Graph Label Smoothing Methods

• Graph label smoothing is a regularization approach that transforms hard one-hot labels into soft, structure-aware targets using neighborhood statistics.
• It encompasses methods like SALS, ALS, and PosteL, which integrate graph-conditioned label distributions via uniform priors, learnable relevance, or posterior estimations.
• The technique extends to smoothing predicted distributions through diffusion and Laplacian regularization, effectively mitigating oversmoothing and enhancing node classification.

Graph label smoothing denotes a family of graph-based supervision and regularization methods in which labels, soft labels, class scores, or closely related label signals are modified using graph structure. In the narrow sense used in node classification, it replaces one-hot targets with graph-informed soft targets derived from neighborhood label statistics, posterior label distributions, or propagated label priors. In a broader literature, the same phrase is also used more loosely for Laplacian smoothing of class-indicator signals, diffusion of predicted labels, and smoothing of representation signals that only indirectly affect predictions. The topic therefore spans both genuine target-space methods and adjacent graph-signal formulations that should be distinguished carefully (Wang et al., 2021, Zhou et al., 2021, Heo et al., 2024, Bontonou et al., 2019).

1. Terminological scope

Standard label smoothing in classification replaces a one-hot target by a mixture with a uniform class distribution, for example

$q^{\mathrm{LS}}=(1-\epsilon)q+\epsilon u_c$

or, equivalently in node classification notation,

$$\mathbf{y}_i^{LS}=(1-\alpha)\mathbf{y}_i+\alpha \mathbf{1}/C.$$

This regularizes overconfidence, but it is graph-agnostic: the residual mass is redistributed uniformly across classes rather than according to graph structure (Wang et al., 2021, Zhou et al., 2021).

Graph label smoothing departs from this uniform construction by making the smoothing distribution depend on neighborhood labels, propagated label statistics, or graph-conditioned posteriors. SALS uses the class distribution among a node’s labeled neighbors; ALS uses propagated neighborhood labels together with a learnable class-relevance matrix; PosteL uses a posterior distribution conditioned on neighboring labels and estimated from global class priors and edgewise class-conditionals (Wang et al., 2021, Zhou et al., 2021, Heo et al., 2024).

The literature also contains several neighboring formulations that are not label smoothing in the standard neural-network sense. The graph smoothness loss of Frecon et al. minimizes the graph smoothness of class-indicator signals on a similarity graph built from network outputs, which is equivalent to maximizing distances between outputs of examples from distinct classes; it replaces cross-entropy but does not soften targets (Bontonou et al., 2019). TDSS in unsupervised graph domain adaptation smooths target-domain node representations rather than labels (Chen et al., 2024). The structurally smoothed graphlet kernel smooths distributions over graphlets rather than node labels (Yanardag et al., 2014). These distinctions are substantive, not terminological accidents.

2. Core mathematical formulations

A central formulation is Laplacian smoothness of a node-wise signal. For a weighted undirected graph with Laplacian $L=D-W$, the standard quadratic form is

$$\mathcal{Q}(L)=\mathrm{tr}(X^\top L X)=\frac{1}{2}\sum_{i,j} w_{i,j}\|x_i-x_j\|^2.$$

When $X$ is replaced by a label matrix $Y$, this becomes the standard label-smoothness penalty

$$\mathrm{tr}(Y^\top L Y)=\frac{1}{2}\sum_{i,j} w_{i,j}\|y_i-y_j\|^2,$$

which encodes the homophily-style prior that connected nodes should have similar labels or class scores (Nobre et al., 2021).

A second canonical formulation is Tikhonov graph smoothing. Given a graph signal $y\in\mathbb{R}^n$, one solves

$$\hat{x} = \argmin_{z\in\mathbb{R}^n} q \left\| y-z \right\|^2 + z^\top L z,$$

with closed-form solution

$$\hat{x} = K y \quad\text{with}\quad K=q(qI+L)^{-1}.$$

In semi-supervised learning, the signal $$\mathbf{y}_i^{LS}=(1-\alpha)\mathbf{y}_i+\alpha \mathbf{1}/C.$$0 may be a class-indicator vector or a class-specific seed-label function, so the smoother acts directly on label signals (Pilavci et al., 2019).

A third formulation treats smoothing as repeated diffusion. For mean aggregation with row-stochastic operator $$\mathbf{y}_i^{LS}=(1-\alpha)\mathbf{y}_i+\alpha \mathbf{1}/C.$$1,

$$\mathbf{y}_i^{LS}=(1-\alpha)\mathbf{y}_i+\alpha \mathbf{1}/C.$$2

This linear message-passing rule captures the finite-step denoising behavior underlying many GNNs, but on connected graphs it also yields oversmoothing: $$\mathbf{y}_i^{LS}=(1-\alpha)\mathbf{y}_i+\alpha \mathbf{1}/C.$$3 Hence sufficiently many smoothing steps collapse node-specific information (Keriven, 2022).

