CLDG++: Dynamic Graph Representation Learning

• CLDG++ is an unsupervised dynamic graph representation learning framework that utilizes timespan views and diffusion to capture local and global structures.
• It employs temporal translation invariance by contrasting node embeddings across timespans, thereby enhancing anomaly detection and embedding quality.
• The framework avoids explicit sequence models by using multi-scale contrastive losses over local-local, local-global, and global-global relationships to boost efficiency and accuracy.

CLDG++ is an unsupervised framework for dynamic graph representation learning that extends CLDG by combining timespan view contrastive learning with graph diffusion and a multi-scale objective over local and global structural views. It is designed for dynamic graphs with timestamped edges, targets both general-purpose node embedding learning and dynamic graph anomaly detection, and avoids explicit sequence models such as RNNs, Transformers, or temporal attention by injecting temporal information through the construction and comparison of timespan views (Xu et al., 26 May 2026). In the lineage described by the authors, CLDG introduced temporal translation invariance as the central inductive bias for contrastive learning on dynamic graphs, while CLDG++ adds global contextual views and cross-scale alignment to improve representation quality and anomaly sensitivity (Xu et al., 2024).

1. Problem setting and antecedents

CLDG++ is formulated for both discrete-time dynamic graphs and continuous-time dynamic graphs. A discrete-time dynamic graph is written as

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

whereas a continuous-time dynamic graph is written as

$$\mathcal{G} = (\mathcal{V}, \mathcal{E}, \mathcal{T}),$$

with $$\mathcal{T} : \mathcal{V} \cup \mathcal{E} \to \mathbb{R}^+$$ assigning timestamps to nodes or edges (Xu et al., 26 May 2026). The representation-learning objective is to learn

$$f: v_i \mapsto \mathbf{z}_i \in \mathbb{R}^d, \quad d \ll |\mathcal{V}|,$$

so that $\mathbf{z}_i$ preserves both structural and temporal information; the anomaly-detection setting further associates nodes with anomaly scores, where higher scores indicate stronger anomalousness (Xu et al., 26 May 2026).

The framework is situated against two limitations of prior graph contrastive learning. First, existing static graph contrastive methods typically construct positive pairs through synthetic augmentations such as node dropping, edge perturbation, feature masking, or graph diffusion, and then optimize InfoNCE or related mutual-information objectives. Second, on dynamic graphs such augmentations can aggravate semantic drift because the graph is already evolving over time; artificial perturbations may remove temporally critical edges or introduce semantically irrelevant links (Xu et al., 2024). CLDG addresses this by using natural temporal evolution rather than synthetic corruption as the source of multiple views, and CLDG++ preserves that design while expanding the notion of view from explicit topology alone to both explicit and diffused structure (Xu et al., 26 May 2026).

In this sense, CLDG++ is not merely a larger variant of CLDG. It retains the local-local timespan contrast mechanism of CLDG but augments it with graph diffusion and a three-part contrastive decomposition over local-local, local-global, and global-global relationships. This suggests a shift from purely temporally aligned local structure to temporally aligned multi-scale structure (Xu et al., 26 May 2026).

2. Temporal translation invariance

The conceptual core of CLDG++ is temporal translation invariance (TTI), defined as the tendency of the identical node to keep similar labels across different timespans. The empirical basis reported by the authors is that, across multiple real dynamic graphs and multiple GNN encoders including GCN, GAT, and GraphSAGE, the predicted labels of the same node tend to remain the same across different timespans (Xu et al., 26 May 2026). The earlier CLDG paper describes the same phenomenon as an empirical regularity observed when classifiers are trained independently on different time spans and the same node’s predicted labels are compared across them (Xu et al., 2024).

Formally, TTI is expressed as approximate invariance of labels and representations across timespans:

$$y_i^t \approx y_i^{t'} \quad \text{and} \quad \mathbf{z}_i^{\mathcal{T}_t} \approx \mathbf{z}_i^{\mathcal{T}_{t'}} \quad \forall\, t,t'.$$

In CLDG and CLDG++, this is not introduced as a separate regularizer; it is encoded directly in the contrastive pairing strategy. Positive pairs always involve the same node across different timespan views, and InfoNCE is used to maximize their similarity while separating them from other nodes in the counterpart view (Xu et al., 26 May 2026).

The importance of TTI is methodological as well as conceptual. It turns temporal persistence into a self-supervised signal. Instead of assuming that a hand-crafted augmentation preserves semantics, the framework assumes that temporal shifts over suitable timespans preserve semantics for many nodes. The authors also make explicit that this assumption is not universal. Nodes that legitimately change roles over time, such as through career change or company pivot, may violate TTI without being anomalous, and in such settings the invariance assumption may need to be relaxed or adapted (Xu et al., 26 May 2026). This caveat is central to interpreting both the representation-learning objective and the anomaly detector.

