Continual Causal Graph Learning

• Continual Causal Graph Learning is the process of updating causal models from evolving data streams while retaining past knowledge and avoiding catastrophic forgetting.
• It employs temporal invariance and variance-based suppression to distinguish stable predictors from spurious features in dynamic environments.
• Graph-aware replay and regularization techniques are crucial for managing structural dependencies and mitigating errors caused by evolving graph topologies.

Searching arXiv for recent and directly relevant papers on continual causal graph learning, online causal model learning, and continual graph learning. Continual causal graph learning denotes the problem of updating structural or causally motivated predictive knowledge from a non-stationary stream while retaining earlier knowledge and avoiding catastrophic forgetting. In the current arXiv literature, the topic is represented less by a single canonical algorithm than by adjacent strands: online discovery of stable causal predictors from temporal non-stationarity, continual learning on sequences of evolving graphs, and topology-aware replay or regularization for graph-structured streams. This suggests that the area is presently best understood as an intersection between causal model learning and continual graph learning, rather than as a standardized framework for recovering time-varying causal DAGs (Javed et al., 2020, Raghavan et al., 2023, Yuan et al., 2023, Yan et al., 12 Dec 2025).

1. Scope and formal setting

A useful starting point is the distinction between continual learning on Euclidean data and continual learning on graphs. The survey literature formulates continual learning as a sequence of tasks $\mathcal{T}=\{t\}_{t=1}^T$ with accumulated loss

$$\mathcal{L}_{1:T}(\theta)=\mathcal{L}_{1:T-1}(\theta)+\mathcal{L}_T(\theta),$$

while emphasizing that graph streams violate the clean conditional-independence assumptions common in ordinary continual learning because current graph data depend on prior graph structure. That dependence is written as

$$p(\theta|D^{1:T}) \propto p(\theta|D^{1:T-1})\,p(D^T|D^{1:T-1},\theta),$$

and later refined into a graph-update form

$$p(\theta|\mathcal{G}^{t-1})p(\Delta \mathcal{G}^t|\mathcal{G}^{t-1}_p,\theta).$$

The same survey also distinguishes dynamic graph learning from continual graph learning: dynamic graph learning focuses on forward transfer to future graph states, whereas continual graph learning additionally requires backward transfer and explicit control of catastrophic forgetting (Yuan et al., 2023).

Within that broader landscape, dynamic/streaming graph work often models the observed structure as a time-indexed sequence

$$\mathcal{G}=\{ \mathcal{G}^{(1)}, \mathcal{G}^{(2)},\dots, \mathcal{G}^{(T)} \}, \qquad \mathcal{G}^{(t)}=(\mathcal{V}^{(t)},\mathcal{E}^{(t)}),$$

with per-step inputs $$\mathcal{D}^{(t)} = (\mathbf{A}^{(t)}, \mathbf{X}^{(t)})$$. The technical difficulty is that topological updates propagate through neighborhoods, so changes in edges or nodes alter embeddings of already-seen vertices and induce forgetting even when no entirely new label space is introduced (Yan et al., 12 Dec 2025).

The papers most directly touching causality do not, however, implement causal graph discovery in the classical sense. "Learning Causal Models Online" is explicitly best understood as continual or online discovery of causal predictors rather than a full-fledged causal graph learning algorithm. Its focus is the online removal of spurious features whose predictive relation to the target changes over time (Javed et al., 2020).

Work	Primary formulation	Relation to continual causal graph learning
"Learning Causal Models Online" (Javed et al., 2020)	Online causal predictor discovery in an MDP	Supplies a temporal stability criterion for spuriousness
"Continual Graph Learning: A Survey" (Yuan et al., 2023)	Taxonomy of continual graph learning	Defines graph-specific forgetting and topology-aware replay issues
"Learning Continually on a Sequence of Graphs -- The Dynamical System Way" (Raghavan et al., 2023)	Dynamical-system and min-max view of graph continual learning	Makes structural nonstationarity explicit
"Condensation-Concatenation Framework for Dynamic Graph Continual Learning" (Yan et al., 12 Dec 2025)	Condensation and selective replay on dynamic graphs	Provides a graph-memory mechanism, but not causal discovery

2. Temporal non-stationarity as a causal signal

The most explicitly causal ingredient in the literature is the use of temporal non-stationarity to distinguish stable from spurious predictors. In the online MDP setting of "Learning Causal Models Online," the agent is in state $s_t$, takes action $a_t$, receives reward or target $y_t$, observes $o_t=e(s_t)$, and learns a predictor $$\mathcal{L}_{1:T}(\theta)=\mathcal{L}_{1:T-1}(\theta)+\mathcal{L}_T(\theta),$$0 online. The objective is not merely low online regret, but a predictor that generalizes to unseen parts of the MDP (Javed et al., 2020).

