Interpretable structural temperature scaling for calibrating graph neural networks and point-cloud segmentation

Chenyi Huang

cs.LGOct 30, 2026

Interpretable structural temperature scaling for calibrating graph neural networks and point-cloud segmentation

Authors: Chenyi Huang

Organizations: Department of Computer Science, University College London, Gower Street, London, WC1E 6BT, United Kingdom

Abstract

Graph-aware calibrators make graph neural networks (GNNs) more trustworthy by predicting a temperature for each node, usually with auxiliary networks whose decisions are hard to inspect. We propose Interpretable Structural Temperature Scaling (ISTS), a twelve-parameter linear model that sets each node's temperature from named descriptors of its confidence, neighbourhood and closeness to labelled nodes, so every temperature traces back to descriptor contributions. ISTS preserves predictions, and cross-validated regularisation lets the calibration data decide how far the model departs from temperature scaling (TS). Its explanations are faithful at the level of node groups: on test nodes, ISTS softens groups that a global temperature leaves over-confident and sharpens under-confident ones (Spearman correlation 0.67 over 246 groups). On seven graphs with two backbones and eleven baselines, ISTS has the best average rank in negative log-likelihood (NLL) and Brier score and lowers NLL relative to TS by 1.8%, more than twice the gain of an identically fitted entropy-based temperature. It ranked first again on four held-out graphs under a protocol fixed in advance. From about 100 calibration nodes on, ISTS is a cheap and inspectable choice, and on point-cloud segmentation it significantly improves NLL, Brier score and the ranking of points for rejection.

Figures & tables

Symbol	Descriptor	Definition
$b$	bias	TS temperature level when $\mathbf{w}=\mathbf{0}$
$\phi_{1}$	confidence	$c_{i}$
$\phi_{2}$	entropy	$H(\mathbf{p}_{i})/\log K$
$\phi_{3}$	agreement	$\frac{1}{d_{i}}\sum_{j\in\mathcal{N}(i)}\mathbb{1}[\hat{y}_{j}=\hat{y}_{i}]$
$\phi_{4}$	relative confidence	$c_{i}-\frac{1}{d_{i}}\sum_{j\in\mathcal{N}(i)}c_{j}$
$\phi_{5}$	similarity	$\frac{1}{d_{i}}\sum_{j\in\mathcal{N}(i)}\cos(\mathbf{p}_{i},\mathbf{p}_{j})$

Table 1: The 11+1 parameters of ISTS: the bias

b

and one weight

w_{j}

per descriptor

\phi_{j}

\delta_{i}

: hop distance to the nearest training node.

Dataset

Model

Uncal.

ETS

HTS

LTS

RBS

CaGCN

GATS

GETS

HoTS

WATS

ISTS

Cora

GCN

62.9

\pm

5.2

56.4

\pm

3.1

57.6

\pm

4.6

55.9

\pm

2.9

55.8

\pm

3.1

56.0

\pm

2.9

57.6

\pm

3.8

65.4

\pm

6.7

55.8

\pm

3.2

65.0

\pm

6.4

54.3

\pm

3.2

56.1

\pm

2.9

55.8

\pm

2.7

Citeseer

GCN

91.8

\pm

8.2

87.6

\pm

3.0

87.3

\pm

2.4

87.0

\pm

2.8

87.2

\pm

2.8

87.2

\pm

2.8

90.0

\pm

7.4

94.5

\pm

5.7

86.7

\pm

2.9

91.6

\pm

4.6

87.2

\pm

3.3

87.4

\pm

2.7

87.1

\pm

2.8

Pubmed

GCN

39.5

\pm

1.2

36.7

\pm

0.6

36.8

\pm

0.6

36.5

\pm

0.7

36.5

\pm

0.7

36.4

\pm

0.8

37.0

\pm

1.0

35.8

\pm

0.6

36.4

\pm

0.6

36.9

\pm

0.6

36.5

\pm

0.6

36.7

\pm

0.6

35.9

\pm

0.7

Computers

GCN

42.8

\pm

1.4

42.8

\pm

1.4

42.8

\pm

1.4

40.5

\pm

1.4

41.4

\pm

1.9

42.3

\pm

2.0

40.4

\pm

1.4

43.2

\pm

3.2

42.2

\pm

1.7

51.7

\pm

7.3

41.9

\pm

1.3

37.6

\pm

1.4

38.5

\pm

1.4

Photo

GCN

28.6

\pm

1.0

28.8

\pm

1.1

28.9

\pm

0.8

27.0

\pm

1.1

28.6

\pm

1.2

28.7

\pm

1.3

27.6

\pm

1.3

40.4

\pm

10.0

28.5

\pm

1.1

44.4

\pm

13.9

28.4

\pm

1.1

26.4

\pm

2.5

27.8

\pm

2.3

GCN

22.1

\pm

1.1

21.4

\pm

0.4

21.7

\pm

0.4

21.4

\pm

0.4

21.4

\pm

0.4

21.4

\pm

0.4

21.5

\pm

0.4

26.4

\pm

6.0

21.4

\pm

0.4

22.7

\pm

0.8

21.2

\pm

0.4

21.4

\pm

0.4

21.4

\pm

0.4

Table 2: Test NLL (

\times 100

; mean

\pm

std over 15 runs) after post-hoc calibration. ECE and Brier score are in Tables S1 and S2 of the supplementary material. Lower is better; best calibrated result in bold and underlined (Uncal. excluded). RBS follows the original definition (equal-width bins); HTS and LTS are fitted with the same pipeline as ISTS. VS and GETS can change predictions; all other calibrators preserve accuracy.