3. Timespan views and architectural components

CLDG++ constructs views by sampling temporal windows rather than perturbing graph structure. For a continuous-time dynamic graph, the total time span is

$\Delta t = \max(\mathcal{T}) - \min(\mathcal{T}).$

Given hyperparameters $v$ and $s$, where $v$ is the number of views and $$\mathcal{G} = (\mathcal{V}, \mathcal{E}, \mathcal{T}),$$0 is the timespan factor, each view covers an interval of length $$\mathcal{G} = (\mathcal{V}, \mathcal{E}, \mathcal{T}),$$1. Center times are sampled as

$$\mathcal{G} = (\mathcal{V}, \mathcal{E}, \mathcal{T}),$$2

with each $$\mathcal{G} = (\mathcal{V}, \mathcal{E}, \mathcal{T}),$$3 lying within the feasible boundary interval, and the $$\mathcal{G} = (\mathcal{V}, \mathcal{E}, \mathcal{T}),$$4-th view keeps edges whose timestamps fall in

$$\mathcal{G} = (\mathcal{V}, \mathcal{E}, \mathcal{T}),$$5

This produces local timespan views

$$\mathcal{G} = (\mathcal{V}, \mathcal{E}, \mathcal{T}),$$6

where each sampled view is denoted

$$\mathcal{G} = (\mathcal{V}, \mathcal{E}, \mathcal{T}),$$7

CLDG++ further constructs a diffusion matrix $$\mathcal{G} = (\mathcal{V}, \mathcal{E}, \mathcal{T}),$$8 for each view to provide a global perspective (Xu et al., 26 May 2026).

Four sampling strategies are specified: sequential, high-overlap, low-overlap, and random. Sequential sampling uses non-overlapping snapshots; high-overlap uses 75% overlap; low-overlap uses 25% overlap; random sampling uses random center times subject to boundary constraints. The reported empirical pattern is that lower-overlap strategies, especially sequential and random, tend to work better, whereas high overlap makes the contrastive task too trivial (Xu et al., 26 May 2026). The earlier CLDG study reports the same pattern and interprets it as evidence that temporal view construction should encourage semantic invariance rather than mere physical near-identity (Xu et al., 2024).

The backbone architecture is built from two encoders in CLDG++: a local encoder $$\mathcal{G} = (\mathcal{V}, \mathcal{E}, \mathcal{T}),$$9 operating on adjacency matrices and a global encoder $$\mathcal{T} : \mathcal{V} \cup \mathcal{E} \to \mathbb{R}^+$$0 operating on diffusion matrices. For a view $$\mathcal{T} : \mathcal{V} \cup \mathcal{E} \to \mathbb{R}^+$$1,

$$\mathcal{T} : \mathcal{V} \cup \mathcal{E} \to \mathbb{R}^+$$2

The two encoders do not share weights, and both are instantiated as multi-layer GCNs in the main implementation, with a 2-layer GCN as the default backbone (Xu et al., 26 May 2026). A shared projection head maps representations to the contrastive space:

$$\mathcal{T} : \mathcal{V} \cup \mathcal{E} \to \mathbb{R}^+$$3

where the projection head consists of a fully connected layer, $$\mathcal{T} : \mathcal{V} \cup \mathcal{E} \to \mathbb{R}^+$$4 normalization, and LeakyReLU (Xu et al., 26 May 2026). Experiments also report that CLDG and CLDG++ work well with GAT and GraphSAGE, which the authors interpret as encoder agnosticism (Xu et al., 26 May 2026).

4. Graph diffusion and the multi-scale contrastive objective

The distinguishing addition in CLDG++ is the use of graph diffusion to expose global contextual correlations between nodes. For each view adjacency matrix $$\mathcal{T} : \mathcal{V} \cup \mathcal{E} \to \mathbb{R}^+$$5 with degree matrix $$\mathcal{T} : \mathcal{V} \cup \mathcal{E} \to \mathbb{R}^+$$6, two diffusion operators are used. The Personalized PageRank diffusion matrix is

$$\mathcal{T} : \mathcal{V} \cup \mathcal{E} \to \mathbb{R}^+$$7

and the heat-kernel diffusion matrix is

$$\mathcal{T} : \mathcal{V} \cup \mathcal{E} \to \mathbb{R}^+$$8

The paper states that PPR is used for most datasets, while the heat kernel is used for Bitcoinotc and TAX51 (Xu et al., 26 May 2026).