A fourth formulation uses class-indicator graph signals explicitly. In graph smoothness loss, the label signal for class $$\mathbf{y}_i^{LS}=(1-\alpha)\mathbf{y}_i+\alpha \mathbf{1}/C.$$4 is a binary indicator $$\mathbf{y}_i^{LS}=(1-\alpha)\mathbf{y}_i+\alpha \mathbf{1}/C.$$5, and the training objective is

$$\mathbf{y}_i^{LS}=(1-\alpha)\mathbf{y}_i+\alpha \mathbf{1}/C.$$6

Because only cross-class edges contribute, minimizing this objective pushes apart outputs of examples from distinct classes while leaving within-class distances unconstrained (Bontonou et al., 2019).

3. Graph-aware soft-target methods for node classification

The most direct strand of graph label smoothing operates in target space: one replaces a hard node label by a node-specific soft distribution computed from graph structure.

Method	Smoothed object	Core construction
SALS (Wang et al., 2021)	Training targets for labeled nodes	Mixture of one-hot target, neighborhood label distribution, and uniform prior
ALS (Zhou et al., 2021)	Training targets in scalable graph training	Propagated labels, learnable class-relevance matrix, and smooth pacing
PosteL (Heo et al., 2024)	Transductive node targets	Posterior $$\mathbf{y}_i^{LS}=(1-\alpha)\mathbf{y}_i+\alpha \mathbf{1}/C.$$7 with global priors and edgewise conditionals

SALS defines, for node $$\mathbf{y}_i^{LS}=(1-\alpha)\mathbf{y}_i+\alpha \mathbf{1}/C.$$8, the neighborhood class ratio

$$\mathbf{y}_i^{LS}=(1-\alpha)\mathbf{y}_i+\alpha \mathbf{1}/C.$$9

the neighborhood label distribution

$L=D-W$0

and the structure-aware target

$L=D-W$1

It changes only the training targets, introduces no extra inference cost, and is applied to GCN, ResGCN, GraphSAGE, GraphSAINT, and SIGN. On the reported transductive benchmarks, GCN + SALS improves from $L=D-W$2 to $L=D-W$3 on Cora, from $L=D-W$4 to $L=D-W$5 on Pubmed, and from $L=D-W$6 to $L=D-W$7 on CoraFull (Wang et al., 2021).

ALS is designed for large-scale graph training with partitioning or subgraph sampling, where mini-batches exhibit label bias. It first propagates training labels by

$L=D-W$8

then refines the propagated label prior with a learnable class-relevance matrix,

$L=D-W$9

and finally constructs

$$\mathcal{Q}(L)=\mathrm{tr}(X^\top L X)=\frac{1}{2}\sum_{i,j} w_{i,j}\|x_i-x_j\|^2.$$0

The full loss includes both cross-entropy against $$\mathcal{Q}(L)=\mathrm{tr}(X^\top L X)=\frac{1}{2}\sum_{i,j} w_{i,j}\|x_i-x_j\|^2.$$1 and a KL term that prevents the learned soft target from degenerating. ALS is reported as best in $$\mathcal{Q}(L)=\mathrm{tr}(X^\top L X)=\frac{1}{2}\sum_{i,j} w_{i,j}\|x_i-x_j\|^2.$$2 out of $$\mathcal{Q}(L)=\mathrm{tr}(X^\top L X)=\frac{1}{2}\sum_{i,j} w_{i,j}\|x_i-x_j\|^2.$$3 backbone-dataset combinations; for example, GraphSAINT on ogbn-products improves from $$\mathcal{Q}(L)=\mathrm{tr}(X^\top L X)=\frac{1}{2}\sum_{i,j} w_{i,j}\|x_i-x_j\|^2.$$4 to $$\mathcal{Q}(L)=\mathrm{tr}(X^\top L X)=\frac{1}{2}\sum_{i,j} w_{i,j}\|x_i-x_j\|^2.$$5, and GraphSAGE on ogbn-mag improves from $$\mathcal{Q}(L)=\mathrm{tr}(X^\top L X)=\frac{1}{2}\sum_{i,j} w_{i,j}\|x_i-x_j\|^2.$$6 to $$\mathcal{Q}(L)=\mathrm{tr}(X^\top L X)=\frac{1}{2}\sum_{i,j} w_{i,j}\|x_i-x_j\|^2.$$7 (Zhou et al., 2021).

