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Dynamic Graph Prompting via Topology-Routed Mixed-Curvature Experts

2026 · arxiv_cs

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Dynamic Graph Prompting via Topology-Routed Mixed-Curvature Experts Quanxin Wang1 , Xuanting Xie1 , Bingheng Li2 , Xingtong Yu3 , Shuo Wang1,4 , Ruiyi Fang5 , Zhao Kang1∗ 1

University of Electronic Science and Technology of China 2 Michigan State University 3 Singapore Management University 4 Tsinghua University 5 Western University tranciewang@gmail.com, x624361380@outlook.com, zkang@uestc.edu.cn

arXiv:2608.06031v1 [cs.LG] 6 Aug 2026

Abstract Dynamic graph prompting freezes a pre-trained temporal backbone and adapts it to label-scarce downstream tasks using lightweight prompts. However, existing methods operate within a single, fixed embedding space. In this work, we reveal that temporal shifts in local clustering and degree heterogeneity actively reorganize the edge curvature spectrum— indicating that the optimal representation geometry dynamically evolves with local topology over time. We formalize this unaddressed mismatch as geometry under-adaptation. To overcome this limitation, we propose CurvPrompt, a topology-routed geometry prompting framework for dynamic graphs. Instead of relying on a single space, CurvPrompt maintains a bank of curvature-diverse Riemannian experts, each paired with a learnable prompt. A topology-aware gate dynamically routes each node–time instance to a sparse subset of experts, constructing a personalized mixed-curvature representation. To ensure parameter efficiency and training stability under extreme label scarcity, CurvPrompt employs soft routing during pre-training to build a continuous topology– geometry mapping, and transitions to hard Top-K routing with uniform weights during downstream adaptation. Extensive experiments across four benchmark datasets show that CurvPrompt significantly advances few-shot link prediction while delivering strong, consistent performance on node classification tasks, validating the necessity of geometry-adaptive prompting.

Introduction Dynamic graphs, in which nodes and edges continuously emerge, vanish, and reconnect over time, arise in a wide range of real-world systems, including social networks, ecommerce platforms, and online education (Kumar, Zhang, and Leskovec 2019; Barros et al. 2021). Dynamic graph neural networks (DGNNs) have emerged as the mainstream technique for modeling such evolving interactions, and are typically pre-trained with temporal link prediction (Xu et al. 2020; Rossi et al. 2020; Yu et al. 2023a). However, this objective leaves a substantial gap from downstream tasks such as node classification, where labeled data are often scarce. Inspired by the success of prompt learning on static graphs (Liu et al. 2023; Sun et al. 2022b, 2023; Fang et al. 2024), recent dynamic graph prompting methods (Yu et al. 2025b; Chen et al. 2024) freeze the pre-trained backbone and adapt it to downstream objectives by tuning only lightweight ∗

Corresponding author.

prompts. How to efficiently transfer a pre-trained dynamic graph model to label-limited downstream tasks has therefore become an important problem. Despite this, these methods largely inherit a fixed-geometry assumption: both the frozen backbone and the prompts operate in a single, predefined embedding space. This assumption collapses under varying local topology: real-world networks exhibit heterogeneous structures—treelike hierarchies, dense communities, and cyclic patterns—that map naturally to distinct constant-curvature geometries (Chami et al. 2019; Nickel and Kiela 2017; Bachmann, Bécigneul, and Ganea 2020). In dynamic graphs, such heterogeneity evolves across both space and time as neighborhoods continuously reconfigure. Figure 1 provides direct empirical evidence on MOOC in (a): mean local clustering falls from 0.88 to 0.71 while degree variance grows fourfold. (b–c): Concurrently, the edge Ollivier–Ricci curvature spectrum reorganizes toward the negative regime from day 0 (-0.02) to day 30 (-0.15), with near-zero edges shrinking from 75% to 39% and negative-curvature edges rising from 17% to 56%. These observations suggest that the locally preferred representation geometry may evolve from approximately Euclidean-like toward more hyperbolic-like structures over time. We term this mismatch geometry under-adaptation: task-level prompts may be updated in existing works, yet the representation geometry remains time-invariant and fails to track topology-induced curvature drift (Wang et al. 2026; Sun et al. 2022a; Yang et al. 2026; Guo et al. 2025). To enable geometry-adaptive prompting for dynamic graphs, two main issues must be addressed. First, designing effective prompts that guide each node–time instance toward its optimal representation geometry remains an open challenge, as it introduces new layers of complexity related to topology-aware conditioning, temporal evolution, and instance-level geometric heterogeneity. Second, such geometry adaptation must remain parameter-efficient. Existing few hyperbolic space models (Yang et al. 2026; Guo et al. 2025; Bachmann, Bécigneul, and Ganea 2020) enable mixed-curvature routing, but train all experts end-to-end per task, incurring heavy computation and overfitting under fewshot labels. These considerations lead to our central question: How can prompts guide each evolving neighborhood toward its optimal representation geometry while preserving

