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Your Temporal Link Predictor Is Blind to Who Is Active: A Missing Factor That Transfers Across Models

Ji Zhang, Zixin Liu, Yiran Ding, Jiayi Wang, Yilu Du, Weijia Xuan

arXiv:2610.04869v1cs.LGcs.SI

TL;DR

Temporal link prediction focuses on destination compatibility, although predicting an interaction also depends on whether its source is active. The paper models source activity separately and combines it with relation scores; its full model ranks first overall, and activity scores transfer across backbones without retraining.

  • Problem

    Temporal link objectives with negatives that keep the positive’s source cancel source activity from the optimal score, leaving the objective without a signal to learn whether that source is active.

    Source: §4.4 · p. 6

  • Method

    SNAM estimates source conditional intensity from decayed interaction counts with a point-process likelihood, and its log intensity can be added to relation scores alongside shared multi-timescale historical states.

    Source: §1 · p. 2

  • Results

    Across 13 datasets and three protocols, the full model ranks first overall against eleven baselines; transfer improves AP across all 20 backbone–dataset combinations, by up to 25 percentage points.

    Source: §1 · p. 2, §6 · p. 10

  • Takeaways & Limitations

    The paper concludes that benchmark protocols determine which factors models can learn and metrics can reveal, motivating audits for factors canceled by negative sampling.

    Source: §6 · p. 10

    The approach is evaluated on existing benchmark datasets, and downstream applications should respect applicable privacy and data-use requirements.

    Source: ETHICS STATEMENT · p. 10

Abstract

from arXiv · show

An interaction has two parts: someone decides to act, and then chooses whom to act on. Temporal link prediction has concentrated on the second, and we show that it is blind to the first by construction: a standard negative keeps the real source and swaps the destination, and we prove that this cancels the source's activity exactly from the optimal score, so no model trained and evaluated this way is ever rewarded for learning it. Under the harder historical and inductive negatives, whose sources differ, the same factor becomes the dominant signal. We model it with Source Node Activity Modeling (SNAM), a self-exciting event intensity fitted by an exact point-process likelihood to decayed interaction counts the history states already contain; it has fewer than 20 parameters. On their own, never looking at the destination, these parameters beat DyGFormer and TPNet on four of five datasets under historical negatives. Added to the frozen scores of TPNet, TGN, DyGFormer and DSRD, four models of different design, without retraining anything, they raise AP on almost every backbone-dataset pair in both settings, by up to 25 points. Our full model ranks first overall against eleven baselines on 13 datasets and three protocols, and on million-event streams trains an epoch 9-100x faster than TPNet and DyGFormer. We conclude that source activity is a blind spot of temporal link prediction, and a cheap, transferable one to close. Code is available at https://github.com/Erutaner/Your-Temporal-Link-Predictor-Is-Blind-to-Who-Is-Active.

1 INTRODUCTION

Temporal link prediction must account for both whether a source is active and which destination is compatible, but standard link objectives do not reward learning source activity. The paper separates these factors and evaluates activity scoring and a combined model.

1 INTRODUCTION

  • Predicting an interaction requires judging both source activity and destination compatibility; pair-focused scoring can favor a historically strong but inactive source.The introduction contrasts A→B, with many past transfers but prolonged inactivity, and recently active C→D.Source: §1 · pp. 1–2
  • Standard negatives keep the source and swap the destination, so source activity cancels from the optimal score and the objective does not reward learning it.The paper motivates learning source activity separately and using negatives with different sources to expose its contribution.Source: §1 · p. 2
  • SNAM estimates source conditional intensity from decayed interaction counts at K timescales using K + 1 shared parameters, while MTHR supplies multi-timescale states to both model factors.CRM scores compatibility using historical node-pair features and a candidate-conditioned encoding of the source’s recent interactions; the final score adds β times SNAM’s log intensity to the relation score.Source: §1 · p. 2
  • Across 13 datasets, the full model ranks first overall across three negative-sampling strategies, with its main advantages under historical and inductive negatives.SNAM alone outperforms DyGFormer and TPNet on four of five datasets under historical negatives; cross-model transfer improves AP across all 20 backbone–dataset combinations by up to 25 percentage points.Source: §1 · p. 2

2 RELATED WORK

Prior temporal graph work models histories, relationships, and event rates, while source activity has generally not been a separate factor in link scores. This paper connects those lines to evaluation protocols that compare candidates across different sources.