This diffusion step changes the representational semantics of a view. The adjacency matrix $$\mathcal{T} : \mathcal{V} \cup \mathcal{E} \to \mathbb{R}^+$$9 captures direct-neighbor topology and is treated as a local view. The diffusion matrix $$f: v_i \mapsto \mathbf{z}_i \in \mathbb{R}^d, \quad d \ll |\mathcal{V}|,$$0 captures soft connectivity through multi-hop propagation and is treated as a global view. The authors state that this helps recover implicit links and can be useful for phenomena such as camouflaged fraud communities (Xu et al., 26 May 2026). The heatmaps described for adjacency and diffusion matrices show that diffusion yields more non-zero interactions and smoother patterns across nodes.

CLDG++ then optimizes a multi-scale contrastive loss over three relation types. For node $$f: v_i \mapsto \mathbf{z}_i \in \mathbb{R}^d, \quad d \ll |\mathcal{V}|,$$1 and timespan views $$f: v_i \mapsto \mathbf{z}_i \in \mathbb{R}^d, \quad d \ll |\mathcal{V}|,$$2 and $$f: v_i \mapsto \mathbf{z}_i \in \mathbb{R}^d, \quad d \ll |\mathcal{V}|,$$3 with $$f: v_i \mapsto \mathbf{z}_i \in \mathbb{R}^d, \quad d \ll |\mathcal{V}|,$$4, the total loss is

$$f: v_i \mapsto \mathbf{z}_i \in \mathbb{R}^d, \quad d \ll |\mathcal{V}|,$$5

All $$f: v_i \mapsto \mathbf{z}_i \in \mathbb{R}^d, \quad d \ll |\mathcal{V}|,$$6 terms are InfoNCE, with the positive pair defined by the same node and the negative set formed by other nodes in the counterpart view (Xu et al., 26 May 2026).

The three components can be grouped as follows:

Component	Positive relation	Function
$$f: v_i \mapsto \mathbf{z}_i \in \mathbb{R}^d, \quad d \ll |\mathcal{V}|,$$7	same node, local view, different timespans	enforces TTI in local space
$$f: v_i \mapsto \mathbf{z}_i \in \mathbb{R}^d, \quad d \ll |\mathcal{V}|,$$8	same node, global view, different timespans	enforces TTI in global space
$$f: v_i \mapsto \mathbf{z}_i \in \mathbb{R}^d, \quad d \ll |\mathcal{V}|,$$9	same node, local and global view, same timespan	aligns local and global structure

In the reported formulation, these terms are summed equally:

$\mathbf{z}_i$0

CLDG is recovered by retaining only $\mathbf{z}_i$1; the paper states that CLDG “exclusively focuses on local-local contrast” (Xu et al., 26 May 2026). The ablation results attribute the largest contribution to the local-local term, but they also report consistent gains from both local-global and global-global terms, supporting the claim that diffusion-based global structure and cross-scale alignment improve representation capacity (Xu et al., 26 May 2026).

5. Training protocol and anomaly detection

The training pipeline per epoch consists of timespan view sampling, diffusion computation for each sampled view, mini-batch neighborhood sampling, encoding through the local and global GNNs, projection to contrastive space, InfoNCE evaluation for all LL, GG, and LG pairs, and optimization with Adam using learning rate $\mathbf{z}_i$2 and weight decay $\mathbf{z}_i$3 (Xu et al., 26 May 2026). The model is trained purely unsupervised. For node classification, embeddings are frozen after pretraining and a linear classifier is fitted on labeled nodes only, following the linear evaluation protocol (Xu et al., 26 May 2026).

CLDG++ is also directly connected to dynamic graph anomaly detection. The underlying premise is that anomalies usually violate temporal translation invariance. After training, the inference stage fixes the model and uses sequential sampling to obtain $\mathbf{z}_i$4 views. For each node $\mathbf{z}_i$5, local representations are collected across views:

$\mathbf{z}_i$6

A consistency distance between two embeddings is defined as

$\mathbf{z}_i$7

that is, one minus cosine similarity. The average pairwise inconsistency for node $\mathbf{z}_i$8 is

$\mathbf{z}_i$9

and the anomaly score is

$$y_i^t \approx y_i^{t'} \quad \text{and} \quad \mathbf{z}_i^{\mathcal{T}_t} \approx \mathbf{z}_i^{\mathcal{T}_{t'}} \quad \forall\, t,t'.$$0

Large mean inconsistency indicates strong change across timespans, and large variance indicates unstable or unpredictable change. Nodes with large $$y_i^t \approx y_i^{t'} \quad \text{and} \quad \mathbf{z}_i^{\mathcal{T}_t} \approx \mathbf{z}_i^{\mathcal{T}_{t'}} \quad \forall\, t,t'.$$1 are therefore treated as anomalous (Xu et al., 26 May 2026).