(a) MOOC: topology drift

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Figure 1: Topology-induced geometry drift on MOOC (1-day windows, 30 days). (a) Mean local clustering falls while degree variance rises. (b) Edge Ollivier–Ricci curvature (ORC) distributions over time from day 0 (back/inner) to day 30 (front/outer) (black: daily mean κ̄(t)). (c) Curvature strata (τ =0.05): near-zero edges shrink while negative-curvature edges expand. parameter-efficient downstream adaptation? To answer this question, we propose CurvPrompt, a topology-routed geometry prompting framework for dynamic graphs. CurvPrompt replaces single-space prompting with a bank of curvature-diverse Riemannian prompt experts and routes each node–time instance to a sparse expert subset based on its multi-resolution temporal topology. This yields personalized mixed-curvature representations and routing-weighted geodesic metrics, enabling both embedding and scoring geometries to adapt jointly to evolving local structures. Additionally, to ensure parameter efficiency, we introduce a soft-to-hard routing strategy, where soft routing learns topology–geometry associations during pre-training, while hard routing reuses the frozen routing mechanism for lightweight downstream prompt tuning. To summarize, our contributions are as follows: • We identify geometry under-adaptation in dynamic graph prompting and address it through topology-conditioned routing over curvature-diverse experts. • To the best of our knowledge, CurvPrompt is the first Riemannian mixture-of-experts framework for graph prompt learning, adapting representation and scoring geometries at the node–time level while remaining parameter-efficient. • Extensive experiments on four benchmarks demonstrate strong and consistent performance across tasks, validating the effectiveness of geometry-adaptive prompting.

Related Work Dynamic graph learning. Many real-world graphs evolve continuously over time, motivating the development of continuous-time dynamic graph learning methods. Existing approaches generally update node representations by propagating and aggregating temporal messages from neighboring nodes (Skarding, Gabrys, and Musial 2021; Barros et al. 2021). Representative techniques include dynamic random

walks for capturing structural evolution (Nguyen et al. 2018; Wang et al. 2021), temporal encoders that fuse structural and temporal information (Xu et al. 2020; Cong et al. 2022; Rossi et al. 2020; Yu et al. 2023a), and temporal point process models for characterizing interaction dynamics (Kumar, Zhang, and Leskovec 2019; Trivedi et al. 2019; Wen and Fang 2022). Despite their success, most of these methods are optimized using link prediction objectives, whereas downstream applications often involve different tasks such as node classification. The resulting objective mismatch limits knowledge transfer and weakens downstream generalization (Chen et al. 2022; Yu et al. 2025b). Graph prompt learning. Prompt learning was originally introduced in natural language processing to reduce the discrepancy between pre-training and downstream objectives (Brown et al. 2020). Instead of updating the entire pretrained model, it adapts task-specific prompts while keeping the backbone fixed. This paradigm has recently achieved promising results on static graphs (Liu et al. 2023; Sun et al. 2023; Fang et al. 2024; Tan et al. 2023; Yu et al. 2023b, 2025c, 2024, 2025a,d, 2023c), but remains relatively underexplored for dynamic graphs. TIGPrompt (Chen et al. 2024), DyGPrompt (Yu et al. 2025b), and DDGPrompt (Peng et al. 2026) adapt temporal prompts for dynamic graphs, yet still operate in a fixed Euclidean space without geometry adaptation. Prompt-expert mixtures have also been explored for static graph foundation models (Wang et al. 2025), but remain Euclidean and non-temporal. Riemannian learning on graphs. Constant-curvature and mixed-curvature models show that hierarchical, communitylike, and cyclic structures favor different geometries (Chami et al. 2019; Gu et al. 2019; Bachmann, Bécigneul, and Ganea 2020). Curvature-stratified evaluation further suggests that model effectiveness is geometry-dependent (Wang et al. 2026). On dynamic graphs, Riemannian methods remain comparatively limited: some adopt time-varying or hyperbolic geometries for temporal graphs (Sun et al. 2022a), while

recent mixture-of-experts approaches learn topology-aware routing over curvature-diverse manifolds for static topological heterogeneity (Guo et al. 2025; Cao et al. 2026) or evolving dynamic topologies (Yang et al. 2026). These models typically train experts and routers end-to-end per task, leaving the Riemannian framework in prompt learning underexplored.