The paper’s central hypothesis is that a feature is spurious if its correlation with the target is not constant over time. For a binary feature $$\mathcal{L}_{1:T}(\theta)=\mathcal{L}_{1:T-1}(\theta)+\mathcal{L}_T(\theta),$$1, spuriousness is formalized through the change of

$$\mathcal{L}_{1:T}(\theta)=\mathcal{L}_{1:T-1}(\theta)+\mathcal{L}_T(\theta),$$2

across temporally distant parts of the same MDP. Stable features are those whose predictive relation to $$\mathcal{L}_{1:T}(\theta)=\mathcal{L}_{1:T-1}(\theta)+\mathcal{L}_T(\theta),$$3 does not drift as the stream unfolds; unstable features are those whose apparent usefulness is regime-dependent. The resulting criterion is a temporal analogue of invariance: instead of comparing multiple static environments, the method uses time itself as the axis along which invariance is tested (Javed et al., 2020).

This framing is closely aligned with causal generalization, but it remains weaker than causal graph discovery. It does not search over directed edges, perform orientation, or estimate structural equations. Rather, it treats stability of the target-conditional relation as evidence that a feature is non-spurious. A plausible implication is that continual causal graph learning, if interpreted narrowly as continual DAG recovery, requires additional machinery beyond temporal invariance alone.

3. Online suppression of spurious features

For a linear predictor over binary features $$\mathcal{L}_{1:T}(\theta)=\mathcal{L}_{1:T-1}(\theta)+\mathcal{L}_T(\theta),$$4, the online learner updates parameters by the usual gradient step,

$$\mathcal{L}_{1:T}(\theta)=\mathcal{L}_{1:T-1}(\theta)+\mathcal{L}_T(\theta),$$5

then tracks an exponentially decayed running mean $$\mathcal{L}_{1:T}(\theta)=\mathcal{L}_{1:T-1}(\theta)+\mathcal{L}_T(\theta),$$6 and variance $$\mathcal{L}_{1:T}(\theta)=\mathcal{L}_{1:T-1}(\theta)+\mathcal{L}_T(\theta),$$7 of each feature weight $$\mathcal{L}_{1:T}(\theta)=\mathcal{L}_{1:T-1}(\theta)+\mathcal{L}_T(\theta),$$8. The updates are

$$\mathcal{L}_{1:T}(\theta)=\mathcal{L}_{1:T-1}(\theta)+\mathcal{L}_T(\theta),$$9

$$p(\theta|D^{1:T}) \propto p(\theta|D^{1:T-1})\,p(D^T|D^{1:T-1},\theta),$$0

with $$p(\theta|D^{1:T}) \propto p(\theta|D^{1:T-1})\,p(D^T|D^{1:T-1},\theta),$$1. These equations are described as an exponentially decayed version of Welford’s method and are applied only when the feature is active (Javed et al., 2020).

The decision rule is a masking or gating mechanism driven by weight variance. In the appendix algorithm, after a warmup period,

$$p(\theta|D^{1:T}) \propto p(\theta|D^{1:T-1})\,p(D^T|D^{1:T-1},\theta),$$2

whereas the main text writes

$$p(\theta|D^{1:T}) \propto p(\theta|D^{1:T-1})\,p(D^T|D^{1:T-1},\theta),$$3

The mask acts through

$$p(\theta|D^{1:T}) \propto p(\theta|D^{1:T-1})\,p(D^T|D^{1:T-1},\theta),$$4

so features with larger-than-average variance are suppressed. The underlying stability test is direct: low variance of the learned weight indicates a stable predictive relation; high variance indicates likely spuriousness (Javed et al., 2020).

The same work extends the method to raw sensory input through Perturbations with Backtracking, which splits the model into a Representation Learning Network and a Prediction Learning Network. The Prediction Learning Network is trained online, the Representation Learning Network is periodically perturbed by setting some parameters to $$p(\theta|D^{1:T}) \propto p(\theta|D^{1:T-1})\,p(D^T|D^{1:T-1},\theta),$$5, $$p(\theta|D^{1:T}) \propto p(\theta|D^{1:T-1})\,p(D^T|D^{1:T-1},\theta),$$6, or $$p(\theta|D^{1:T}) \propto p(\theta|D^{1:T-1})\,p(D^T|D^{1:T-1},\theta),$$7, and the perturbation is retained if either the running loss decreases or the sum of the variances decreases. This converts the variance statistic into a representation-search signal rather than merely a feature-selection signal (Javed et al., 2020).

