CTGraph: Graph-Based Methods in Diverse Contexts

• CTGraph is a self-supervised graph representation method that encodes students’ learning journeys via curriculum DAGs using Graph Isomorphism Networks.
• The term also interprets models for traffic flow prediction, CT reconstruction, and irregular-time forecasting, highlighting methodological commonalities across domains.
• By leveraging graph topology for structural encoding, CTGraph improves interpretability and facilitates actionable insights in diverse applications.

CTGraph most explicitly denotes a self-supervised, graph-level representation learning approach for profiling student behaviors and performance in curriculum-based online learning systems. In that formulation, a student’s learning journey within a topic is encoded as a directed graph derived from the curriculum DAG, and a graph-level embedding is learned to jointly capture content coverage, learning intensity, and concept proficiency (Xiao et al., 26 Aug 2025). Across the provided literature, however, the term is not unique to one stabilized architecture. It also appears as an interpretive or contextual label for a Counterfactual Graph Transformer for traffic flow prediction (Yang et al., 2023), a CM-CC-CM realization of the graph linear canonical transform (Li et al., 2024), graph-structured optimization in sparse-view CT reconstruction (Yuluo et al., 4 Aug 2025), and a continuous-time temporal graph model framed as a CTGraph instance (Oskarsson et al., 2023).

1. Scope and naming

Across the provided sources, CTGraph denotes more than one graph-based construct. The most direct and formal use is in curriculum-based Intelligent Tutoring Systems, where CTGraph is introduced as a graph-level representation learner for student profiling (Xiao et al., 26 Aug 2025). Other sources use the label as a conceptual alignment rather than as the authors’ original model name. In the traffic-flow paper, for example, the paper states that the model introduced is called CGT, and that “CTGraph” is an interpretation of the same concept in the sense of a counterfactual graph-based transformer for traffic flow prediction (Yang et al., 2023). In the irregular-time forecasting paper, the actual model name is TGNN4I, but the supplied description presents it “through the lens” of CTGraph (Oskarsson et al., 2023).

Usage in provided sources	Core object	Citation
CTGraph	Self-supervised graph-level representation learning for student profiling in curriculum-based ITSs	(Xiao et al., 26 Aug 2025)
CTGraph interpreted as CGT	Counterfactual Graph Transformer for traffic flow prediction	(Yang et al., 2023)
CTGraph in graph signal processing	CM-CC-CM-based graph linear canonical transform	(Li et al., 2024)
CTGraph in sparse-view CT	Graph structures over Gaussian primitives for reconstruction	(Yuluo et al., 4 Aug 2025)
CTGraph lens on TGNN4I	Continuous-time temporal graph model for irregular data	(Oskarsson et al., 2023)

This suggests that CTGraph functions less as a single canonical model name than as a context-dependent label for graph-centric methods whose primary contribution is structural encoding, structural explanation, or structural regularization.

2. CTGraph as curriculum-aligned student graph encoding

In the ITS formulation, CTGraph addresses widening performance gaps, multifaceted student profiling, and alignment to curriculum structures (Xiao et al., 26 Aug 2025). The formal goal is: given a set of student graphs aligned to a topic’s curriculum DAG, learn a fixed-dimensional embedding for each student that captures both local concept-level attributes and global path structure. For a topic $\phi$ and student $u$, the representation is written as $y(G^u_\phi) \in \mathbb{R}^d$.

The curriculum structure for topic $\phi$ is a directed acyclic graph

$G_\phi = (V_\phi, E_\phi),$

where $V_\phi$ are concepts $\Theta = \{\theta_1,\dots,\theta_t\}$ and each directed edge encodes a prerequisite or ordering relation (Xiao et al., 26 Aug 2025). For each student and concept, CTGraph uses a multivariate learning-tracing vector

$$x^\theta(u) = x_1^\theta(u) \oplus x_2^\theta(u) \oplus x_3^\theta(u),$$

where $x_1^\theta(u)$ is average accuracy on all attempts covering $\theta$, $u$0 is total number of attempts on $u$1, and $u$2 is median week number in the academic calendar when the student attempted $u$3 (Xiao et al., 26 Aug 2025).

The student curriculum-based learning graph is

$u$4

a directed graph whose nodes are the subset of concepts in $u$5 that the student actually engaged with. Each node carries the tracing vector $u$6 as attributes, and the edge set preserves prerequisite-consistent transitions via nearest-successor relations inherited from the curriculum DAG (Xiao et al., 26 Aug 2025).