Methodology In this section, we present CurvPrompt, a dynamic Riemannian mixture-of-prompt-experts framework for temporal graph learning. The proposed framework contains four key components: (1) curvature-diverse Riemannian experts, (2) expert-specific prompts together with node–time conditional prompts, (3) a topology-aware temporal gating mechanism, and (4) mixed-curvature representation alignment.

Problem Formulation Dynamic graph. We consider a continuous-time dynamic graph G = (V, E, T ), where V is the node set, E is the temporal interaction set, and T is the time domain. Each event (vi , vj , t) ∈ E represents an interaction between nodes vi and vj at time t. Each node v is associated with a temporal feature vector xt,v ∈ Rd at time t. We use Nu (t) = {(v, t′ ) : (v, u, t′ ) ∈ E, t′ < t}

(1)

to denote the historical temporal neighborhood of node u before time t. A time encoder TE : T → Rdt maps a timestamp to a temporal feature. A common implementation is r 1 TE(t) = cos(ω1 t), sin(ω1 t), . . . , dt (2) cos(ωdt /2 t), sin(ωdt /2 t) , d /2

t where {ωi }i=1 are learnable frequencies (Xu et al. 2020; Cong et al. 2022).

Constant-curvature manifolds. Let Mdκh denote a dh dimensional constant-curvature manifold. Depending on the sign of κ, it corresponds to a spherical space when κ > 0, a hyperbolic space when κ < 0, and a Euclidean space when κ = 0. We adopt the κ-stereographic model as a unified formulation. The exponential map expκ (·) and logarithmic map logκ (·) transform representations between the manifold and the tangent space (Bachmann, Bécigneul, and Ganea 2020).

Diverse Riemannian Experts Temporal graphs may contain hierarchical, cyclic, community-like, and approximately Euclidean structures at different time periods. A single Euclidean embedding space is therefore insufficient to faithfully represent all temporal substructures. To address this issue, we replace the Euclidean GNN experts in a conventional mixture-of-experts architecture with Riemannian experts operating in different constantcurvature spaces. The expert bank is denoted by E = {EH1 , EH2 , . . . , EEi , . . . , ESi } ,

(3)

where EHi is a hyperbolic expert with κ < 0, EE is a Euclidean expert with κ = 0, and ESi is a spherical expert with κ > 0. Riemannian expert update. We instantiate the experts on a Temporal Graph Network (TGN) backbone (Rossi et al. 2020). Let st,v denote the memory-augmented node state at (v, t): st,v = memv (t) + xt,v , (4) where memv (t) is the TGN memory of node v at time t and xt,v is the node feature. Let mt,v be the mean-pooled neighborhood message obtained by projecting concatenated neighbor, time-interval, and edge features (using only interactions in Nv (t)). An expert Eϵ with curvature κϵ applies a single-layer update: zϵt,v = expκϵ σ Wϵ,s logκϵ (st,v ) (5) + Wϵ,n logκϵ expκϵ (mt,v ) + ct,v + pϵ . where σ denotes GELU, pϵ is an expert-specific prompt, and ct,v is a shared node–time condition bias produced by a time condition-net (TCN) and a node condition-net (NCN) (Yu et al. 2025b). Diversity thus comes from the expert bank (multiple curvatures and prompts), while each expert itself uses a lightweight single-layer Riemannian update. Curvature diversity. Experts of the same geometry type are initialized with different curvature values to capture different degrees of geometric bending. The expert curvatures are learnable during pre-training.

Expert Prompts and Node–Time Conditioning Beyond the curvature-diverse expert bank, we introduce prompt and conditioning mechanisms that specialize experts and adapt node–time interfaces. Lightweight structure and time prompts on the TGN backbone serve as feature adapters, while our focus is the expert-specific prompts and the shared node–time condition bias used by both experts and the router. Expert-Specific Prompt Each expert Eϵ is equipped with a learnable prompt pϵ ∈ Rdh , injected additively into the tangent-space update in Eq. (5). This gives every curvature space an independent steer, promoting expert specialization without changing the expert backbone weights at downstream time. Time Prompt and Condition Bias A shared time prompt pT rescales the time encoding, ftT = pT ⊙ TE(t). Inspired by conditional prompt learning (Zhou et al. 2022; Yu et al. 2025b), we then build a node–time condition bias with dual bottleneck networks: a time condition-net (TCN) and a node condition-net (NCN), ct,v = TCN(ftT ) + NCN(xSt,v ),

(6)

where xSt,v denotes the (optionally structure-prompted) node feature. Unlike multiplicative dual-prompt pipelines, ct,v is added into both the expert update (Eq. (5)) and the router input below, so expert computation and expert selection share the same node–time context.

