A Diagnostic Protocol for GNNs in Electricity Load Forecasting: Separating Parameter Sharing from Neighborhood Access

Eloi Campagne; Yvenn Amara-Ouali; Yannig Goude; Itai Zehavi; Argyris Kalogeratos

cs.AIOct 26, 2026

A Diagnostic Protocol for GNNs in Electricity Load Forecasting: Separating Parameter Sharing from Neighborhood Access

Authors: Eloi Campagne, Yvenn Amara-Ouali, Yannig Goude, Itai Zehavi, Argyris Kalogeratos

Organizations: CB · LMO

Abstract

A graph neural network (GNN) forecasting a collection of load series combines two mechanisms, sharing parameters across series and reading neighboring series, and a gain over a non-graph model may come from either; telling them apart decides whether a graph is worth maintaining. We propose a model diagnostic protocol that compares the GNN with three reference models: a per-node network, a network shared across nodes without neighbors, and the same GNN without edges. Degree-preserving rewiring tests whether neighbor identity matters. After a tuned benchmark on regional load in France and on low-voltage feeders in the UK, we apply the protocol to deliberately short node histories, as when non-stationary demand leaves only recent observations relevant: the feeders, within one week, and a public dataset of client consumption series. On the feeders, sharing parameters recovers most of the error gap between node-specific models and GNNs, about two thirds against a regularized one. Adding the edges to an otherwise identical backbone brings a smaller gain, half or more of which disappears under rewiring. On the client series, the neighbor mean degrades the per-node model on the shorter window and the edges' gain is not consistently distinguishable from zero; on the regions of France, where each node's own history is already informative, the ranking between per-node models and GNNs changes with tuning. A synthetic study points to noisy local measurements as one regime where neighbors help. In an exploratory online aggregation, graph and non-graph forecasters appear complementary, and the weight assigned to GNNs is not proportional to their contribution. In none of the three datasets do the edges account for most of a GNN's advantage over node-specific models, so a forecasting gain should be credited to the graph only after comparison with per-node and shared references.

Figures & tables

Panel of Figure 1 — Figure 1: The four node-wise models on the sharing (rows) and access (columns) axes. Each cell shows the series reaching the forecast $\hat{y}_{3}$ , the matrix producing it and its parameter count for $N$ nodes, $d$ features, and $H$ hidden units; each segment is one $d\times H$ block.

Block	Datasets	Training window	Models	Fixed or distinctive settings
Architecture benchmark (§ 6.1 ; Q1)	regions-fr ; weave-uk	$1090$ d; $9$ d	$8$ tuned GNNs, non-graph baselines, GNN and shared-MLP seed ensembles, and GNN mixtures	Split and horizon within each dataset; inputs as specified
Per-node reference (§ 6.2 ; Q1, Q4)	regions-fr	$54$ – $1090$ d	Per-node MLP ( $\pm$ neighbor mean) and all-node MLP; identity-graph and SAGE GNN	SAGE and hidden width
Synthetic regimes (§ 6.3 ; Q4)	Synthetic denoising data	Training fraction varied	Same 4 node-wise models	Base generator, one factor varied
Larger datasets (§ 6.4 ; Q2–Q4)	electricity ; weave-uk	$11$ / $60$ d; $30$ h/ $10$ d	Per-node, shared, all-node MLP; per-node and shared MLP $+$ neighbors; SAGE/ChebConv GNN, empty-graph GNN, rewired graphs	Hidden width $64$
Complementarity (§ 6.5 ; exploratory)	regions-fr ; synthetic data	Jan $2017$ –Jun $2018$	$6$ GNNs, feedforward network, additive model, ML-Poly mixture	Own split, $1$ seed per expert

Table 1: The experimental blocks presented in Section 6 that address the questions of Section 1 .