A central preprocessing step is node absorption. Concept nodes without logs for student $u$7 are removed, and the remaining nodes are reconnected using nearest successor links consistent with $u$8’s DAG. The resulting graph is a sparse, student-specific subgraph with dense node attributes, intended to improve contrastive learning stability and discriminativeness (Xiao et al., 26 Aug 2025). For a graph with $u$9 nodes, the node feature matrix is $y(G^u_\phi) \in \mathbb{R}^d$0 with $y(G^u_\phi) \in \mathbb{R}^d$1, and no edge features are used; edges are untyped, directed prerequisite-consistent transitions (Xiao et al., 26 Aug 2025).

3. Encoder architecture and self-supervised objective

CTGraph uses Graph Isomorphism Networks as the encoder backbone because of their high discriminative power for graph structures (Xiao et al., 26 Aug 2025). With initial node features $y(G^u_\phi) \in \mathbb{R}^d$2, GIN iteratively aggregates neighbor information:

$y(G^u_\phi) \in \mathbb{R}^d$3

$y(G^u_\phi) \in \mathbb{R}^d$4

In implementation, these are learned via GIN layers, with injective multiset aggregation followed by an MLP (Xiao et al., 26 Aug 2025).

Graph-level representations follow an InfoGraph-style construction. Per-node, multi-scale patch embeddings are concatenated as

$y(G^u_\phi) \in \mathbb{R}^d$5

and a sum READOUT over nodes forms the graph embedding

$y(G^u_\phi) \in \mathbb{R}^d$6

The reported configuration uses $y(G^u_\phi) \in \mathbb{R}^d$7 GIN layers, hidden dimension $y(G^u_\phi) \in \mathbb{R}^d$8 per layer, and therefore a $y(G^u_\phi) \in \mathbb{R}^d$9-dimensional graph embedding (Xiao et al., 26 Aug 2025).

Training is self-supervised through mutual-information-based contrastive learning between local patch representations and the graph-level representation. Positive pairs are $\phi$0 for a node within its own graph; negative pairs are formed by combining the same local patch with graph-level embeddings from other student graphs (Xiao et al., 26 Aug 2025). The paper gives the topic-level objective as

$\phi$1

with a discriminator $\phi$2 implemented as a feedforward network and a standard logistic objective following InfoGraph (Xiao et al., 26 Aug 2025).

The embedding is intentionally unified rather than factorized. CTGraph produces a single embedding per graph; there are no separate heads for different aspects. Content coverage is implicit in which concept nodes remain after node absorption, learning intensity is present in attempt counts, and concept proficiency is present in accuracy, all of which are propagated through GIN and pooled into the graph embedding (Xiao et al., 26 Aug 2025).

4. Profiling functions, probes, and empirical findings

Although CTGraph is trained without supervision, the study quantifies the three central aspects with explicit probes derived from node attributes and graph structure (Xiao et al., 26 Aug 2025). Content coverage is

$\phi$3

learning intensity is

$\phi$4

and concept proficiency is the attempt-weighted accuracy

$\phi$5

where $\phi$6 and $\phi$7 (Xiao et al., 26 Aug 2025). For timing, the study visualizes

$\phi$8

The reported dataset is real-world mathematics ITS data from Adaptemy, with a curriculum containing 26 topics. Results are shown for Algebra I, Algebra II, Functions I, and Fractions (Xiao et al., 26 Aug 2025). To ensure sufficient path diversity and stable contrastive training, CTGraph trains on students whose concept coverage satisfies

$\phi$9

Optimization uses Adam with learning rate $G_\phi = (V_\phi, E_\phi),$0 and batch size $G_\phi = (V_\phi, E_\phi),$1 (Xiao et al., 26 Aug 2025).

The study emphasizes qualitative and exploratory analysis rather than comparative metrics. PCA visualizations of the 96-dimensional embeddings show clear clusters of students with similar paths and performance, and outliers representing learners with low accuracy, atypical practice intensity, or late engagement (Xiao et al., 26 Aug 2025). In Functions I, outliers form a separate cluster with markedly different timing and lower proficiency. In Fractions, the nearest neighbors of a selected struggling student share low accuracy around specific successor concepts, such as high accuracy on Concept 1 followed by substantial drop on Concepts 2 and 5. In Algebra II, several students have lower average accuracy than the majority, but some extend coverage into more advanced concepts with lower performance there while retaining relatively solid mastery of the first 12 concepts (Xiao et al., 26 Aug 2025).

