How Much Graph Is Too Much? Graph Construction and Exposure in Spatio-Temporal Forecasting

Saj Maru; Qiyu Han; Krishna Mamidala

stat.MLOct 25, 2026

How Much Graph Is Too Much? Graph Construction and Exposure in Spatio-Temporal Forecasting

Authors: Saj Maru, Qiyu Han, Krishna Mamidala

Abstract

Spatio-temporal graph neural networks combine a node's history with information from graph neighbors. The central question is whether graph neighbors provide predictive information beyond each node's own history. Forecast accuracy depends on both the choice of neighbors and the way graph messages enter the temporal model. We study these choices as graph construction and graph exposure on U.S. state unemployment, traffic speed, and electricity demand. Our diagnostic design uses two comparisons: varying exposure while holding the domain graph fixed, and varying construction while holding late gated integration fixed. A GRU using only each node's own history and randomized graph controls help assess what the graph contributes. Under a common change forecasting protocol, the same domain graph can improve or worsen error relative to the GRU across integration designs. Late integration alone is insufficient, while adding a fusion gate yields lower mean error in all three tasks. With late gated integration fixed, a graph based on residual correlations has the lowest mean error among the tested graphs, although randomized graphs perform similarly on unemployment. The findings support evaluating graph construction and integration together, with conclusions tied to the comparisons actually performed.

Figures & tables

Figure 1. Shared variation and residual spatial dependence. PC1 is the variance share of the first principal component. Residual Moran’s

I

measures spatial autocorrelation after common component removal ( Moran, 1950 ) . The right panel reports the share of timesteps for which it is significant.

Graph	Construction and purpose
Domain prior	State borders, road proximity, or grid ties. Tests the forecasting value of the domain graph.
Random degree matched	Random neighbors with each node’s outgoing degree preserved. Tests whether the identity of its neighbors matters.
Raw correlation kNN	The $k=5$ neighbors with the highest positive correlations in the raw series. Tests neighbor selection based on shared levels.
Residual correlation kNN	The $k=5$ neighbors selected after differencing and mean removal. Tests neighbor selection based on residual changes.
Rewired 100%	All domain edges rewired with graph density preserved. Tests the value of the original topology at the same density.

Table 1. Graph construction rules and the question each comparison addresses.

Method	Insertion stage	Frequency	Fusion gate	Update sketch
GRU	None	None	No	$H_{t}=\mathrm{GRUCell}(x_{t},H_{t-1})$
GCN $\rightarrow$ GRU	Before temporal encoding	Each step	No	$H_{t}=\mathrm{GRUCell}(M_{A}(x_{t}),H_{t-1})$
T-GCN	Within GRU gates	Each step	No	GRU gates use $M_{A}([x_{t},H_{t-1}])$
GRUCell $\rightarrow$ GCN	After each memory update	Each step	No	$H^{\prime}_{t}=\mathrm{GRUCell}(x_{t},H_{t-1})$ , $H_{t}=M_{A}(H^{\prime}_{t})$
Late GRU $\rightarrow$ GCN	After the full window	Once	No	$H^{T}=\mathrm{GRU}(X_{t-L+1:t})$ , $H=M_{A}(H^{T})$
Late + gate	After the full window	Once	Yes	Local and graph representations combined by Eq. 3

Table 2. Graph exposure strategies. Frequency counts graph applications within one input window. The fusion gate combines local and graph representations.

	Unemployment (BLS LAUS)	Traffic (METR-LA)	Electricity (EIA-BA)
What it measures	Monthly unemployment rate per U.S. state	Vehicle speed on LA freeways	Hourly electricity demand per balancing authority
Why a graph seems natural	Interstate labor mobility links state labor markets	Congestion propagates along road segments	Power flows across AC grid interconnections
Nodes	51 U.S. states	207 road sensors	52 balancing authorities
Domain graph	109 edges (state borders)	1,515 edges (road proximity)	204 edges (AC grid ties)
Frequency	Monthly	5-minute	Hourly
Time span	Jan 2007–Dec 2024	Mar–Jun 2012	Jan 2019–Apr 2026

Table 3. Dataset summary. Three domains are selected to represent distinct graph-signal regimes.

Coupling	Unemp.	Traffic	Elec.
GRU (no graph)	1.63 $\pm$ .40	7.74 $\pm$ .22	6.18 $\pm$ .08
GCN $\to$ GRU	1.68 $\pm$ .90	7.95 $\pm$ .13	9.90 $\pm$ 2.74
T-GCN	1.42 $\pm$ .73	8.09 $\pm$ .09	10.43 $\pm$ 2.97
GRUCell $\to$ GCN	1.85 $\pm$ .73	8.08 $\pm$ .08	10.73 $\pm$ 2.99
Late GRU $\to$ GCN	1.57 $\pm$ .33	7.96 $\pm$ .12	9.21 $\pm$ 2.46
Late + gate	1.36 $\pm$ .17	7.64 $\pm$ .14	5.27 $\pm$ .07

Table 4. Coupling results (domain graph fixed). WMAPE (%) mean

\pm

std over 3 seeds. Bold = best learned model per dataset.

Late gated fusion:
Graph / Method	Unemp.	Traffic	Elec.
GRU (no graph)	1.63 $\pm$ .40	7.74 $\pm$ .22	6.18 $\pm$ .08
Domain prior	1.36 $\pm$ .17	7.64 $\pm$ .14	5.27 $\pm$ .07
Random (deg.-matched)	1.27 $\pm$ .03	7.81 $\pm$ .05	5.31 $\pm$ .06
Raw-correlation kNN	1.25 $\pm$ .07	7.48 $\pm$ .17	5.11 $\pm$ .03
Resid.-correlation kNN	1.23 $\pm$ .09	6.87 $\pm$ .16	5.00 $\pm$ .04

Table 5. Graph construction (late gated fusion) and reference methods. WMAPE (%) mean

\pm

std over 3 seeds. Bold = best per dataset.

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Electricity forecasting often involves spatially related signals observed over regions, substations, and feeders, and Graph Neural Networks (GNNs) provide a natural way to represent these relations. Building a complete GNN forecasting experiment is nonetheless laborious, because graph construction, model selection, training, aggregation, and interpretation sit in incompatible tools. We present GraphToolbox, an open-source Python framework that unifies these stages in one configurationdriven pipeline built on PyTorch Geometric. It offers data-driven graph construction, an adapter that instantiates and trains 51 of the 65 PyTorch Geometric convolutions together with the recurrent cells of PyTorch Geometric Temporal, online expert aggregation, forecasting interpretability, and significance testing on cached forecasts. We evaluate the pipeline in two case studies. On French regional load, the 48 convolutions included in the complete forecasting sweep fall in a band from 1.14% to 1.60% error, online aggregation lowers this to 0.98%, and the graph models improve on classical additive and boosting baselines. On net-load, direct graph models are less accurate than a classical additive model, while forecasting each physical component separately improves them without closing that gap. Both comparisons use the same experimental interface, illustrating the role of GraphToolbox in systematic architectural evaluation.

Abstract

Spatiotemporal GNNs

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