Graph Double Exponential Smoothing (GDES)

• GDES is a forecasting module that integrates a non-homogeneous Poisson process with double exponential smoothing for dynamic link prediction in supply chains.
• It employs a two-step smoothing process enhanced by directed graph neural networks to capture both historical data and cross-edge effects.
• The method demonstrates high interpretability and an AUC of 93.84% on large-scale datasets, ensuring reliable and explainable forecasts.

Graph Double Exponential Smoothing (GDES) is a forecasting module introduced for dynamic link prediction in supply-chain networks, where firms may form or dissolve directed supplier–buyer relationships over time. In the formulation of "Graph Neural Poisson Models for Supply Chain Relationship Forecasting," GDES sits on top of a one-step non-homogeneous Poisson process (NHPP) estimate of edge intensities and integrates graph neural networks (GNNs) with a nonparametric double exponential smoothing approach to predict the probability of future supply edge formations. The model is presented as interpretable because it decomposes intensity increments into contributions from the current edge’s historical data and influences from neighboring edges in the supply chain network. On a large-scale supply chain dataset with 87,969 firms, the reported performance is an AUC of 93.84% in dynamic link prediction (Xiang et al., 16 Aug 2025).

1. Problem setting and modeling objective

GDES is defined on a dynamic directed supply-chain graph

$G=(V,A(\tau),X), \qquad \tau=1,\ldots,t,$

where $V=\{1,\ldots,n\}$ is the set of firms, $A(\tau)\in\{0,1\}^{n\times n}$ is the directed adjacency at time $\tau$, and $X\in\mathbb{R}^{n\times d}$ collects node features such as text embeddings (Xiang et al., 16 Aug 2025).

For each ordered pair $i\to j$, the observed temporal signal is the counting process

$N_{ij}(t)=\sum_{\tau=1}^{t} A_{ij}(\tau),$

which is modeled as a non-homogeneous Poisson process with intensity $\lambda_{ij}(t)$. At time $t$, the current intensity $\hat{\lambda}_{ij}(t)$ is estimated by fitting the NHPP, for example via linear, sigmoid, or Gaussian basis on $V=\{1,\ldots,n\}$0. The forecasting objective is then to predict $V=\{1,\ldots,n\}$1 for dynamic link prediction.

Within this setup, GDES does not replace the NHPP layer. Instead, it augments the one-step NHPP estimate $V=\{1,\ldots,n\}$2 with a multivariate double-exponential-smoothing layer whose trend update incorporates information propagated across the graph via a directed GNN. Its output is $V=\{1,\ldots,n\}$3, the forecasted future intensity.

2. Double-exponential smoothing over directed edges

GDES collects all edge intensities into the vector

$V=\{1,\ldots,n\}$4

where $V=\{1,\ldots,n\}$5 directed edges. At each time step $V=\{1,\ldots,n\}$6, it maintains two smoothed vectors,

$V=\{1,\ldots,n\}$7

The first recursion is a one-step smoothing of the current NHPP estimate:

$V=\{1,\ldots,n\}$8

The second recursion is a two-step smoothing in which the update is graph-enhanced:

$V=\{1,\ldots,n\}$9

Here $A(\tau)\in\{0,1\}^{n\times n}$0 is the smoothing coefficient, and $A(\tau)\in\{0,1\}^{n\times n}$1 is a GNN-driven increment term.

From these two smoothed quantities, GDES forms the level and trend for each directed edge as in Holt’s method:

$A(\tau)\in\{0,1\}^{n\times n}$2

$A(\tau)\in\{0,1\}^{n\times n}$3

The one-step forecast of the Poisson intensity is then

$A(\tau)\in\{0,1\}^{n\times n}$4

This construction makes explicit that GDES is a double-exponential smoothing mechanism defined not on a scalar series but on a vector of directed-edge intensities, with the second smoothing term modified by graph-derived cross-edge information (Xiang et al., 16 Aug 2025).

3. Directed GNN mechanism for cross-edge influence

The graph component of GDES is designed to capture correlated Poisson intensity functions of supply edges, reflecting the assumption that interdependent economic dynamics link upstream and downstream firms. To implement this, the model defines a learnable vector $A(\tau)\in\{0,1\}^{n\times n}$5 through directed message passing (Xiang et al., 16 Aug 2025).

Two initial node-embedding matrices are introduced:

$A(\tau)\in\{0,1\}^{n\times n}$6

where $A(\tau)\in\{0,1\}^{n\times n}$7 and $A(\tau)\in\{0,1\}^{n\times n}$8 correspond to “supply” and “demand” modes. For layers $A(\tau)\in\{0,1\}^{n\times n}$9, GDES applies two directed-graph propagation steps using adjacency at time $\tau$0:

$\tau$1

$\tau$2

The adjacency operators are defined directionally. $\tau$3 if $\tau$4 in $\tau$5, corresponding to out-edges of $\tau$6, while $\tau$7 if $\tau$8 in $\tau$9, corresponding to in-edges of $X\in\mathbb{R}^{n\times d}$0. The matrices $X\in\mathbb{R}^{n\times d}$1 and $X\in\mathbb{R}^{n\times d}$2 are learnable parameters, and $X\in\mathbb{R}^{n\times d}$3 is a nonlinearity such as ReLU.