A notable design choice is that the anomaly discriminator uses local embeddings, even though CLDG++ trains with both local and global information. The authors state that the training stage benefits from global information, which can make anomalies more detectable, including structural anomalies in cliques and camouflaged fraud rings (Xu et al., 26 May 2026). A plausible implication is that global-context training improves the geometry of the local embedding space used during scoring, without requiring the anomaly score itself to be defined over multiple embedding types.

6. Empirical results, efficiency, and limitations

The evaluation reported for CLDG++ uses seven dynamic graphs from multiple domains: DBLP, Bitcoinotc, BITotc, BITalpha, TAX, TAX51, and Reddit (Xu et al., 26 May 2026). For node classification, the comparison includes supervised baselines such as LP, GCN, GAT, and GraphSAGE; dynamic unsupervised baselines such as CAW, TGAT, DySAT, and MNCI; and static contrastive baselines such as DGI, GRACE, MVGRL, and CCA-SSG (Xu et al., 26 May 2026). The key summary is that CLDG outperforms all unsupervised baselines on almost all metrics, and CLDG++ further improves over CLDG on 10 of 14 node-classification metrics across the seven datasets (Xu et al., 26 May 2026). On DBLP, for example, GraphSAGE as a supervised baseline attains 72.36% accuracy and 71.99% weighted F1, while CLDG++ as an unsupervised method attains 72.94% accuracy and 72.69% weighted F1 (Xu et al., 26 May 2026).

For anomaly detection, anomalies are synthetically injected following standard practice, including structural cliques and attribute anomalies (Xu et al., 26 May 2026). The paper reports that CLDG already outperforms all baselines and that CLDG++ achieves the best AUC on all seven datasets (Xu et al., 26 May 2026). On DBLP, the best baseline listed is SAD with 78.24% AUC, CLDG reaches 84.89% AUC, and CLDG++ reaches 86.41% AUC (Xu et al., 26 May 2026). Qualitative visualization on BITalpha is described as showing that CLDG++ accurately catches synthetic structural anomalies that CoLA misses.

Efficiency is a second major empirical claim. The encoder time complexity is

$$y_i^t \approx y_i^{t'} \quad \text{and} \quad \mathbf{z}_i^{\mathcal{T}_t} \approx \mathbf{z}_i^{\mathcal{T}_{t'}} \quad \forall\, t,t'.$$2

with parameter space complexity

$$y_i^t \approx y_i^{t'} \quad \text{and} \quad \mathbf{z}_i^{\mathcal{T}_t} \approx \mathbf{z}_i^{\mathcal{T}_{t'}} \quad \forall\, t,t'.$$3

The paper contrasts this with dynamic graph methods that integrate node or edge embeddings into parameters and thereby incur additional $$y_i^t \approx y_i^{t'} \quad \text{and} \quad \mathbf{z}_i^{\mathcal{T}_t} \approx \mathbf{z}_i^{\mathcal{T}_{t'}} \quad \forall\, t,t'.$$4 parameter footprint (Xu et al., 26 May 2026). Empirically, CLDG has a constant parameter count of about 0.05M across datasets, and CLDG++ increases this only slightly to 0.06M (Xu et al., 26 May 2026). Training-time comparisons show that CLDG is 30–900x faster than sequence-based methods on large datasets; on TAX51, it is described as about 946x faster than CAW and 158x faster than TGAT (Xu et al., 26 May 2026). The earlier CLDG paper summarizes average reductions of about 2,001.86 times in parameter count and 130.31 times in training time across seven datasets (Xu et al., 2024).

The principal limitations are explicit. CLDG and CLDG++ model temporal evolution implicitly through contrastive learning across timespans rather than through fine-grained sequence modeling, so they do not exploit event order or other detailed sequence patterns (Xu et al., 26 May 2026). TTI may fail in domains where node roles legitimately change over time, in which case the framework may penalize genuine evolution rather than isolate anomalies (Xu et al., 26 May 2026). CLDG++ also requires computing diffusion matrices per view, which adds preprocessing cost; for very large graphs and many views this remains non-trivial, even though the authors characterize it as far cheaper than sequence models (Xu et al., 26 May 2026).

Taken together, CLDG++ occupies a specific position in dynamic graph learning: it treats time itself as the source of contrastive views, formalizes node-level temporal persistence through temporal translation invariance, and enriches that persistence signal with diffusion-based global context. Its reported gains in node classification, anomaly detection, and efficiency follow from that combination rather than from explicit temporal sequence modeling (Xu et al., 26 May 2026).