PosteL replaces uniform smoothing by a posterior target conditioned on neighboring labels: $$\mathcal{Q}(L)=\mathrm{tr}(X^\top L X)=\frac{1}{2}\sum_{i,j} w_{i,j}\|x_i-x_j\|^2.$$8 The likelihood is factorized by conditional independence over neighbors, and the edgewise conditional label distributions are estimated empirically from labeled edges. A uniform-noise term is then added, and the final soft label is

$$\mathcal{Q}(L)=\mathrm{tr}(X^\top L X)=\frac{1}{2}\sum_{i,j} w_{i,j}\|x_i-x_j\|^2.$$9

PosteL is architecture-agnostic, applied to MLP, GCN, GAT, APPNP, ChebNet, GPR-GNN, and BernNet, and improves the hard-label baseline in $X$0 out of $X$1 model-dataset settings. The reported GCN gains include $X$2 on Cornell, $X$3 on Chameleon, and $X$4 on Squirrel (Heo et al., 2024).

4. Propagation-based smoothing of predictions and label signals

A second strand smooths predicted label distributions or class-score signals rather than the supervision targets themselves. In NDLS-L, the smoothed object is the soft predicted label distribution $X$5. Standard propagation is

$X$6

but NDLS replaces a graph-wide fixed depth by a node-specific local smoothing iteration

$X$7

where $X$8 and $X$9. The final node-wise smoothed prediction is

$Y$0

This post-processing view yields large improvements for MLP baselines; on Cora, MLP improves from $Y$1 to $Y$2, and on ogbn-papers100M from $Y$3 to $Y$4 (Zhang et al., 2021).

Elastic GNN reframes graph propagation as graph denoising with an elastic combination of $Y$5- and $Y$6-based smoothness: $Y$7 The resulting Elastic Message Passing scheme smooths class-related signals while preserving local discontinuities. Relative to APPNP, it increases the ratio of average difference on wrong edges to average difference on correct edges from $Y$8 to $Y$9 on Cora, from $$\mathrm{tr}(Y^\top L Y)=\frac{1}{2}\sum_{i,j} w_{i,j}\|y_i-y_j\|^2,$$0 to $$\mathrm{tr}(Y^\top L Y)=\frac{1}{2}\sum_{i,j} w_{i,j}\|y_i-y_j\|^2,$$1 on CiteSeer, and from $$\mathrm{tr}(Y^\top L Y)=\frac{1}{2}\sum_{i,j} w_{i,j}\|y_i-y_j\|^2,$$2 to $$\mathrm{tr}(Y^\top L Y)=\frac{1}{2}\sum_{i,j} w_{i,j}\|y_i-y_j\|^2,$$3 on PubMed, indicating stronger within-class smoothing and less blurring across class boundaries (Liu et al., 2021).

Random-spanning-forest methods provide a computational route to exact Laplacian smoothing. For the Tikhonov smoother $$\mathrm{tr}(Y^\top L Y)=\frac{1}{2}\sum_{i,j} w_{i,j}\|y_i-y_j\|^2,$$4, the identity

$$\mathrm{tr}(Y^\top L Y)=\frac{1}{2}\sum_{i,j} w_{i,j}\|y_i-y_j\|^2,$$5

links the smoothing matrix to root-location probabilities in a random spanning forest. This yields unbiased Monte Carlo estimators for the smoothed signal, including the tree-average estimator

$$\mathrm{tr}(Y^\top L Y)=\frac{1}{2}\sum_{i,j} w_{i,j}\|y_i-y_j\|^2,$$6

In semi-supervised learning, the class labels are encoded as one scalar signal per class, smoothed classwise, and the final class is chosen by $$\mathrm{tr}(Y^\top L Y)=\frac{1}{2}\sum_{i,j} w_{i,j}\|y_i-y_j\|^2,$$7 (Pilavci et al., 2019).

5. Adjacent literatures often conflated with graph label smoothing

Several important papers use graph smoothness of label-related signals but are not target-smoothing methods. The graph smoothness loss of Frecon et al. builds a $$\mathrm{tr}(Y^\top L Y)=\frac{1}{2}\sum_{i,j} w_{i,j}\|y_i-y_j\|^2,$$8-nearest-neighbor graph on the current network outputs, defines one binary class-indicator signal per class, and minimizes

$$\mathrm{tr}(Y^\top L Y)=\frac{1}{2}\sum_{i,j} w_{i,j}\|y_i-y_j\|^2,$$9

Because only edges connecting different classes contribute, the loss encourages large between-class distances but does not explicitly encourage small within-class distances. The paper is explicit that this is not standard label smoothing and that the training “does not prevent inputs from the same class to be mapped to distant locations in the output domain” (Bontonou et al., 2019).