Table 3

Figure 2: regions-fr . Left: national RMSE of the

4

node-wise models at one fixed configuration (width

64

, untuned) as the training set contracts,

3

seeds (Table C.2 , Appendix). Right: at the full sample, each family at the fixed configuration (hollow) and tuned (filled, Table 3 ), beside the tuned pooled XGBoost and the bottom-up graph mixture.

Figure 3: RMSE over nodes and test snapshots, in standardized load units, of the

4

node-wise models on the synthetic denoising data,

3

seeds per point with the spread shaded; each plot varies one factor around the base configuration.

	electricity , $319$ clients (RMSE in MW)			weave-uk , $500$ feeders (RMSE in kWh)
Model	RMSE	MAPE (%)	nRMSE	RMSE	MAPE (%)	nRMSE
Short window: $11$ days ( electricity ), $30$ hours ( weave-uk )
Per-node MLP	$22.3$ ( $0.7$ )	$2.34$ ( $0.11$ )	$0.3030$ ( $0.0002$ )	$346.8$ ( $3.8$ )	$9.73$ ( $0.14$ )	$1.0509$ ( $0.0013$ )
Per-node MLP $+$ neighbors	$24.1$ ( $2.6$ )	$2.72$ ( $0.34$ )	$0.3201$ ( $0.0036$ )	$323.2$ ( $2.0$ )	$8.89$ ( $0.08$ )	$0.9995$ ( $0.0010$ )
Shared MLP	$19.4$ ( $1.3$ )	$2.10$ ( $0.15$ )	$0.2804$ ( $0.0014$ )	$329.7$ ( $11.0$ )	$9.02$ ( $0.31$ )	$0.9074$ ( $0.0103$ )
All-node MLP	$26.0$ ( $2.4$ )	$2.89$ ( $0.23$ )	$0.4527$ ( $0.0048$ )	$421.8$ ( $82.3$ )	$12.45$ ( $2.75$ )	$1.0609$ ( $0.0583$ )

Table 4: The node-wise models at hidden width

64

on the full datasets,

3

seeds, under the two system scores and the per-node error (nRMSE). The GNN uses ChebConv (

K=3

) on electricity at

60

days and weave-uk at

30

hours, SAGE otherwise, as selected on validation. Equal width does not match depth or total parameter count. weave-uk window lengths include the validation portion; electricity lengths are fitting windows, followed by

2

validation days.

Panel of Figure 4 — Figure 4: The two decompositions of the per-node gap on the $60$ -day electricity window (blue) and the $30$ -hour weave-uk window (green) of Table 4 . Vertices give per-node errors; each arrow points to the better model and gives the error decrease. Both routes reach the same total, with increments differing by $0.0248$ on weave-uk ; arrows into the GNN also change the backbone, so this difference is not a pure factorial interaction.

Appendix figures & tables23 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 8 — Figure A.1: Example of a message-passing layer in a GNN. $V_{\ell}$ , $E_{\ell}$ , and $U_{\ell}$ respectively refer to node, edge, and global level at layer $\ell$ . $\phi$ and $\rho$ are respectively update and propagation functions.

Figure B.1: Weights ML-Poly plays over the test year $2019$ . The two left columns are regions-fr and the synthetic dataset whose residual covariance follows the spatial graph (unobservable by the experts); the third has none, its residuals being independent across nodes. Single-step forecasting is above and sequence-to-sequence forecasting below. The $6$ GNN experts are summed into one plum band, the feedforward network appearing in blue and the additive model in orange. The dashed line marks $0.75$ , the share the $6$ hold among the $8$ by count alone, so a band sitting on it reports an allocation that cannot separate its experts and not an accuracy the edges bought. The strip beneath each plot carries the departure from that share, plum where the mixture buys the GNN experts more than counting them would give and gray where it buys them less.