2 RELATED WORK

  • Historical encoders and structural or temporal features model interaction histories and source–candidate relationships; this work combines reusable aggregates with candidate-conditioned recent-interaction encoding.Prior methods include walks, memory updates, neighborhood aggregation, sequence encoders, temporal paths, and time or frequency features (Wang et al., 2021; Kumar et al., 2019; Rossi et al., 2020; Xu et al., 2020; Liu et al., 2025; Xiong & Sakellariou, 2026; Yu et al., 2023; Cong et al., 2023; Li et al., 2025; Tieu et al., 2025; Luo & Li, 2022; Souza et al., 2022; Lu et al., 2024; Chung et al., 2025; Tian et al., 2024).Source: §2 · pp. 2–3
  • Temporal point-process graph methods model event intensities, often folding a node’s rate into pair functions; TREND adds a node-level term as an auxiliary loss rather than in the link score.The paper instead models source rate separately with K + 1 parameters over decayed counts, adding it to link scores for transfer across backbones without retraining.Source: §2 · p. 3
  • Benchmark results depend on data splits, negative sampling, and evaluation procedures, while activity and recency heuristics can provide predictive information without complex relation encoders.The paper distinguishes cross-source discrimination from candidate ranking for a fixed source.Source: §2 · p. 3

3 PRELIMINARIES

The preliminaries represent a temporal graph as ordered interaction records and define prediction using the history before a query time. They also introduce time blocks for partitioning event records.

3 PRELIMINARIES

  • A temporal graph is an ordered sequence of interaction records, each containing source, destination, timestamp, and optional features; repeated node pairs are allowed.Source: §3 · p. 3
  • The interaction history at query time t consists of all records strictly preceding t.Source: §3 · p. 3
  • Temporal link prediction asks whether a given node pair interacts at query time t, given the preceding history.Source: §3 · p. 3
  • A time block partitions ordered interaction records into contiguous, non-overlapping blocks; each block’s start time is its earliest interaction timestamp.Source: §3 · p. 3

4 METHODOLOGY

The method separates source activity from candidate-relation compatibility, builds both from shared multi-timescale history, and combines their scores for prediction.

4.1 MULTI-TIMESCALE HISTORICAL REPRESENTATION

  • Multi-Timescale Historical Representation aggregates decayed interactions at block start and caches the resulting neighborhood states for reuse within the block.Exact query-time updates incorporate newly observed interactions without repeatedly processing pre-block history.Source: §4.1 · p. 4

4.2 CANDIDATE RELATION MODELING

  • Candidate Relation Modeling combines node-pair historical features with candidate-conditioned encodings of recent source interactions to score compatibility.Its node-pair features include direct interactions, representation matching, and structural associations; computations use only history preceding the candidate time.Source: §4.2 · p. 5

4.3 SOURCE NODE ACTIVITY MODELING

  • SNAM estimates each source’s conditional interaction intensity from decayed counts using K + 1 learnable parameters shared across source nodes.Its computation does not depend on other trainable parameters, enabling reuse across models with consistent historical statistics and timescales.Source: §4.3 · pp. 5–6

4.4 SCORE INTEGRATION AND OPTIMIZATION

  • The final score combines a relation score with β times SNAM’s log intensity, while separate losses train the relation and activity branches.The two losses act on disjoint parameter sets.Source: §4.4 · p. 6
  • The activity branch uses a point-process likelihood because the link objective’s source-preserving negatives cancel source activity from the optimal discriminator.The activity likelihood uses observed event times and the intervals without events; the exponential kernel permits a closed-form integral without negative sampling or temporal discretization.Source: §4.4 · p. 6
  • SNAM can be trained independently and its log intensity added to an existing model’s score without retraining that model.Parameter reuse requires consistent historical statistics, time units, and decay-scale configurations.Source: §4.4 · p. 7

5 EXPERIMENTS

Across 13 benchmarks, the full model has the best overall average rank under the three negative-sampling strategies, with its main advantages under historical and inductive negatives. The experiments also examine efficiency, SNAM’s contribution, timescale sensitivity, and feature ablations.