TDSS in unsupervised graph domain adaptation is likewise not label smoothing. Its regularizer,

$y\in\mathbb{R}^n$0

acts on target-domain node representations over sampled neighborhoods. The intended effect is local consistency of target representations under structural shift, not smoothing of one-hot labels or logits (Chen et al., 2024).

The structurally smoothed graphlet kernel smooths multinomial distributions over graphlets. Its core objects are graphlet counts $y\in\mathbb{R}^n$1, the normalized graphlet histogram

$y\in\mathbb{R}^n$2

and a Kneser-Ney-style redistribution of probability mass across graphlets via a DAG relating size-$y\in\mathbb{R}^n$3 and size-$y\in\mathbb{R}^n$4 graphlets. The paper did not use edge or node labels in the experiments (Yanardag et al., 2014).

Distributed graph learning with smooth data priors addresses a different problem: learning the graph itself from signals assumed smooth on the unknown graph. Its smoothness objective is again

$y\in\mathbb{R}^n$5

If one sets $y\in\mathbb{R}^n$6, this becomes a label-smoothness prior, but the paper itself studies unsupervised graph structure inference rather than label smoothing or node classification (Nobre et al., 2021).

A related Bayesian literature smooths node-wise latent signals by placing priors built from powers of $y\in\mathbb{R}^n$7. In “Optimal Bayesian Smoothing of Functional Observations over a Large Graph,” the latent quantity is a function at each node rather than a class label, but the mechanism remains Laplacian spectral shrinkage of high graph frequencies (Roy et al., 2021).

6. Theory, operating regimes, and limitations

The dominant theoretical theme is that smoothing is beneficial only at finite depth. For mean aggregation, repeated graph averaging initially denoises by shrinking non-principal directions faster than principal ones in regression, and by shrinking nodes within communities faster than the communities collapse together in classification. Yet for connected graphs, the same process converges to a non-informative constant profile. The paper “Not too little, not too much” therefore proves both the existence of a useful finite $y\in\mathbb{R}^n$8 in specific models and the inevitability of oversmoothing in the limit (Keriven, 2022).

This finite-step perspective is consistent with node-adaptive propagation schemes. NDLS makes smoothing depth node-specific because high-degree nodes in dense regions smooth quickly, whereas low-degree nodes in sparse regions need more steps. The locality bound

$y\in\mathbb{R}^n$9

formalizes that useful smoothing depth is driven by local graph structure rather than a single global propagation budget (Zhang et al., 2021).

Direct target-space methods also have explicit operating assumptions. SALS relies on labeled neighbors in the training set and uses only their class histogram, so its neighborhood signal weakens when labeled neighbors are scarce (Wang et al., 2021). PosteL assumes that the conditional distributions $$\hat{x} = \argmin_{z\in\mathbb{R}^n} q \left\| y-z \right\|^2 + z^\top L z,$$0 vary meaningfully across classes; the paper identifies Actor as a case where these conditionals are relatively similar and gains are smaller (Heo et al., 2024). ALS is motivated by positive label correlation and mini-batch label bias in scalable graph training, so its strongest benefits appear in partitioned or sampled large-graph regimes (Zhou et al., 2021).

The broader smoothing literature supplies additional statistical structure. Random-spanning-forest estimators are unbiased for the exact Laplacian smoother, and the tree-average estimator has variance

$$\hat{x} = \argmin_{z\in\mathbb{R}^n} q \left\| y-z \right\|^2 + z^\top L z,$$1

which is lower than that of the root-value estimator (Pilavci et al., 2019). In the Bayesian functional setting, the graph-Laplacian prior attains the minimax rate

$$\hat{x} = \argmin_{z\in\mathbb{R}^n} q \left\| y-z \right\|^2 + z^\top L z,$$2

and a random-series prior achieves the minimax rate up to a logarithmic factor, while credible regions of optimal order can have arbitrarily high frequentist coverage in the fixed-smoothness setting (Roy et al., 2021).

Taken together, these results delimit the topic precisely. Graph label smoothing is not a single method but a spectrum of constructions ranging from graph-aware soft targets, through propagation of predicted label distributions, to Laplacian regularization of class-indicator signals. What unifies them is the assumption that label-related information should vary more smoothly over a graph than one-hot supervision alone would suggest. What separates them is the object being smoothed: targets, predictions, class-indicator signals, representations, graphlet distributions, or even the graph structure itself.