Dataset	Setting	Best expert	Uniform	ML-Poly	Without the GNN experts	Without the two others
regions-fr	single-step	$1249$	$1144$	$970$	$+132$ ( $14\%$ )	$+117$ ( $12\%$ )
regions-fr	seq-to-seq	$1047$	$893$	$844$	$+162$ ( $19\%$ )	$+90$ ( $11\%$ )
Synthetic structured	single-step	$917$	$838$	$799$	$+79$ ( $10\%$ )	$+43$ ( $5\%$ )
Synthetic structured	seq-to-seq	$800$	$784$	$766$	$+34$ ( $4\%$ )	$+1$ ( $0\%$ )
Synthetic independent	single-step	$658$	$656$	$624$	$+23$ ( $4\%$ )	$+24$ ( $4\%$ )
Synthetic independent	seq-to-seq	$604$	$576$	$574$	$+19$ ( $3\%$ )	$+4$ ( $1\%$ )

Appendix

Table B.1: National root mean squared error in MW over the test year, and what each family of experts is worth inside the mixture. The

4

cells above the rule have spatially correlated loads, the two below independent ones given their own covariates; ‘seq-to-seq’ abbreviates the sequence-to-sequence setting. The last two columns give the error ML-Poly loses when the

6

GNNs are withdrawn from the pool, and when the feedforward and the additive model are, in MW and as a share of the mixture’s own error. The datasets sit at different load levels, so the shares compare across rows where the MW values do not.

Figure B.2: Mean absolute percentage error of each region’s own forecast over the test year, single-step setting. Left, GatedGraph, the most accurate of the $6$ GNN experts region by region; center, the additive model; right, ML-Poly run region by region, each region carrying its own mixture and its own weights. The color scale is shared. Regions carry their official codes: ARA Auvergne-Rhône-Alpes, BFC Bourgogne-Franche-Comté, BRE Bretagne, CVL Centre-Val de Loire, GES Grand Est, HDF Hauts-de-France, IDF Île-de-France, NAQ Nouvelle-Aquitaine, NOR Normandie, OCC Occitanie, PAC Provence-Alpes-Côte d’Azur, PDL Pays de la Loire.

Panel of Figure 12 — Figure C.1: Sequence-to-sequence evaluation protocol ( $48$ half-hourly steps, stride $h=48$ ). Input windows $i_{k}$ (blue) each feed a forecast window $f_{k}$ (orange), and consecutive forecast windows are contiguous, so each prediction spans $w=48$ lags into $h=48$ future steps. Training, validation and testing all use this stride, so every forecast origin falls on a day boundary and output $k$ always denotes slot $k$ of the day. A stride of one would place the origin at an arbitrary half-hour and force a single $48$ -output head to represent $48$ phase-shifted mappings, which is a different learning problem and a different test set; every model of the benchmark, graph or not, is trained and scored under the stride of this figure; Section 6.4 states its own protocol.

Panel of Figure 13 — Figure C.2: Temporal splits for training, validation, and testing on regions-fr and weave-uk .

Features	Best	Best to worst
Calendar and weather	$1141$	$586$
$+$ temperatures of the target slots	$1133$	$367$
$+$ 48 load lags	$\mathbf{865}$	$120$
Both blocks	$\mathbf{768}$	$119$

Appendix

Table C.1: Validation RMSE in MW on regions-fr , by feature set. The middle column is the best of

18

GNN configurations (

6

convolutions at three widths), and the right column is the gap between the best and the worst of those. It uses

1

seed, so the column reads as a screening pass and not as a ranking.

Training days	Per-node MLP	All-node MLP	Identity-graph GNN	SAGE GNN
$54$ ( $5\%$ )	$1948$ ( $69$ )	$2432$ ( $269$ )	$2038$ ( $103$ )	$2080$ ( $112$ )
$109$ ( $10\%$ )	$1412$ ( $28$ )	$1687$ ( $105$ )	$1763$ ( $101$ )	$1738$ ( $55$ )
$272$ ( $25\%$ )	$1090$ ( $41$ )	$1179$ ( $25$ )	$1292$ ( $17$ )	$1206$ ( $41$ )
$545$ ( $50\%$ )	$958$ ( $5$ )	$1014$ ( $29$ )	$1113$ ( $31$ )	$983$ ( $27$ )
$1090$ ( $100\%$ )	$876$ ( $3$ )	$921$ ( $3$ )	$918$ ( $7$ )	$891$ ( $19$ )

Appendix

Table C.2: National RMSE (MW) on regions-fr as the training set contracts, SAGE convolution, days drawn at random,

3

seeds.