TIME Pre-train

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Top-K Expert Mixture (weighted sum) Figure 2: Overview of CurvPrompt. Left: temporal topology shifts (e.g., rising clustering) indicate changing curvature preferences. Center: a shared node–time condition ct,v drives Soft Top-K routing in pre-training and Hard Top-K routing downstream, while a frozen Temporal Graph Network (TGN) backbone provides st,v ; selected curvature-diverse Riemannian experts (H/E/S) with prompts pϵ are mixed. Right: Scoring, loss and few-shot tasks NC and LP.

Topology-Aware Temporal Gating Different temporal substructures fit different curvatures. We therefore route each node–time instance to a sparse subset of Riemannian experts with a topology-aware gate. Multi-resolution topology encoding. Given the retrieved temporal neighbors of v at time t (ordered by recency), we encode multi-scale local structure by mean-pooling the r most recent neighbors for each neighborhood size r ∈ R (e.g., {2, 5, 10}) and fusing the pooled features with the backbone state st,v :    Tt,v = σ Wfuse st,v Poolr {st′ ,u }(u,t′ )∈Nv (t)  . r∈R

(7) Here Poolr averages the first r neighbors in the retrieved list; r denotes a neighborhood size, not a hop-based radius. Conditioned Top-K routing. The router consumes topology together with the condition bias: ℓt,v = ϕgate (Dropout(Tt,v + ct,v )) ∈ RM . During pre-training we use soft Top-K routing  ϵ   P exp(ℓt,v /τg ) , ϵ ∈ Top -K, (ϵ) j gt,v = j∈Top -K exp(ℓt,v /τg )  0, otherwise,

contain hierarchical, community-like, and approximately flat patterns. During few-shot downstream tuning we switch to (ϵ) hard Top-K with uniform weights gt,v = 1/K on the selected experts, so that the model stably reuses the geometry selection learned in pre-training under scarce labels. Orthogonality, distortion and load-balancing losses further encourϵ age geometry-aware, non-collapsed routing. We write Wt,v (ϵ)

for the post-Top-K weight gt,v .

Mixture and Alignment Personalized mixed-curvature embedding. After each expert produces zϵt,v (Eq. (5)), we aggregate the selected experts in the shared ambient coordinates of the κstereographic model and apply a linear alignment to the backbone state: M X ϵ ẑt,v = Wt,v zϵt,v , ht,v = Walign ẑt,v . (10) ϵ=1

(8)

Geometry is specialized inside each expert; the gate forms a personalized mixed-curvature view for the same node–time instance.

(9)

Geometric scoring. For pairwise comparison we use a P ϵ routing-weighted curvature κ̄t,v = W κ and the ϵ t,v ϵ geodesic distance dκ̄ (ht,u , ht,v ).

Soft routing is also consistent with the mixed-geometry nature of temporal neighborhoods, which may simultaneously

Learning Objectives We adopt temporal link prediction using binary cross-entropy (BCE) for pre-training and reuse compatible task objectives

for downstream NC/LP, together with auxiliary regularizers: embedding distortion for geometry-aware routing and load balancing. Pre-Training Objective For each positive temporal interaction (v, a, t), we sample a negative node b that does not interact with v at time t, resulting in a tuple (v, a, b, t). Candidate pairs are scored with a lightweight decoder that takes the mixed-curvature embeddings and the routing-weighted geodesic distance dκ̄ (ht,u , ht,v ), yielding predicted probabilities ŷv,a , ŷv,b ∈ (0, 1). We optimize BCE: h X i Lpre = − log ŷv,a + log 1 − ŷv,b . (11) (v,a,b,t)∈Dpre

The distortion loss aligns Riemannian distances with local graph proximity. We sample pairs S in each mini-batch and use a neighborhood proxy g̃ (0/1/2/capped constant for identical, adjacent, shared-neighbor, and other pairs): 1 X LD = ℓδ dκ̄ (ht,u , ht,v ), g̃(u, v) , (12) |S| (u,v)∈S

where ℓδ is the smooth L1 loss. To avoid expert under-utilization, we apply a loadbalancing loss that pulls the average gate distribution toward uniform: 1 Lbal = MSE ḡ, 1 , (13) M where ḡ ∈ RM is the batch-averaged gate weight over M experts. To encourage expert specialization, we impose a soft orthogonality constraint on the expert prompts, where P = [p1 , . . . , pM ]⊤ stacks the expert prompts: 2

Lortho = PP⊤ − I F .

(14)

The overall pre-training objective is Lpre-total = Lpre + λ1 LD + λ2 Lbal + λ3 Lortho .