	avg. rank			time (s)	params
Method	ECE	NLL	Brier	Cora/Ph.	Cora
TS	8.14	8.64	7.29	0.1/0.2	1
VS	8.29	6.86	5.86	0.7/0.3	14
ETS	4.71	4.71	6.93	0.1/0.2	4
HTS	4.64	5.07	6.00	0.2/0.4	2
LTS	6.57	5.86	5.43	0.3/0.5	8

Table 3: Average rank over the 14 settings (lower is better), mean fitting time in seconds on one CPU thread (Cora / Physics, averaged over backbones) and number of fitted parameters on Cora. Split-level W/M/L counts against every baseline are in Table S5 of the supplementary material.

Variant	ECE (%)	$\Delta$ NLL vs. TS (%)
full (11 descriptors)	1.89	$-1.80$
confidence only	2.02	$-0.63$
w/o neighbourhood	1.89	$-1.56$
w/o LP group	1.90	$-1.69$
w/o distance	1.94	$-1.73$
TS (no descriptors)	2.14	$+0.00$

Table 4: Descriptor ablation, averaged over the 14 settings: test ECE and relative change in test NLL with respect to TS (negative is better). All variants use the same fitting pipeline. Further variants and HTS/LTS: Table S3 of the supplementary material.

	avg. rank			$\Delta$ NLL
Method	NLL	Brier	ECE	vs. TS (%)
ISTS	1.17	1.50	8.17	$-5.23$
VS	5.00	3.67	9.00	$-2.43$
LTS	5.67	4.67	5.50	$-2.12$
HTS	6.00	6.00	4.33	$-1.54$
HoTS	6.67	7.17	7.17	$-0.68$

Table 5: Independent confirmation on CoraFull, DBLP and WikiCS (six settings, 15 runs each), run under a protocol fixed before any result was seen: average rank among the 13 calibrators and mean relative NLL change with respect to TS. All methods: Table S17.

Figure 1: Faithfulness of ISTS: per setting and group of test nodes (grouped by one descriptor), the gap left by one global temperature (confidence minus accuracy) against ISTS’s mean change of log temperature. Points in the upper-right (over-confident, softened) and lower-left (under-confident, sharpened) quadrants are corrected in the right direction.

	original				label smoothing
Method	ECE	NLL	Brier	AURC	ECE	NLL	Brier	AURC
Uncal.	0.64	16.89	9.13	0.773	1.03	18.15	9.41	0.868
TS	0.26 ↑	16.83 ↓	9.12 ↓	0.773	0.79 ↓	17.86 ↓	9.45 ↓	0.862
VS	0.45 ↓	18.51 ↓	9.34 ↓	0.885	0.93 ↓	19.07 ↓	9.72 ↓	1.090
ETS	0.26 ↑	16.83 ↓	9.12 ↓	0.773	0.79 ↓	17.86 ↓	9.45 ↓	0.862
HTS	0.26 ↑	16.83 ↓	9.12 ↓	0.773	0.42 ↑	17.72 ↓	9.41 ↓	0.856

Table 6: Per-point calibration of DGCNN on the 2874 ShapeNetPart test shapes (5.8M points), trained without (“original”) or with label smoothing (

\epsilon=0.2

). Calibrators are fitted on 400 validation shapes; mean over 3 backbones

\times

5 calibration draws. ECE and AURC in %, NLL and Brier

\times 100

. ↓ / ↑ : ISTS better/worse, 95% shape-level paired bootstrap interval of the difference excludes zero. RBS follows the original definition; HTS and LTS use the ISTS fitting pipeline. VS changes predictions; all others preserve accuracy.

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Model calibration aligns predicted confidence or model outputs with observed outcomes and measurements to support more reliable decisions.

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HoTS: Homophily-Aware Temperature Scaling for Graph Neural Network Calibration

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UNIST · Pusan National University · UNIST LinqAlpha

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Open paper AskPDF arXiv

For graph node classification, calibrated class probabilities are needed when confidence scores, usually the maximum predicted class probability, are used to rank predictions, defer uncertain nodes to human review, or control risk. Existing post-hoc calibrators either apply one global temperature or use graph-aware modules without a principled structural form. We study how local graph structure should enter node-level calibration. Our first results show that a logit-only temperature rule is insufficient when nodes with identical logits but different local homophily require different optimal temperatures. We then analyze a population-concentration contextual stochastic block model with Gaussian features and a one-layer linear GCN. Under equidistant class means, the Bayes posterior over class-template scores is a temperature-scaled softmax whose inverse-temperature is governed by a homophily-dependent signal strength. In the positive-signal homophilic regime, the resulting temperature decreases approximately inversely with normalized local homophily. This law motivates Homophily-aware Temperature Scaling (HoTS), a simple post-hoc calibrator that assigns each node a positive scalar temperature from entropy-based logit concentration and estimated local homophily. HoTS has three temperature parameters, preserves the predicted class, and learns the strength of the structural correction from calibration data. Across 18 node-classification benchmarks, two GNN backbones, and eight calibration baselines, HoTS achieves the best mean Expected Calibration Error (ECE) of 4.79%, the best average rank, and the most reliable confidence ranking in selective classification. Code is available at https://github.com/inu0104/HoTS.

Abstract

Model Calibration

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