Figure C.3: Table C.2 read against the training window, on a logarithmic day axis. Left: the national error of the

4

node-wise models, the per-node MLP leading at every volume. Right: the same table as increments over that reference model, so a curve is the price of one restriction and a higher value is a costlier one. The access restriction falls from

484

45

MW as the window opens and the edge cost from

132

15

MW; weight sharing costs

42

–

351

MW with no monotone trend.

Window	Nodes	MLP			GNN
Window	Nodes	Per-node	Shared	All-node	SAGE	ChebConv
Per-node error, nRMSE
$30$ h	$10$	$1.0790$ ( $0.0194$ )	$0.9846$ ( $0.0578$ )	$0.9333$ ( $0.0135$ )	$0.9444$ ( $0.0170$ )	$0.9632$ ( $0.0049$ )
$30$ h	$40$	$1.0216$ ( $0.0161$ )	$0.8746$ ( $0.0298$ )	$0.8952$ ( $0.0100$ )	$0.8719$ ( $0.0405$ )	$0.8637$ ( $0.0304$ )
$30$ h	$160$	$1.0733$ ( $0.0190$ )	$0.9315$ ( $0.0302$ )	$0.9402$ ( $0.0244$ )	$0.8949$ ( $0.0256$ )	$0.8994$ ( $0.0164$ )
$30$ h	$500$	$1.0509$ ( $0.0013$ )	$0.9074$ ( $0.0103$ )	$1.0609$ ( $0.0583$ )	$0.8760$ ( $0.0015$ )	$0.8808$ ( $0.0047$ )

Appendix

Table E.1: weave-uk as the dataset is thinned, three draws of nodes and

1

seed each, the same

5

-nearest graph rebuilt inside every draw. The per-node error is computed after standardizing targets separately by node and horizon using training moments, then taking the root mean squared error over test origins and horizons and averaging over nodes. A value of

1

denotes an error on the training-standard-deviation scale, not a test-set mean-forecast baseline. This score and MAPE compare across node counts, whereas a raw RMSE grows with the number of feeders summed and is given at

500

of them in Table 4 , whose fixed width of

64

this table shares. The stds over the draws in parentheses; at

500

feeders the draw is the whole dataset, so the three repetitions of that row vary the fit alone. Gray shades mark the top three models of each row, darkest being the best. The GNN improves on the per-node reference model at every sampled size. Its gain over the shared MLP grows with node count on the

10

-day window and does not order with node count on the

30

-hour one; the limited repetitions do not identify a threshold on either.

Dataset and window	Forecast change: rewired $-$ true			Edge gain	Graph diagnostic: true $\rightarrow$ rewired
Dataset and window	RMSE	MAPE points	nRMSE	nRMSE	$z_{\text{residual}}$	Two-hop nodes
electricity , $11$ d	$+2.82$ MW	$+0.31$	$+0.0046$	$+0.0033$	$23.3\rightarrow 1.5$	$39.5\rightarrow 72.8$
electricity , $60$ d	$-3.29$ MW	$-0.35$	$-0.0082$	$+0.0002$	$21.0\rightarrow 1.9$	$43.7\rightarrow 76.3$
weave-uk , $30$ h	$-33.6$ kWh	$-0.40$	$+0.0036$	$+0.0069$	$13.1\rightarrow 1.6$	$135.1\rightarrow 144.0$
weave-uk , $10$ d	$+24.3$ kWh	$+1.21$	$+0.0178$	$+0.0191$	$10.3\rightarrow 0.7$	$135.1\rightarrow 144.0$