(15)

Downstream Prompt-Tuning Objective During downstream adaptation, we freeze the Riemannian experts, curvatures, topology encoder, router, TCN/NCN, and the pretrained time encoder, and update only the prompts and the task head. This preserves the geometric and routing knowledge from pre-training while keeping downstream adaptation parameter-efficient. Temporal node classification. Let Ddown = {(vi , yi , ti )} denote a labeled downstream dataset. We obtain class predictions ŷvi ,ti from a task decoder over the mixed embeddings and optimize BCE: Ldown = BCE ŷvi ,ti , yi . (16) Temporal link prediction. For downstream temporal link prediction, we reuse the same BCE objective as in Eq. (11). The overall prompt-tuning objective is primarily the task loss; auxiliary regularizers may be retained with small weights when beneficial: Ldown-total = Ldown + λ1 LD + λ2 Lbal .

(17)

Experiments Experimental Setup Datasets We evaluate four public dynamic graph datasets based on continuous-time temporal interaction networks: Wikipedia (Wiki), Reddit, MOOC (Kumar, Zhang, and Leskovec 2019), and Genre (Huang et al. 2023). Tasks and evaluation protocol To ensure a fair comparison, we evaluate temporal node classification and temporal link prediction under both transductive and inductive settings following DyGPrompt (Yu et al. 2025b). Given a chronologically ordered event stream, we use the first 80% of events for pre-training (once per dataset) and the remaining 20% for downstream adaptation, further split into 1%/1%/18% as the training pool, validation pool, and test set. Each few-shot task samples 30 events from the training pool ( 0.01% of the full dataset). We report AUC-ROC (%) averaged over 100 independently sampled tasks with five random seeds. Implementation details We build CurvPrompt on a TGN backbone (Rossi et al. 2020) with a node memory module and M Riemannian experts with Top-K routing under diverse curvatures in the κ-stereographic model, and assign diverse curvatures from positive, near-zero to negative to different experts. Each expert is equipped with an expert-specific prompt. The gating network is conditioned on multi-resolution temporal neighborhoods (neighborhood sizes r ∈ {2, 5, 10}). Dual node/time prompts and dual condition-nets are combined with these expertspecific prompts. Pre-training uses temporal link prediction with BCE, regularized by prompt orthogonality, embedding distortion and load balancing losses. We score candidate links with a lightweight MLP that takes node embeddings and their geodesic distance as input. In downstream tuning, we freeze the Riemannian experts, curvatures, topology encoder, router, TCN/NCN, and the time encoder, and optimize prompt-related parameters (time/expert prompts, structure prompts, and the task head). Baselines We compare CurvPrompt against state-of-theart approaches across four categories: (1) Conventional DGNNs: GCN-ROLAND, GAT-ROLAND (You, Du, and Leskovec 2022), TGAT (Xu et al. 2020), TGN (Rossi et al. 2020), TREND (Wen and Fang 2022), and GraphMixer (Cong et al. 2022); (2) Dynamic graph pre-training: DDGCL (Tian et al. 2021) and CPDG (Bei et al. 2024); (3) Static graph prompting: GraphPrompt (Liu et al. 2023) and ProG (Sun et al. 2023) (with DGNN backbones); (4) Dynamic graph prompting: TIGPrompt (Chen et al. 2024) and DyGPrompt (Yu et al. 2025b). DyGPrompt is the strongest baseline and serves as our primary comparison. We report the best numbers from the original papers where available, or re-run them under our unified protocol.

Overall Performance Comparison We compare CurvPrompt against these baselines on all four datasets. Table 1 reports the AUC-ROC (%) for node classification (NC), transductive link prediction (LP-Trans), and inductive link prediction (LP-Indu). Overall, CurvPrompt achieves the best results on all 8 link-prediction settings and

on 2 of 4 node-classification settings; it is highly competitive on the remainder. Link prediction. CurvPrompt obtains the best AUCROC on every LP split, with gains ranging from +0.5% (Reddit transductive, vs. TGN-DYGPrompt) to +13.9% (MOOC transductive, vs. TGN-DYGPrompt). The advantage is most pronounced on MOOC and Genre. On inductive LP, several non-prompt baselines (e.g., TGN, TGAT, ROLANDstyle models) remain near the random-guess regime on Genre, whereas CurvPrompt sustains strong generalization. Combined with its gains over other prompting baselines, this suggests that prompt-based multi-curvature routing further helps under few-shot adaptation. Compared with DyGPrompt, these gains show that prompt-guided selection of mixed-curvature spaces is effective. The large margin over GraphMixer further highlights the benefit of our topologyrouted mixture-of-experts design. Node classification. NC remains harder than LP for all methods, with larger variances and lower absolute AUC. CurvPrompt leads on Reddit (75.60) and MOOC (79.15); on Wiki it ranks second (76.17, behind TGAT-DYGPrompt 82.09). On Genre NC, all methods cluster in a narrow band (46.33–52.03), with CurvPrompt (50.27) lying within this range; this flatness could suggest that Genre’s relatively stable local topology offers less opportunity for geometry-adaptive encoding to differentiate.