Similarity-based cohorts are defined in the original embedding space with cosine similarity,

$G_\phi = (V_\phi, E_\phi),$2

and the paper further proposes a latent-direction cohort construction: given endpoint students with embeddings $G_\phi = (V_\phi, E_\phi),$3 and $G_\phi = (V_\phi, E_\phi),$4, compute $G_\phi = (V_\phi, E_\phi),$5 and then retrieve students whose embeddings are closest to $G_\phi = (V_\phi, E_\phi),$6 (Xiao et al., 26 Aug 2025). These operations are used to identify struggling students, reveal comparable learning trajectories, and localize where proficiency drops within the prerequisite structure.

A recurrent limitation is that the paper does not present quantitative baselines such as DKTs or Transformers, formal precision/recall/F1 for struggling identification, or statistical tests; it demonstrates interpretability and practical utility primarily through visualizations and case studies (Xiao et al., 26 Aug 2025).

5. Counterfactual-transformer interpretation in traffic flow prediction

A separate usage aligns CTGraph with the Counterfactual Graph Transformer, or CGT, for traffic flow prediction (Yang et al., 2023). The source makes an explicit naming clarification: the paper does not use the exact name “CTGraph”; the model introduced is called CGT, and “CTGraph” is treated as the same concept in the sense of a counterfactual graph-based transformer for traffic flow prediction (Yang et al., 2023).

The traffic formulation operates on a sensor graph $G_\phi = (V_\phi, E_\phi),$7, where $G_\phi = (V_\phi, E_\phi),$8 is the sensor set, $G_\phi = (V_\phi, E_\phi),$9 captures connectivity, and $V_\phi$0 is an adjacency matrix defined by distance between sensor locations. Over past $V_\phi$1 steps, the input is $V_\phi$2, and the model predicts the next $V_\phi$3 steps through

$V_\phi$4

The backbone is a PDFormer-based Graph Transformer with geographic attention, semantic attention, GCN using the Laplacian of $V_\phi$5, and a temporal transformer mining long-range bidirectional temporal patterns (Yang et al., 2023).

Interpretability is introduced through a spatial mask $V_\phi$6 over graph structure and a temporal mask $V_\phi$7 over input sensor features. Masked inputs are

$V_\phi$8

The counterfactual objective combines a prediction-divergence term and a perturbation-size term,

$V_\phi$9

with

$\Theta = \{\theta_1,\dots,\theta_t\}$0

The intended criterion is “minimal perturbation that changes prediction most,” and the learned explanations correspond to dominant subgraphs and dominant time slices (Yang et al., 2023).

The paper reports experiments on PeMS04, PeMS07M, and PeMS08. After explanation embedding and retraining, the resulting model improves forecasting performance relative to PDFormer on two of the three datasets in MAE, MAPE, and RMSE, and improves RMSE on all three. For example, on PeMS04, PDFormer reports $\Theta = \{\theta_1,\dots,\theta_t\}$1, while CGT-retrained-$\Theta = \{\theta_1,\dots,\theta_t\}$2-$\Theta = \{\theta_1,\dots,\theta_t\}$3 reports $\Theta = \{\theta_1,\dots,\theta_t\}$4; on PeMS07M, the corresponding values are $\Theta = \{\theta_1,\dots,\theta_t\}$5 versus $\Theta = \{\theta_1,\dots,\theta_t\}$6; on PeMS08, they are $\Theta = \{\theta_1,\dots,\theta_t\}$7 versus $\Theta = \{\theta_1,\dots,\theta_t\}$8 (Yang et al., 2023). Explanation quality is evaluated with Fidelity, Explanation Size, Sparsity, and $\Theta = \{\theta_1,\dots,\theta_t\}$9, and the source characterizes CGT-Explainer as yielding sparse, actionably small counterfactuals with large $$x^\theta(u) = x_1^\theta(u) \oplus x_2^\theta(u) \oplus x_3^\theta(u),$$0 (Yang et al., 2023).

6. Signal-processing, CT-reconstruction, and continuous-time interpretations

In graph signal processing, CTGraph is used for the CM-CC-CM-based graph linear canonical transform. The transform generalizes the graph Fourier transform and graph fractional Fourier transform through a parameter matrix

$$x^\theta(u) = x_1^\theta(u) \oplus x_2^\theta(u) \oplus x_3^\theta(u),$$1

and realizes the graph linear canonical transform by chirp multiplication, chirp convolution, and chirp multiplication in the graph spectral domain (Li et al., 2024). The stated advantages are that the method is irrelevant to sampling periods and requires no oversampling, because it avoids scaling transforms and Iwasawa dilation. Simulation results indicate that the CM-CC-CM-GLCT achieves similar additivity to the CDDHFs-GLCT and exhibits better reversibility (Li et al., 2024). In the paper’s arithmetic accounting, the complexity for $$x^\theta(u) = x_1^\theta(u) \oplus x_2^\theta(u) \oplus x_3^\theta(u),$$2 is approximately $$x^\theta(u) = x_1^\theta(u) \oplus x_2^\theta(u) \oplus x_3^\theta(u),$$3 real multiplications, compared with about $$x^\theta(u) = x_1^\theta(u) \oplus x_2^\theta(u) \oplus x_3^\theta(u),$$4 real multiplications for CDDHFs-GLCT (Li et al., 2024).