5.1 EXPERIMENTAL SETTINGS

  • Experiments cover 13 temporal link prediction benchmarks and 11 baselines, with chronological train, validation, and test splits and random, historical, and inductive evaluation negatives.Evaluation reports AP and AUC-ROC; main experiments use means and standard deviations over five seeds, with stated exceptions.Source: §5.1 · p. 7

5.2 OVERALL EFFECTIVENESS AND EFFICIENCY

  • Average rank 2.44 gives the full model the best overall transductive result across three negative-sampling strategies, with SNAM improving historical and inductive average ranks from 4.42 and 4.31 to 1.50 and 1.54.Without SNAM, the model still outranks TPNet (4.33 vs. 4.55); under random negatives, neither configuration clearly beats the strongest baseline.Source: Table 1 · p. 8, §5.2 · p. 8
  • TPNet and DyGFormer require approximately 9.2× and 60× the model’s per-epoch training time, respectively, and 1.7× and 27× its full-test-set scoring time.The LastFM comparison reports medians over three runs; test scoring is timed under random negatives.Source: §5.2 · p. 8

5.4 PARAMETER SENSITIVITY ANALYSIS

  • Several timescales bring large gains over one, while adding more beyond about ten changes little.The comparison is reported for transductive AP.Source: Table 3 · p. 9, §5.4 · p. 9

5.3 SNAM: DOES THE ACTIVITY-BASED APPROACH REALLY WORK?

  • The SNAM studies assess source-activity transfer across backbones and the effect of source-activity terms on transductive test AP under historical and inductive negatives.The supplied table captions identify these scopes but do not provide the corresponding cell values.Source: Table 2 · p. 9, Table 5 · p. 9

5.5 ABLATION STUDY

  • Dropping candidate-conditioned encoding or random-projection features changes the full score by under one point on every stream, while dropping counts costs up to two and removing all three costs at most four.The reported ablations indicate that the multi-timescale backbone and activity term, rather than engineered input features, account for the advantage.Source: §5.5 · pp. 9–10

6 CONCLUSION AND FUTURE WORK

The conclusion argues that temporal link prediction’s negative-sampling design can cancel factors held fixed between positives and negatives, including source activity. It recommends auditing benchmarks for such cancellations and aligning objectives and protocols with deployment comparisons.

6 CONCLUSION AND FUTURE WORK

  • Random destination negatives preserve source and time, cancelling their shared distribution from the optimal score and leaving absolute source activity unconstrained.With historical and inductive negatives, source nodes can differ, making activity informative; SNAM uses K + 1 parameters, and the full model ranks first overall against eleven baselines on 13 datasets.Source: §6 · p. 10
  • Any factor held fixed between a positive and its negatives drops out of the optimal score, so the sampler determines what a model can learn and a metric can see.The paper notes that negatives drawn at the positive timestamp also cancel global activity, motivating benchmark audits and protocols matched to deployment comparisons.Source: §6 · p. 10

AI USE STATEMENT

The authors report using generative AI for several research and writing tasks, as well as for scientific figures and manuscript editing. Four human authors independently reviewed LLM-generated code for data leakage and misleading implementation practices.

AI USE STATEMENT

  • Generative AI assisted with proof writing, methodology and experiment feedback, implementation, translation, qualitative and thematic analysis, figure creation or modification, and manuscript editing.The authors state they reviewed all AI-assisted work and take responsibility for the final content.Source: AI USE STATEMENT · p. 10

ETHICS STATEMENT

The work evaluates a methodological approach on existing temporal link prediction benchmarks and identifies no ethical concerns specific to the methodology. The authors note that downstream applications should respect applicable privacy and data-use requirements.