Ablation Study We ablate the key components of CurvPrompt on Reddit and MOOC (Table 2). Removing any module degrades performance relative to the full model, indicating that each design choice contributes. (i) The strongest degradation occurs under w/o Expert (i.e., neither multi-expert routing nor multi-κ specialization). LP drops sharply, approaching chance level (e.g., MOOC LP-Old 84.05→56.32 ), indicating that the curvature-specialized expert mixture is the main contributor to interaction modeling. The NC impact is datasetdependent: Reddit NC falls substantially (75.60→66.43 , the largest among NC ablations), whereas MOOC NC remains nearly unchanged (79.15→78.41 ). (ii) w/o Balance yields the second-largest LP degradation on both datasets, indicating that gate-load balancing is necessary to keep multi-κ routing effective and avoid expert under-utilization. (iii) w/o GE (i.e., disabling Lortho and LD ) and w/o Expert Prompt both cause moderate, consistent drops, showing that geometric regularization and per-expert prompts are useful. (iv) w/o Topology leaves LP nearly unchanged on Reddit (< 0.1), but causes clear drops on MOOC (≈ 6 points) and consistently hurts NC on both datasets, implying that topology encoding supplies important condition/routing signals. Overall, CurvPrompt benefits from the synergy of topology-conditioned routing, mixed-curvature experts with specialized prompts, balanced gating, and geometric regularization. However, w/o Expert simultaneously removes expert capacity, routing, and curvature specialization; it therefore cannot isolate whether the gains come from MoE capacity or from mixed-curvature geometry.

Figure 3: Reddit expert gate usage over 31 days. Each stacked region is the daily mean gate weight of one curvature expert. Routing mass shifts from κ=1.0 toward κ=0.2, while hyperbolic experts remain minority.

Necessity of mixed curvature. To disentangle MoE capacity from geometric diversity, we further compare three geometry configurations with matched expert count and routing (Table 3): Euclidean MoE (all κ=0), Single Curvature (shared non-Euclidean κ), and Mixed Curvature (CurvPrompt). Euclidean MoE already outperforms w/o Expert on LP, confirming that the expert mixture itself is beneficial; yet it remains clearly below CurvPrompt on both datasets. Replacing all-Euclidean experts with a shared curvature improves LP/NC, showing that non-Euclidean geometry helps beyond MoE capacity alone. CurvPrompt further improves over Single Curvature, indicating that the gains are not merely from using a curved space, but from multi-κ specialization across experts.

Expert Analysis To examine whether CurvPrompt adapts expert allocation over the interaction timeline, we freeze a pretrained model and run causal inference on Reddit, recording gate weights for every edge. Edges are then aggregated by day to obtain the mean usage of each curvature expert (Fig. 3). For clearer visualization, we use a four-expert probe (κ∈{−1.8, −0.5, 0.2, 1}, Top-K=1), not the main Reddit setting. The resulting stacked usage reveals a clear temporal shift. Early in the timeline, the high-curvature spherical expert (κ=1) accounts for a substantial share of routing decisions, while hyperbolic experts (κ∈{−1.8, −0.5}) remain secondary. Over the subsequent days, usage gradually concentrates on the mild spherical expert (κ=0.2), which becomes dominant by the end of the period; meanwhile, both κ=1 and the hyperbolic experts decline. This trend indicates that the learned gate does not collapse to a fixed expert mixture. Instead, it redistributes mass across complementary curvature regimes as interactions progress, supporting the necessity of multi-κ MoE routing for temporal graphs.