After $X\in\mathbb{R}^{n\times d}$4 layers, the model forms the match score matrix

$X\in\mathbb{R}^{n\times d}$5

This is combined with an observed transportation-cost matrix $X\in\mathbb{R}^{n\times d}$6 by element-wise division:

$X\in\mathbb{R}^{n\times d}$7

The off-diagonal entries of $X\in\mathbb{R}^{n\times d}$8 are then flattened into a vector of length $X\in\mathbb{R}^{n\times d}$9 via a flattening operator $i\to j$0, yielding

$i\to j$1

The role of this construction is specific: $i\to j$2 supplies the trend update in the second smoothing equation. This means that the graph component enters GDES not merely as a score for edge existence, but as the graph-enhanced increment term that modulates the smoothed intensity dynamics.

4. Final forecasting equation and component interpretation

Collecting the NHPP estimate, the two smoothing recursions, and the Holt-style level–trend decomposition yields the full GDES forecasting pipeline for each directed edge $i\to j$3:

$i\to j$4

$i\to j$5

$i\to j$6

$i\to j$7

$i\to j$8

$i\to j$9

The paper specifies the meaning of each term. $N_{ij}(t)=\sum_{\tau=1}^{t} A_{ij}(\tau),$0 is the current Poisson intensity estimate; $N_{ij}(t)=\sum_{\tau=1}^{t} A_{ij}(\tau),$1 and $N_{ij}(t)=\sum_{\tau=1}^{t} A_{ij}(\tau),$2 are the one- and two-step smoothed vectors; $N_{ij}(t)=\sum_{\tau=1}^{t} A_{ij}(\tau),$3 and $N_{ij}(t)=\sum_{\tau=1}^{t} A_{ij}(\tau),$4 are the level and trend components; and $N_{ij}(t)=\sum_{\tau=1}^{t} A_{ij}(\tau),$5 is the predicted intensity, which is used as the probability for link formation (Xiang et al., 16 Aug 2025).

A common oversimplification would be to view the method as relying only on historical event counts. The stated motivation is the opposite: forecasting future intensities is hindered by the limitations of historical data alone, so GDES introduces graph-propagated information into the trend update. This suggests that the method is intended to combine edge-local temporal evidence with network-level dependence rather than treat link formation as conditionally independent across edges.

5. Training and inference procedure

The algorithmic procedure takes as input historical adjacency $N_{ij}(t)=\sum_{\tau=1}^{t} A_{ij}(\tau),$6, event counts $N_{ij}(t)=\sum_{\tau=1}^{t} A_{ij}(\tau),$7, node features $N_{ij}(t)=\sum_{\tau=1}^{t} A_{ij}(\tau),$8, and transport costs $N_{ij}(t)=\sum_{\tau=1}^{t} A_{ij}(\tau),$9. The smoothed states are initialized as

$\lambda_{ij}(t)$0

or small (Xiang et al., 16 Aug 2025).

For each training iteration, the sequence of operations is fixed. First, the current NHPP fit is computed:

$\lambda_{ij}(t)$1

Second, one-step smoothing is applied:

$\lambda_{ij}(t)$2

Third, the directed GNN constructs $\lambda_{ij}(t)$3 through the supply-mode and demand-mode message-passing recursions, followed by

$\lambda_{ij}(t)$4

Fourth, two-step smoothing is computed:

$\lambda_{ij}(t)$5

Fifth, the level and trend are updated:

$\lambda_{ij}(t)$6

Sixth, the forecast is produced:

$\lambda_{ij}(t)$7

The optimization objective is

$\lambda_{ij}(t)$8

Backpropagation then updates $\lambda_{ij}(t)$9. At inference time, steps 1 through 6 yield $t$0, from which edges with high $t$1 are predicted as future links.

6. Interpretability, empirical result, and relation to the supply-chain setting

The paper characterizes the model as interpretable because the intensity increment is decomposed into two sources: the current edge’s historical data, represented through the NHPP estimate and its one-step smoothing, and influences from neighboring edges, represented through the directed GNN term $t$2 (Xiang et al., 16 Aug 2025). In this sense, the trend component is not an opaque residual but the consequence of an explicit graph-enhanced update.

The supply-chain context is central to the design. The model is built for a dynamic directed graph of firms and assumes that the Poisson intensity functions of supply edges are correlated, aligning with the non-homogeneous nature of the process. The distinction between upstream and downstream firms is encoded by the separate supply-mode and demand-mode embeddings and by the use of $t$3 and $t$4. The inclusion of the observed transportation-cost matrix $t$5 in the term $t$6 further ties the graph score to a domain-specific observed matrix rather than to learned embeddings alone.

Empirically, the reported evaluation is on a large-scale supply chain dataset with 87,969 firms, where the overall approach achieves an AUC of 93.84% in dynamic link prediction. The stated significance is that the method captures complex supply chain interactions for accurate forecasting. A plausible implication is that GDES is intended not merely as a generic temporal smoother, but as a domain-structured forecasting layer for dynamic relational event data in which edge intensities are both temporally evolving and network-dependent.