ETHICS STATEMENT

  • The evaluation is limited to existing benchmark datasets, and downstream applications should respect applicable privacy and data-use requirements.The authors identify no ethical concerns specific to the proposed methodology.Source: ETHICS STATEMENT · p. 10

REPRODUCIBILITY STATEMENT

The paper emphasizes reproducibility through released artifacts that rebuild reported results and rerun experiments, alongside documented, auditable evaluation and baseline procedures. Its mathematical foundation presents source activity modeling as a compact multi-timescale Hawkes-type intensity independent of learned relation representations.

REPRODUCIBILITY STATEMENT

  • Released result files rebuild every table and figure byte for byte, and released scripts rerun every experiment from public benchmark data.A retraining on another machine reproduced six AP values within 0.35 points; reevaluation of a shipped checkpoint reproduced them to five decimals.Source: REPRODUCIBILITY STATEMENT · p. 10
  • Evaluation protocols, dataset statistics, configurations, seeds, and baseline provenance are documented, while information-leakage corrections and transfer procedures are specified for audit.The split, negative samplers, and metric code are unmodified DyGLib components; theoretical claims are proved in the appendices.Source: REPRODUCIBILITY STATEMENT · pp. 10–11

A STATISTICAL MOTIVATION AND MATHEMATICAL FOUNDATIONS OF SOURCE ACTIVITY MODELING

  • SNAM uses a multi-timescale exponential Hawkes-type intensity with K + 1 shared parameters and models source activity independently of learned relation representations.The paper derives the point-process formulation and likelihood from first principles.Source: A STATISTICAL MOTIVATION AND… · p. 14

A.1 FROM NEGATIVE SAMPLING TO SOURCE ACTIVITY CORRECTION

This section explains how negative-sampling choices affect whether source activity is informative, and how SNAM uses MTHR’s multi-timescale exponential activity states to estimate source intensity with an exact point-process likelihood and independently of relation models. It also specifies temporal bases and Candidate Relation Modeling features that make historical information explicit for candidate pairs.

A.1 FROM NEGATIVE SAMPLING TO SOURCE ACTIVITY CORRECTION

  • With random-destination negatives, the shared source-time distribution cancels exactly, so uniform destination sampling rewards conditional destination preference, not absolute source activity.A model may encode activity-related history, but its discrimination score is not required to recover the source event rate.Source: §A.1 · p. 18
  • When hard negatives can change the source, activity can help distinguish real from artificial edges, but protocol-specific sampling and normalization mean activity alone need not recover the optimal score.SNAM supplements the relation score with source intensity; β is selected on validation, and this term affects comparisons across sources or query times rather than destination rankings within a fixed source-time query.Source: §A.1 · p. 19
  • Across 13 datasets, validation selected β = 0 in 123 of 130 random-negative cases, but β > 0 in 111 historical- and 116 inductive-negative cases.The training objective creates the source-activity blind spot; historical and inductive protocols expose it by comparing different sources, and the activity correction transfers across backbones.Source: §A.1 · pp. 19–20

A.2 THE SHARED MATHEMATICAL FOUNDATION OF SOURCE ACTIVITY AND MULTI-TIMESCALE HISTORICAL REPRESENTATION

  • MTHR’s historical aggregation yields SNAM’s multi-timescale decayed activity counts when each interaction contributes a unit count, so no separate history encoder is needed.Partner embeddings and activity counts are two uses of the same aggregation operator, differing in the interaction content being accumulated.Source: §A.2 · p. 21
  • With only K + 1 shared trainable parameters, SNAM models activity as positive self-excitation, but cannot represent activity suppression or guarantee long-term stationarity.Its capacity comes from multi-timescale historical counts rather than node-specific parameters; this intensity form is an explicit modeling assumption compatible with MTHR’s exponential states.Source: §A.2 · p. 25
  • SNAM reuses MTHR’s historical activity counts and timescales to estimate source intensity, without depending on the relation branch’s learned representations.The shared exponential counts can be updated exactly through each query time, including every preceding within-block interaction, without rescanning the full history.Source: §A.2 · pp. 22, 26