Table 1: AUC-ROC (%) evaluation of temporal node classification and link prediction. The best result is bolded and the runner-up is underlined. Methods

Wiki

Node Classification Reddit MOOC

Genre

Wiki

Transductive LP Reddit MOOC

Genre

Wiki

Inductive LP Reddit MOOC

Genre

GCN-ROLAND GAT-ROLAND TGAT TGN TREND GraphMixer

58.86±10.3 62.81±9.88 67.00±5.35 50.61±13.6 69.92±9.27 65.43±4.21

48.25±9.57 47.95±8.42 53.64±5.50 49.54±6.23 64.85±4.71 60.21±5.36

49.93±6.74 50.01±6.34 59.27±4.43 50.33±4.47 66.79±5.44 63.72±4.98

46.33±3.97 47.26±3.49 51.26±2.31 50.72±2.31 50.34±1.62 50.15±1.49

49.61±3.12 52.34±1.82 55.78±2.03 72.48±0.19 63.24±0.71 59.73±0.35

50.01±2.53 50.04±1.98 62.43±1.86 67.37±0.07 80.42±0.45 61.88±0.11

49.82±1.44 55.74±3.71 51.49±1.30 54.60±0.80 58.70±0.78 52.42±1.38

49.15±3.74 47.69±2.81 69.11±3.89 86.46±2.84 52.78±1.14 60.83±3.25

49.60±2.37 52.29±1.97 48.21±1.55 74.38±0.29 50.15±0.90 51.34±0.84

49.90±1.64 49.85±2.35 57.30±0.70 69.81±0.08 65.13±0.54 57.64±0.31

49.16±2.48 54.01±2.16 51.42±4.27 54.62±0.72 57.52±1.01 51.16±2.59

47.25±2.97 49.38±2.72 48.38±4.72 87.17±2.68 45.31±0.43 56.32±3.08

DDGCL CPDG

65.15±4.54 43.56±6.41

55.21±6.19 65.92±6.25

62.34±5.13 50.32±5.06

50.91±2.08 49.89±1.34

54.96±1.46 52.86±0.64

61.68±0.81 59.72±2.53

55.62±0.32 53.82±1.50

68.49±5.31 49.71±2.64

47.98±1.11 47.37±2.23

55.90±1.13 56.40±1.17

55.18±2.73 53.58±2.10

42.70±3.26 40.01±3.59

GraphPrompt ProG

73.78±5.62 60.86±7.43

60.89±6.37 68.60±5.64

64.60±5.76 63.18±4.79

51.28±2.43 51.46±2.38

55.67±0.26 92.28±0.21

67.46±0.31 93.32±0.06

51.07±0.75 58.73±1.58

86.78±3.14 86.24±2.87

48.46±0.28 89.75±0.28

59.18±0.49 90.69±0.08

50.27±0.58 56.42±1.95

87.45±2.57 85.43±3.16

TGAT-TIGPrompt TGN-TIGPrompt

69.21±8.88 44.80±5.45

67.70±9.64 63.75±5.60

73.90±6.68 55.42±3.60

51.38±2.72 50.84±2.75

59.54±1.41 82.04±2.03

78.45±1.44 83.26±2.38

51.69±1.24 65.00±4.73

69.71±4.16 86.25±2.43

49.52±0.85 81.75±1.97

65.66±2.68 79.51±2.58

51.58±4.02 64.98±4.61

48.34±3.28 86.19±3.06

TGAT-DyGPrompt TGN-DyGPrompt

82.09±6.43 74.47±3.44

73.50±6.47 74.00±3.10

77.78±5.08 69.06±3.89

52.03±2.24 51.97±2.16

69.88±0.18 94.33±0.12

90.76±0.09 96.82±0.06

53.92±0.97 70.17±0.75

72.04±4.71 87.02±1.63

52.58±0.23 92.22±0.19

75.20±0.17 95.69±0.08

53.29±0.87 69.77±0.66

50.82±3.67 87.63±1.97

CurvPrompt

76.17±4.89

75.60±4.75

79.15±3.20

50.27±0.48

95.70±0.04

97.30±0.09

84.05±0.31

94.40±0.21

93.56±0.05

96.33±0.09

83.80±0.31

93.93±0.18

Table 2: Ablation study on Reddit and MOOC evaluated by AUC-ROC (%).

CurvPrompt CurvPrompt (w/o Topology) CurvPrompt (w/o Expert Prompt) CurvPrompt (w/o GE) CurvPrompt (w/o Balance) CurvPrompt (w/o Expert)

100 96.1

MOOC

LP-Trans

LP-Indu

LP-Trans

LP-Indu

97.30 97.22 95.49 95.71 93.42 87.29

96.33 96.31 93.89 94.24 91.92 83.98

75.60 73.06 73.53 72.53 74.68 66.43

84.05 78.11 77.17 76.63 72.74 56.32

83.80 77.77 77.10 77.02 72.63 56.27

79.15 75.57 75.61 76.75 77.10 78.41

Table 3: Geometry attribution on Reddit and MOOC by AUC-ROC (%). All variants keep the same expert count and gating; only the curvature configuration changes.