In sparse-view CT reconstruction, the CTGraph perspective is a graph built over Gaussian primitives in GR-Gaussian. Nodes are Gaussian centers, edges are symmetric $$x^\theta(u) = x_1^\theta(u) \oplus x_2^\theta(u) \oplus x_3^\theta(u),$$5-nearest-neighbor relations with $$x^\theta(u) = x_1^\theta(u) \oplus x_2^\theta(u) \oplus x_3^\theta(u),$$6, and weights are

$$x^\theta(u) = x_1^\theta(u) \oplus x_2^\theta(u) \oplus x_3^\theta(u),$$7

The graph supports a Pixel-Graph-Aware Gradient Strategy and a Laplacian regularizer

$$x^\theta(u) = x_1^\theta(u) \oplus x_2^\theta(u) \oplus x_3^\theta(u),$$8

with a Denoised Point Cloud Initialization Strategy based on 3D Gaussian filtering using $$x^\theta(u) = x_1^\theta(u) \oplus x_2^\theta(u) \oplus x_3^\theta(u),$$9 as the optimal value in experiments (Yuluo et al., 4 Aug 2025). The reported gains over baselines are PSNR improvements of $x_1^\theta(u)$0 dB and $x_1^\theta(u)$1 dB, and SSIM gains of $x_1^\theta(u)$2 and $x_1^\theta(u)$3 on X-3D and real-world datasets, respectively (Yuluo et al., 4 Aug 2025).

A further interpretive usage presents TGNN4I as a CTGraph model for irregular temporal data (Oskarsson et al., 2023). In that description, each node has a time-continuous latent state $x_1^\theta(u)$4 that evolves between observation events according to a linear ODE with explicit solution,

$x_1^\theta(u)$5

and is updated at observation times via GRU gates that integrate neighborhood information through GNN layers (Oskarsson et al., 2023). The reported advantage is prediction at arbitrary time steps, with empirical validation on traffic and climate data.

7. Limitations and open questions

The most immediate limitation is terminological. The provided sources do not support a single, domain-independent definition of CTGraph. In the ITS paper, CTGraph is a named graph-level encoder for student profiling (Xiao et al., 26 Aug 2025). In the traffic paper, the exact model name is CGT rather than CTGraph (Yang et al., 2023). In the irregular-time forecasting paper, the formal model is TGNN4I rather than CTGraph (Oskarsson et al., 2023). This naming overlap can obscure whether a reference concerns a student-profiling encoder, a counterfactual explainer, a graph canonical transform, or graph-aware CT reconstruction.

Within the ITS formulation itself, several limitations are explicit. The attribute set is restricted to three aggregates; training on students with $x_1^\theta(u)$6 may bias learned representations toward more engaged learners; evaluation is unsupervised and largely qualitative; generalizability is assessed only on mathematics topics from one ITS; and privacy and fairness remain central concerns because student profiles should support equitable interventions and avoid stigmatization (Xiao et al., 26 Aug 2025).

The traffic interpretation inherits a different set of constraints. Residual dataset bias may remain, spatial and temporal masks are optimized separately because of disparity in perturbation ranges, thresholding continuous masks may introduce sensitivity, and cross-city or cross-sensor generalization requires further validation (Yang et al., 2023). The sparse-view CT interpretation likewise remains sensitive to hyperparameters such as $x_1^\theta(u)$7, $x_1^\theta(u)$8, $x_1^\theta(u)$9, and $\theta$0, and extremely sparse views or limited-angle settings remain ill-posed despite graph-aware gradient amplification (Yuluo et al., 4 Aug 2025).

A plausible implication is that the durable commonality across CTGraph variants is methodological rather than nominal: graph structure is used to encode prerequisite order, sensor dependence, spectral coupling, spatial adjacency, or irregular temporal interaction, and that structure is then exploited for representation learning, explanation, transform design, or regularization. Under that reading, CTGraph is best understood not as one model family with a single architecture, but as a recurrent pattern in which graph topology is treated as the primary carrier of inductive bias.