A.3 INDEPENDENT LEARNING AND CLOSED-FORM LIKELIHOOD OF SOURCE ACTIVITY

  • SNAM learns source activity directly from real interaction sources and times, using both observed events and event-free intervals without constructing negative node pairs.The observed-event term rewards high intensity at event times, while cumulative intensity constrains rates over the full observation window.Source: §A.3 · p. 28
  • The full interaction intensity separates into a source event rate and a conditional destination choice, allowing activity and relation modeling to receive supervision matched to their distinct objectives.Because SNAM’s exponential intensity shares the historical kernels, its training-objective time integral can also be computed exactly.Source: §A.2 · p. 26, §A.3 · p. 28
  • SNAM’s shared exponential states enable exact cumulative-intensity integration and independent source-activity learning that can be reused across relation encoders under consistent history statistics.The activity loss updates only SNAM’s parameters, while the relation loss does not train them; exact integration requires no temporal sampling or numerical integration. The exactness claim concerns the model’s integral, not artifacts from discretized or tied timestamps.Source: §A.3 · pp. 31–32

B.1 EXPONENTIAL KERNELS AND EXACT STATE PROPAGATION

  • Under continuity, positivity, normalization, and strict decay assumptions, the exponential kernel is the unique decaying kernel that permits exact scalar propagation of arbitrary linear historical states.The same multiplicative property supports propagating summarized history to later query times without revisiting earlier interactions.Source: §B.1 · p. 34

B.2 MULTIPLE TIMESCALES AND LOGARITHMICALLY SPACED DECAY RATES

  • A single decay rate covers only a limited temporal range, so MTHR uses multiple logarithmically spaced rates to allocate finite timescales across widely varying histories.Larger rates emphasize recent events, smaller rates retain distant history, and logarithmic spacing provides approximately uniform relative resolution across orders of magnitude.Source: §B.2 · pp. 34–35

B.3 REPRESENTING SLOWLY DECAYING PATTERNS WITH EXPONENTIAL MIXTURES

  • A positive continuous mixture of exponential kernels can represent power-law decay exactly, while finite log-spaced rates approximate that mixture.This lets the overall response decay more slowly over a range than any individual exponential kernel.Source: §B.3 · p. 35
  • Exponential basis functions can represent slowly decaying temporal influences while retaining exact state propagation, but finite mixtures incur truncation and discretization error.The resulting learned representations are more general than a single positive exponential mixture.Source: §B.3 · p. 35

B.4 TIMESCALE CONFIGURATION

  • The timescale grid is derived from training timestamps, fixed before training, and spans durations from the shortest to the longest characteristic time constant.The model learns how to combine historical representations across the available resolutions.Source: §B.4 · p. 36

C DETAILED CONSTRUCTION OF CANDIDATE RELATION MODELING

  • CRM combines complete-history pair features with MTHR endpoint representations and uses only interactions preceding the candidate query time.Its pair-history features include interaction and structural information, plus recent source interactions measured for relevance to the candidate destination.Source: C DETAILED CONSTRUCTION OF CANDIDATE… · p. 36

C.1 NODE-PAIR FEATURES FROM THE COMPLETE HISTORY

  • Complete-history node-pair features combine decayed pair-interaction counts, endpoint activity, low-rank historical-representation matching, and historical edge-feature matching when available.The components are concatenated across timescales; Table 11 summarizes their composition.Source: §C.1 · pp. 36, 38, §C.2 · p. 41, Table 11 · p. 41

C.2 STRUCTURAL FEATURES FROM FIXED RANDOM PROJECTIONS

  • CRM’s fixed random projections compress weighted neighborhood overlap into node-count-independent inner products that estimate structural similarities and walk-based relationships.For bipartite graphs, cross-order projections can capture indirect associations; clipping and compression make the final transformed features no longer unbiased estimates of the original walk sums.Source: §C.2 · pp. 39–40

C.3 CANDIDATE-CONDITIONED RECENT INTERACTION ENCODING

  • CRM encodes up to 20 recent source interactions relative to each candidate destination, then applies nonlinear event encoding and permutation-invariant summation.Temporal weights preserve interaction age, while structural overlap, representation compatibility, and observed event features add complementary information; the resulting representation depends on the candidate destination.Source: §C.3 · pp. 41–42

C.4 RELATION SCORING ARCHITECTURE

The relation-scoring network combines node representations, historical pair features, and a candidate-conditioned encoding of recent source interactions. Its nonlinear and direct linear paths produce a scalar relation score that is combined with SNAM’s independent source intensity.