AUC (%)

Method

(a) Top-K

85 80

M = 2, k = 1 M = 4, k = 2 M = 4, k = 3 M = 6, k = 3 M = 6, k = 2

(b) Riemannian experts(M=2, K=2) 100

96.8

MOOC

LP-Trans

LP-Indu

LP-Trans

LP-Indu

97.30 95.64 94.56

96.33 94.23 92.84

75.60 70.65 69.18

84.05 75.68 68.92

83.80 75.74 69.02

79.15 78.56 76.57

AUC (%)

CurvPrompt (Mixed Curvature) Single Curvature Euclidean MoE

96.7 94.0

95.2

95 Method

97.3 94.8

92.8

91.8

91.4

All-hyp.

Hyp.+Euc.

90 85 80

All-sphere

Sph.+Euc.

Sph.+Hyp.

Hyperparameter Sensitivity Sensitivity to Top-K and curvature. Fig. 4 examines Top-K routing and the Riemannian experts on Reddit. In Fig. 4(a), while varying (M, K), each expert’s curvature κ is drawn from a pre-defined multi-type pool (positive, near-zero, and negative values). AUC stays in a narrow band (94.0–97.3), showing that Top-K is robust under such randomized κ assignments. Fig. 4(b) fixes (M =2, K=2) and compares different κ experts, where the spread is larger (91.4–96.8): the all-spherical experts performs best, while hyperboliccontaining experts are weaker, consistent with the learned routing behavior on Reddit, where gate mass concentrates on the mild spherical expert while retaining non-trivial weight on hyperbolic experts (Fig. 3). This spread far exceeds that across Top-K settings in (a), indicating that the geometric composition of the expert is the dominant factor. Since the optimal expert is dataset-dependent and unknown a priori, committing to any fixed geometry is brittle. Overall, the curvature composition of the expert bank is more consequential than the exact Top-K setting—and across both panels, diversity, rather than any single geometry, is the robust choice.

Figure 4: Sensitivity analysis on Reddit (LP). (a) AUC (%) under different Top-K settings (M, K), where each expert’s κ is drawn from a pre-defined multi-type pool; the default configuration is shown in bold. (b) AUC (%) of explicit twoexpert κ compositions under fixed M =2, K=2.

Conclusion In this paper, we identify geometry under-adaptation in dynamic graph prompting and propose CurvPrompt, a topology-routed Riemannian mixture-of-experts framework. CurvPrompt utilizes a bank of curvature-diverse experts with learnable prompts and routes each node–time instance via a topology-aware gate, employing soft Top-K routing during pre-training and hard Top-K routing during downstream adaptation for parameter-efficient geometry adaptation. Extensive experiments demonstrate that CurvPrompt achieves the best few-shot link prediction and strong node classification across four benchmarks.

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Appendix A Additional Experimental Details A.1 Environment All experiments are conducted in the following environment: • Operating system: Ubuntu 22.04 LTS • CPU: 13th Gen Intel Core i9-13900K • GPU: 2× NVIDIA GeForce RTX 3090 (24 GB) • Memory: 128 GB RAM

A.2 Datasets details Table 4 summarizes the datasets used in our experiments. Wikipedia, Reddit, and MOOC are bipartite interaction networks with binary labels and 172-dimensional features over a 30-day span, while Genre is a larger-scale user–genre network from TGB with multi-class labels and a much longer time span.

A.3 Sensitivity to Distortion and Balance Weights Figure 5 illustrates the effect of varying (λ1 , λ2 ) on the MOOC dataset. Performance is sensitive to the relative scale of the two auxiliary weights. The best performance is achieved with moderate distortion regularization (λ1 ≈ 0.02) combined with a matched balance weight (λ2 ∈

Figure 5: Sensitivity of the auxiliary loss weights on MOOC (LP-Trans AUC, %). We sweep λ1 (distortion) and λ2 (balance) over a 5 × 5 grid while keeping all remaining hyperparameters fixed. Table 4: Summary of datasets. Dataset

Nodes

Edges Classes Labels Feat. dim Time span

Wikipedia 9,227 157,474 Reddit 11,000 672,447 MOOC 7,144 411,749 Genre 1,505 17,858,395

2 2 2 474

217 366 4,066 984

172 30 days 172 30 days 172 30 days 86 1,500 days

{0.01, 0.02, 0.05}), which is also adopted as the default configuration. In contrast, mismatched settings—particularly a very small distortion weight paired with a larger balance weight, or a large balance weight without sufficient distortion regularization—lead to noticeable performance degradation. These results indicate that both auxiliary objectives are beneficial when properly balanced, although the method is not insensitive to arbitrary weight choices.

Dynamic Graph Prompting via Topology-Routed Mixed-Curvature Experts

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