Empirically, the online Colored MNIST benchmark makes the logic concrete. Digit class is the causal signal, color is a spurious feature, and the color-label correlation is reversed between a Seen MDP and an Unseen MDP. Reported accuracies show the characteristic tradeoff: Online Learning attains about $$p(\theta|D^{1:T}) \propto p(\theta|D^{1:T-1})\,p(D^T|D^{1:T-1},\theta),$$8 seen and about $$p(\theta|D^{1:T}) \propto p(\theta|D^{1:T-1})\,p(D^T|D^{1:T-1},\theta),$$9 unseen, Oracle IRM about $$p(\theta|\mathcal{G}^{t-1})p(\Delta \mathcal{G}^t|\mathcal{G}^{t-1}_p,\theta).$$0 seen and about $$p(\theta|\mathcal{G}^{t-1})p(\Delta \mathcal{G}^t|\mathcal{G}^{t-1}_p,\theta).$$1 unseen, Online IRM about $$p(\theta|\mathcal{G}^{t-1})p(\Delta \mathcal{G}^t|\mathcal{G}^{t-1}_p,\theta).$$2 seen and about $$p(\theta|\mathcal{G}^{t-1})p(\Delta \mathcal{G}^t|\mathcal{G}^{t-1}_p,\theta).$$3 unseen, and the proposed method about $$p(\theta|\mathcal{G}^{t-1})p(\Delta \mathcal{G}^t|\mathcal{G}^{t-1}_p,\theta).$$4 seen and about $$p(\theta|\mathcal{G}^{t-1})p(\Delta \mathcal{G}^t|\mathcal{G}^{t-1}_p,\theta).$$5 unseen. The paper further reports that the online variance metric $$p(\theta|\mathcal{G}^{t-1})p(\Delta \mathcal{G}^t|\mathcal{G}^{t-1}_p,\theta).$$6 and Oracle IRM gradients both identify color as spurious, whereas an IID or shuffled version of $$p(\theta|\mathcal{G}^{t-1})p(\Delta \mathcal{G}^t|\mathcal{G}^{t-1}_p,\theta).$$7 fails. Experience replay does not fix Online IRM because replay destroys the temporal information needed to identify non-stationarity (Javed et al., 2020).

4. Structural nonstationarity in continual graph learning

On the graph side, continual learning requires explicit treatment of nonstationary vertices, edges, and features. The survey literature isolates two graph-specific difficulties. The first is sample-level dependency: in message-passing models, a node representation depends on its neighbors, so forgetting cannot be reduced to errors on independent samples. The second is graph-level dependency: new subgraphs perturb old representations, while old subgraphs provide context for interpreting new data. The survey therefore treats topological updates, feature updates, and continuous evolution as first-class sources of forgetting rather than peripheral nuisances (Yuan et al., 2023).

A more formal treatment appears in "Learning Continually on a Sequence of Graphs -- The Dynamical System Way," which represents graph streams using a vertex-edge random graph formalism. For $$p(\theta|\mathcal{G}^{t-1})p(\Delta \mathcal{G}^t|\mathcal{G}^{t-1}_p,\theta).$$8 and $$p(\theta|\mathcal{G}^{t-1})p(\Delta \mathcal{G}^t|\mathcal{G}^{t-1}_p,\theta).$$9, the vertex stochastic process is written as

$$\mathcal{G}=\{ \mathcal{G}^{(1)}, \mathcal{G}^{(2)},\dots, \mathcal{G}^{(T)} \}, \qquad \mathcal{G}^{(t)}=(\mathcal{V}^{(t)},\mathcal{E}^{(t)}),$$0

the edge stochastic process as

$$\mathcal{G}=\{ \mathcal{G}^{(1)}, \mathcal{G}^{(2)},\dots, \mathcal{G}^{(T)} \}, \qquad \mathcal{G}^{(t)}=(\mathcal{V}^{(t)},\mathcal{E}^{(t)}),$$1