C.4 RELATION SCORING ARCHITECTURE

  • CRM combines node representations, historical pair features, and a candidate-conditioned encoding of the source’s recent interactions to score candidate relations.SNAM supplies an independent source-activity correction at the final scoring stage, rather than participating in the relation network’s internal feature fusion.Source: §C.4 · pp. 42–43
  • A nonlinear path and a direct linear path jointly produce two outputs that define one scalar scoring function, not separate edge and no-edge classes.The positive-negative activation split lets the nonlinear path process positive and negative preactivations separately.Source: §C.4 · p. 42
  • Zero-initialized projections let the added pairwise and recent-interaction channels begin by leaving the base scoring head unchanged and contribute gradually during training.Dropout of 0.2 is applied to concatenated node representations and complete-history pairwise features, but not to the recent-interaction encoding at this stage.Source: §C.4 · p. 42
  • Training uses smooth aggregation followed by binary cross-entropy with random destination negatives, while evaluation uses the base relation score.The training score is a smooth approximation to the evaluation score, with a difference that varies across samples.Source: §C.4 · pp. 42–43

D COMPUTATIONAL COMPLEXITY AND EFFICIENT TRAINING

The model reduces repeated historical computation by sharing block-start states, updating them within blocks, and propagating gradients through fixed-depth sparse operations rather than temporal BPTT. Its block count balances reconstruction and query-time costs, while SNAM reuses existing activity counts.

D.1 SHARED BLOCK-START COMPUTATION

  • Block-start historical states are shared across candidate queries, so graph aggregation is performed once per block rather than separately for each candidate.For B_c positive interactions and an equal number of random negatives, approximately 2B_c candidate pairs reuse the states.Source: §D.1 · pp. 43, 45

D.2 QUERY-TIME COMPUTATION AND WITHIN-BLOCK UPDATES

  • Query-time processing retrieves and decays cached states, handles strictly earlier within-block records, and builds candidate-specific features; activity counts are updated exactly.Source-side states can be reused for matched random destination negatives, while destination representations and candidate-specific features are computed separately.Source: §D.2 · p. 45

D.3 GRADIENT PROPAGATION WITHOUT TEMPORAL BPTT

  • MTHR propagates gradients through sparse matrix operations in a fixed-depth graph, avoiding the event-history-length computation graph required by full temporal BPTT.Each block’s graph is released after its backward pass and optimizer update, without retaining graphs from previous blocks.Source: §D.3 · pp. 47–48

D.4 OVERALL COMPLEXITY, MEMORY USAGE, AND THE BLOCK-SIZE TRADE-OFF

  • The block count trades reduced within-block record processing against more frequent block-start reconstruction, while SNAM adds only K + 1 parameters and reuses MTHR activity counts.Finer blocks can lower peak scoring memory by up to 3× on large streams; the reported epoch-time overhead of within-chunk records falls from up to 3× at C = 48 to 1.1–1.3× at C = 192.Source: §D.4 · pp. 49–50, Figure 6 · p. 49
  • On LastFM and Flights prefixes, training-epoch and test-scoring times grow at most linearly with stream length, while fixed per-block overhead dominates on short prefixes.LastFM keeps C = 192 blocks at every prefix length; Flights uses 5, 17, and 90 blocks at 10%, 30%, and 100% of events, respectively.Source: Table 13 · p. 50

E DATASETS AND EVALUATION PROTOCOLS

The evaluation covers thirteen temporal interaction streams with chronological splits and event-time history visibility. It compares random, historical, and inductive negative-sampling protocols, with batch-level AP and ROC-AUC metrics.

