and potential applications of computing node values, such as identifying influential nodes, quantifying model brittleness, detecting poisoned data, and others.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. What is the homophily ratio of the used datasets? Datasets with low homophily ratios, like Texas and Wisconsin [1], should also be considered to enhance the claim, especially Key Findings 1.\n\n2. Dataset information is missing in most Figures' captions and the corresponding textual content, making it hard for readers to follow the experimental settings.\n\n3. I notice that the Appendix includes a wide range of experiments, including analyses of detecting influential nodes or abnormal nodes. I suggest that authors should add corresponding notes in the main paper to facilitate the readers, especially considering the considerable \nlength of this paper.\n\n[1] GBK-GNN: Gated Bi-Kernel Graph Neural Networks for Modeling Both Homophily and Heterophily. WWW 2022.", "broader_impact_concerns": "None.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "mfSCHphrMb", "reviewer": "Reviewer_dCNo", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strengths\n1. Data valuation in graph neural networks is an interesting and under-explored area.\n2. This paper presents several experiments across several downstream tasks.\n\n---\n\n### Weaknesses\n1. The contribution of this work is limited. It can be acknowledged that the authors do not aim to propose novel techniques and instead focus on analyzing existing data valuation methods for GNNs. However, the study neither proposes an objective and measurable standard for benchmarking these methods nor provides actionable insights that could serve as concrete guidelines for future research. Therefore, despite some interesting empirical observations, the contribution is insufficient for this journal.\n\n2. Some conclusions in the Discussion section do not appear well-supported by the current experiments. For example, in the first paragraph, the authors claim that \"connectivity patterns amplify directional influence\" seems to rely solely on the observed asymmetric influence between classes. However, this asymmetry is both expected and intuitive and is commonly observed even in standard i.i.d. settings. Without additional evidence, it is difficult to conclude that connectivity patterns are the underlying cause of this phenomenon. In addition, the discussion on poisoning in the second paragraph—specifically the statement \"This redundancy suggests opportunities for potential pruning, but also vulnerabilities\"—is somewhat confusing. If nodes behave as collective units, one would expect the effect of a single poisoned node to be diluted due to redundancy. Therefore, the claim that redundancy introduces vulnerabilities requires further evidence to be convincing.\n\n\n3. The experimental settings need further clarification. For instance, in Figure 8, how are the 500 nodes selected for removal? According to Table 1, all datasets contain fewer than 200 training nodes. Does this imply that test nodes are also removed? If so, it would be problematic to compare test accuracy across different runs, as the test set is no longer consistent. \n\n\n4. The authors claim that the previous papers consider only training nodes in a transductive setting, ignoring in such a way the interactions between unlabeled and labeled nodes\" (Page 12), which is inaccurate. For example, Chen et al. explicitly consider labeled and unlabeled data, as illustrated in Figure 1 of their paper. The figure shows that unlabeled nodes, represented as smaller nodes, are included in the influence estimation process through edge removal, capturing their interactions with other nodes in the graph.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper analyzes and evaluates existing data valuation methods on graph neural networks at the node level. It summarizes the experimental results and compares the performance of different valuation methods in a semi-supervised setting.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Please clearly state the settings for each experiment in Section 4. For instance, I am a bit confused about what model/data value are used in Finding I.\n\n2. The manuscript's overall structure is clear, but the writing could be improved for better clarity.\n   1. Several long sentences should be broken up with commas to improve grammatical correctness and enhance readability.\n   2. Abbreviations like \"SGC\" should be defined upon first use.\n   3. Minor typos, such as \"alos\" in the first paragraph of page 5, should be corrected.", "broader_impact_concerns": "## Broader Impact Concerns\nThis paper compares different data valuation methods for semi-supervised graph neural networks. I did not see any concerns regarding its broader impact.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)