Appendix

Table F.1: Paired degree-preserving rewiring control, averaged over

3

seeds, against the true graph stored as a simple graph. Forecast columns report rewired minus true performance, so positive values are degradations; the last forecast column gives the edge increment at constant backbone on that simple graph. Structural columns show how the residual-alignment score and two-hop neighborhood size change when only the edge identities are randomized.

	electricity		weave-uk
Model	$11$ d	$60$ d	$30$ h	$10$ d
Persistence	$0.5573$	$0.5768$	$0.9776$	$0.9600$
Per-node MLP	$0.3030$	$\mathbf{0.2776}$	$1.0509$	$0.8956$
Per-node ridge	$\mathbf{0.2909}$	$0.2839$	$\mathbf{0.9519}$	$\mathbf{0.8662}$
Shared ridge	$0.2980$	$0.2942$	$0.8922$	$0.8286$
Shared MLP	$0.2804$	$0.2770$	$0.9074$	$0.8013$

Appendix

Table F.2: Per-node error (nRMSE) of the decomposition’s models beside stronger node-specific references, on the cells of Table 4 . Ridge regressions are fitted in closed form on the same standardized inputs; the per-node ridge selects its penalty node by node on the validation window, the shared ridge on the pooled validation error. Persistence repeats the same slot of the previous day. The last two rows give the share of the gap between the stronger node-specific reference and the GNN that the shared MLP and the shared ridge recover.

	electricity		weave-uk
Model	$11$ d	$60$ d	$30$ h	$10$ d
Per-node MLP	$0.3030$	$0.2776$	$1.0509$	$0.8956$
Per-node MLP $+$ neighbors	$0.3201$	$0.2752$	$0.9995$	$0.8762$
Shared MLP	$0.2804$	$0.2770$	$0.9074$	$0.8013$
Shared MLP $+$ neighbors	$0.2933$	$0.2886$	$0.8941$	$0.7707$
GNN	$0.2775$	$0.2752$	$0.8808$	$0.7745$

Appendix

Table F.3: Per-node error (nRMSE) of the MLP family crossed on sharing and neighbor access, beside the GNN, on the cells of Table 4 ;

3

seeds, same inputs, graph, width, and training routine. The neighbor mean is that of the per-node MLP with neighbors.

Window end	Total gap	Sharing (share)	Neighbors, unshared	Edges, same backbone
Jul 2019	$+0.026$	$+0.023$ ( $89\%$ )	$-0.017$	$+0.005$ $[-0.001,+0.011]$
Jun 2019	$+0.017$	$+0.014$ ( $80\%$ )	$-0.013$	$-0.001$ $[-0.005,+0.002]$
May 2019	$+0.046$	$-0.023$ ( $-50\%$ )	$-0.011$	$+0.004$ $[+0.000,+0.007]$
Apr 2019	$+0.028$	$+0.031$ ( $111\%$ )	$-0.012$	$+0.001$ $[-0.004,+0.006]$
Mar 2019	$+0.017$	$+0.004$ ( $23\%$ )	$-0.022$	$-0.001$ $[-0.005,+0.002]$

Appendix

Table F.4: Increments of per-node error on the

11

-day electricity cell over

5

disjoint windows,

3

seeds; seed-and-node

95\%

intervals in brackets. The first row is the window of Table 4 .

Figure G.1: Explanation graphs for the GAT convolutional layer on regions-fr .

Figure G.2: t-SNE projections of the final

64

-dimensional latent representations of the matched shared MLP and SAGE GNN on the regions-fr test set. Every third test day and all

12

regions are retained, giving

1452

identical observations in each plot; the upper row colors them by region and the lower row by quarter of the test period. The projections are fitted separately with perplexity

40

and a fixed seed, so only within-plot neighborhoods are read.

Figure G.3: Geographic distance against mean same-day latent distance for all

66

pairs of regions, in the original standardized representation spaces. The dashed line is a linear visual guide; the reported statistic is Spearman’s rank correlation, with a two-sided

p

-value obtained from

9999

permutations of the region labels to preserve the dependence among pairwise distances. The

5

pairs separated by less than

200

km are labeled.