and the resulting dynamic graph as a probability space $$\mathcal{G}=\{ \mathcal{G}^{(1)}, \mathcal{G}^{(2)},\dots, \mathcal{G}^{(T)} \}, \qquad \mathcal{G}^{(t)}=(\mathcal{V}^{(t)},\mathcal{E}^{(t)}),$$2. A task over an interval $$\mathcal{G}=\{ \mathcal{G}^{(1)}, \mathcal{G}^{(2)},\dots, \mathcal{G}^{(T)} \}, \qquad \mathcal{G}^{(t)}=(\mathcal{V}^{(t)},\mathcal{E}^{(t)}),$$3 is

$$\mathcal{G}=\{ \mathcal{G}^{(1)}, \mathcal{G}^{(2)},\dots, \mathcal{G}^{(T)} \}, \qquad \mathcal{G}^{(t)}=(\mathcal{V}^{(t)},\mathcal{E}^{(t)}),$$4

with cumulative cost

$$\mathcal{G}=\{ \mathcal{G}^{(1)}, \mathcal{G}^{(2)},\dots, \mathcal{G}^{(T)} \}, \qquad \mathcal{G}^{(t)}=(\mathcal{V}^{(t)},\mathcal{E}^{(t)}),$$5

The global objective is expressed as

$$\mathcal{G}=\{ \mathcal{G}^{(1)}, \mathcal{G}^{(2)},\dots, \mathcal{G}^{(T)} \}, \qquad \mathcal{G}^{(t)}=(\mathcal{V}^{(t)},\mathcal{E}^{(t)}),$$6

This viewpoint models graph continual learning as a dynamical system whose value function evolves under changes in features, connectivity, and parameters (Raghavan et al., 2023).

To solve that problem, the paper defines a two-player sequential min-max game,

$$\mathcal{G}=\{ \mathcal{G}^{(1)}, \mathcal{G}^{(2)},\dots, \mathcal{G}^{(T)} \}, \qquad \mathcal{G}^{(t)}=(\mathcal{V}^{(t)},\mathcal{E}^{(t)}),$$7

with a composite loss

$$\mathcal{G}=\{ \mathcal{G}^{(1)}, \mathcal{G}^{(2)},\dots, \mathcal{G}^{(T)} \}, \qquad \mathcal{G}^{(t)}=(\mathcal{V}^{(t)},\mathcal{E}^{(t)}),$$8

The maximizing player simulates worst-case task shifts, while the minimizing player adapts the GNN to reduce forgetting. Under Lipschitz and bounded-gradient assumptions, the paper claims existence of a local minimax or Stackelberg equilibrium and proves gradient convergence for the maximizing and minimizing players at rate $$\mathcal{G}=\{ \mathcal{G}^{(1)}, \mathcal{G}^{(2)},\dots, \mathcal{G}^{(T)} \}, \qquad \mathcal{G}^{(t)}=(\mathcal{V}^{(t)},\mathcal{E}^{(t)}),$$9 and $$\mathcal{D}^{(t)} = (\mathbf{A}^{(t)}, \mathbf{X}^{(t)})$$0, respectively (Raghavan et al., 2023).

For continual causal graph learning, the significance of this line of work is structural rather than directly causal. It provides explicit machinery for evolving node sets, edge sets, and graph features over time, but it does not define causal adjacency, intervention semantics, or identifiability constraints. This suggests that graph continual learning supplies the nonstationary substrate on which a causal graph learner would have to operate.

5. Memory, replay, and graph-aware forgetting

Replay and memory compression occupy a central place in continual graph learning because graph updates affect only parts of the old structure, yet those local changes can propagate widely through message passing. The survey organizes existing methods into regularization-based, replay-based, and architecture-based families. Weight-based regularization includes TWP, which combines a loss-based importance score with a topology-based importance score. Replay-based methods include ContinualGNN, which samples nodes according to perturbation propagation and reservoir-style criteria, and ER-GNN, which proposes prototype, coverage, and influence maximization criteria. The survey’s general conclusion is that graph-aware replay is especially aligned with graph-level dependency because it can explicitly focus on perturbed regions rather than treating the memory buffer as an IID store (Yuan et al., 2023).

A more specialized instance is the "Condensation-Concatenation Framework for Dynamic Graph Continual Learning," abbreviated CCC. CCC first condenses historical graph snapshots into compact semantic graphs while aiming to preserve the original label distribution and topological properties, then trains a dynamic GNN on the condensed historical sequence to obtain historical embeddings, identifies nodes affected by structural changes in the current graph, and concatenates historical embeddings with current graph representations selectively. The condensed graph is written as

$$\mathcal{D}^{(t)} = (\mathbf{A}^{(t)}, \mathbf{X}^{(t)})$$1

with edges determined by cosine similarity,

$$\mathcal{D}^{(t)} = (\mathbf{A}^{(t)}, \mathbf{X}^{(t)})$$2

Historical embeddings are generated using a dynamic GNN, illustrated with EvolveGCN,

$$\mathcal{D}^{(t)} = (\mathbf{A}^{(t)}, \mathbf{X}^{(t)})$$3

and fused with current embeddings by

$$\mathcal{D}^{(t)} = (\mathbf{A}^{(t)}, \mathbf{X}^{(t)})$$4

Selective replay is restricted to an influence region around structural modifications,

$$\mathcal{D}^{(t)} = (\mathbf{A}^{(t)}, \mathbf{X}^{(t)})$$5

so only nodes deemed affected by graph updates receive historical concatenation (Yan et al., 12 Dec 2025).

CCC also refines the forgetting measure to fit dynamic graphs: $$\mathcal{D}^{(t)} = (\mathbf{A}^{(t)}, \mathbf{X}^{(t)})$$6 where $$\mathcal{D}^{(t)} = (\mathbf{A}^{(t)}, \mathbf{X}^{(t)})$$7 denotes nodes correctly predicted at task $$\mathcal{D}^{(t)} = (\mathbf{A}^{(t)}, \mathbf{X}^{(t)})$$8 and $$\mathcal{D}^{(t)} = (\mathbf{A}^{(t)}, \mathbf{X}^{(t)})$$9 denotes nodes incorrectly predicted at task $s_t$0. Performance Mean is

$s_t$1

Across Arxiv, Paper100M, DBLP, and Elliptic, CCC is reported to achieve the lowest FM in every case and competitive PM, often near the best baseline. On Arxiv and Paper100M it has the best PM and best FM; on DBLP it does not have the highest PM but has the best FM; on Elliptic it has near-best PM and best FM (Yan et al., 12 Dec 2025).

For continual causal graph learning, the transfer of these ideas is conceptual. Condensation suggests a memory mechanism for compressed historical structure; selective replay around changed regions suggests a way to revisit variables whose neighborhoods have shifted; refined forgetting metrics suggest how one might track structural degradation over time. But the preserved object in CCC is predictive and topological information, not causal adjacency or causal effect.

6. Limits, misconceptions, and open directions

A recurrent misconception is to equate continual causal graph learning with any one of online causal prediction, dynamic graph learning, or graph continual learning. The literature explicitly resists that conflation. "Learning Causal Models Online" learns predictors, feature masks, potentially useful representations, and an implicit distinction between stable and unstable predictors, but it does not recover a causal DAG, identify causal parents or children, estimate structural equations, or perform graph search or edge orientation. Its criterion is stability of predictive relations over time, not a causal graph test in the classical sense (Javed et al., 2020).

The same caution applies on the graph side. CCC is not a causal graph learning method: it does not infer causal edges, causal mechanisms, or interventions; its condensation is similarity-based; its selective replay is heuristic; it has no explicit treatment of concept drift versus structural drift; and its experiments are limited to node classification rather than downstream causal tasks, edge causality, or time-varying causal discovery. The paper itself frames its relevance to causal graph learning as indirect, noting that a true continual causal graph learning framework would require a causal discovery objective, interventional or counterfactual reasoning, causal identifiability constraints such as acyclicity or temporal ordering, a causal notion of forgetting, and explicit mechanism-change modeling (Yan et al., 12 Dec 2025).

The survey of continual graph learning broadens the list of unresolved issues. It identifies graph robustness, compatibility beyond GNNs, multi-modality, heterogeneous graphs, and better topology-aware design as open problems, and notes that many studies remain concentrated on relatively simple benchmarks such as node classification on citation networks (Yuan et al., 2023). The dynamical-system paper supplies smoothness-based equilibrium and convergence guarantees for continual graph learning, but those guarantees are not causal identifiability guarantees; they concern optimization of a graph-continual-learning game under bounded gradients, not recovery of invariant mechanisms (Raghavan et al., 2023).

Two operational lessons emerge clearly. First, temporal order can itself be the signal that distinguishes stable from spurious relations: in the online causal-predictor setting, shuffling destroys the very evidence needed for causal discrimination (Javed et al., 2020). Second, preserving historical graph structure requires replay and regularization mechanisms that are topology-aware rather than IID. Taken together, these points suggest that a mature continual causal graph learning framework would need to combine temporal invariance tests, explicit structural discovery objectives, and graph-aware memory over evolving relational domains.