Figure G.4: t-SNE projections of the final $64$ -dimensional states of the shared MLP and SAGE GNN on weave-uk at $500$ feeders. All feeders and every 12th test origin are retained, giving $6500$ identical observations per plot; the upper row colors them by $8$ spectral clusters of the fixed graph and the lower row by quarter of the test window. The projections are fitted separately with perplexity $50$ and a fixed seed, so only within-plot neighborhoods are read.

Figure G.5: Two projections of the GAT attention matrices on regions-fr , first layer and first head, one point per half-hour of the test year and colored by its position in that year. The partial separation of successive periods places the attention matrices along the seasonal cycle, which moves temperature, daylight, and activity together.

Figure G.6: PCA of the attention matrices on the synthetic dataset with two coupling modes $\mathbf{A}_{1}$ and $\mathbf{A}_{2}$ , colored by the mode in effect. In (a), the mode is handed to the model as a binary input and the two clusters separate at once. In (b) to (d), it is only implied by the joint value of two binary inputs, and the separation is built layer by layer: the first layer mixes the two modes, the second splits them partially, the third recovers them.

Panel of Figure 28 — Figure G.7: Why pruning is performed separately for every target node. A global ranking mostly separates node degrees because attention is normalized over each incoming neighborhood. Within-node pruning fixes the retained degree and changes only the identity of the retained edges. Green edges are kept.

Edges kept	Count	Highest attention	Random	Lowest attention
All edges	$5000$	$0.8763$	$0.8763$	$0.8763$
One half	$2412$	$0.8768$ ( $+0.06\%$ )	$0.8835$ ( $+0.82\%$ )	$0.8912$ ( $+1.70\%$ )
One quarter	$1235$	$0.8862$ ( $+1.07\%$ )	$0.8938$ ( $+1.17\%$ )	$0.9079$ ( $+1.87\%$ )
One tenth	$654$	$0.8931$ ( $+0.78\%$ )	$0.9031$ ( $+1.04\%$ )	$0.9218$ ( $+1.53\%$ )
No neighbors	$0$	$0.8956$ ( $+0.28\%$ )	$0.8956$ ( $-0.83\%$ )	$0.8956$ ( $-2.84\%$ )

Appendix

Table G.1: Direct test nRMSE, lower is better, of the per-node MLP augmented by the mean features of its retained neighbors. Values are averaged over

3

seeds. Each value in parentheses gives the relative change from the row above to the row below,

100(E_{\text{below}}-E_{\text{above}})/E_{\text{above}}

. Red positive values denote an increase in error; green negative values denote an improvement. GATv2 ranks the edges on the full retained graph; it does not produce the errors in the table. The full-graph and no-neighbor values are repeated across the three columns because edge ranking is irrelevant when all or no edges are retained. Seed dispersion is

0.0003

–

0.0033

Dataset	Model	Graph	Batch size	Layers	Width	Learning rate
regions-fr	GCN	Space	$8$	$3$	$68$	$2\cdot 10^{-3}$
	SAGE	DTW	$8$	$2$	$196$	$1\cdot 10^{-3}$
	GAT	Space	$8$	$2$	$75$	$2\cdot 10^{-3}$
	GATv2	Space	$8$	$3$	$155$	$6\cdot 10^{-4}$
	Transformer	Correlation	$8$	$2$	$88$	$2\cdot 10^{-3}$
	TAG	Space	$8$	$3$	$98$	$4\cdot 10^{-3}$

Appendix

Table H.1: Hyperparameters of the GNN models for regions-fr and weave-uk in the sequence-to-sequence forecasting setup. The Graph column gives the construction of Section 4.1 retained by validation; on regions-fr the dense constructions share one (near-)complete support, which SAGE, GAT, GATv2, and Transformer read identically (Section 4.1 ).

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