E DATASETS AND EVALUATION PROTOCOLS

  • Evaluation uses thirteen interaction streams, with released-file properties including event features, bipartite structure, temporal spans, and timestamp steps reported alongside temporal statistics.The released-file spans for MOOC and LastFM are approximately 30 days and 4.3 years, differing from durations in the benchmark table.Source: E DATASETS AND EVALUATION PROTOCOLS · pp. 50–51, Table 14 · p. 51

E.1 DATASET STATISTICS AND TEMPORAL STRUCTURE

  • The timescale count K is determined from training timestamps, while tied-event and repeated-test-pair statistics characterize strict history visibility and recurring relationships.Events at the query timestamp are excluded regardless of file order; repeated test-pair shares are approximately 100% for Social Evolution and UN Vote, but considerably lower for Canadian Parliament.Source: §E.1 · pp. 51–52

E.2 TEMPORAL SPLITS, INDUCTIVE SETTING, AND HISTORY VISIBILITY

  • Chronological 70/15/15 train-validation-test splits are used, and inductive evaluation holds out a sampled 10% of nodes from training.History is available only for interactions strictly earlier than the query time, including during streaming validation and test.Source: §E.2 · pp. 52–53

E.3 NEGATIVE SAMPLING AND BATCH-LEVEL EVALUATION

  • Historical negatives are previously observed pairs absent from the current positive batch, while inductive negatives are first observed during the current evaluation period and later sampled when not positive.Both protocols compare complete pairs and can therefore involve sources different from the matched positives, unlike random destination negatives.Source: §E.3 · p. 53
  • AP and ROC-AUC are computed within evaluation batches of at most 200 positives and an equal number of negatives, then averaged across batches.Although a source-activity term leaves matched same-source random-negative ordering unchanged, it can alter batch metrics by reordering examples across sources and query times.Source: §E.3 · pp. 53–54

F IMPLEMENTATION DETAILS AND REPRODUCIBILITY

  • The implementation appendix describes training configurations, baseline-result sources, corrections to released baseline implementations, SNAM transfer, and reproducibility procedures.Source: F IMPLEMENTATION DETAILS AND… · p. 54

F.1 TRAINING CONFIGURATION

The model uses shared architectural settings across datasets, with timescales determined from training timestamps and chronological event blocks. Training configuration specifies optimization, negative sampling, validation selection, and seed counts.

F.1 TRAINING CONFIGURATION

  • Across datasets, the model uses 128-dimensional node embeddings, rank-16 bilinear features, 256-dimensional random projections, and a recent-interaction encoder capped at 20 events.The encoder uses a 64-dimensional hidden representation; per-stream settings, including whether embeddings are trainable, are listed separately.Source: §F.1 · p. 54, Table 17 · p. 55
  • The number of exponential timescales K is determined from training timestamps, and chronological blocks are formed by event count or discrete time step depending on the dataset.For coarse timestamp streams, blocks do not split events sharing a timestamp; streams with more than 96 time steps merge consecutive steps into at most 96 blocks.Source: §F.1 · p. 55
  • Training uses AdamW with one optimizer update per block; SNAM is optimized independently with learning rate 0.05 and no weight decay.The relation branch uses a base learning rate of 10^-3, while specified zero-initialized output projections use ten times that rate.Source: §F.1 · p. 55
  • Each positive training interaction receives one random destination negative resampled every epoch, and checkpoints are selected using validation AP and AUC with early stopping.Main-model experiments use five seeds; baseline values are either reported by Lu et al. (2024) or retrained with released code and corrected implementations as indicated in the tables.Source: §F.1 · pp. 55–56, Table 18 · p. 56

F.2 BASELINE RESULTS AND RETRAINING

The appendix documents which baseline results were reused or retrained, the corrections made to released implementations, the frozen-backbone SNAM transfer procedure, and reproducibility checks. It also identifies the scope of the supplementary inductive AUC results.

F.2 BASELINE RESULTS AND RETRAINING

  • Four TPNet datasets failed the stated tolerance and use retrained means, while the other nine retain published values.The tolerance is max(1.0, 2σ) percentage points on each of six AP measurements; other baseline results are identified as published or corrected retraining runs.Source: §F.2 · pp. 56–57

F.3 CORRECTIONS TO RELEASED BASELINE IMPLEMENTATIONS

  • Baseline corrections address query-label and timestamp leakage, cross-query batch dependence, and state restoration while retaining original architectures and objectives.Corrections affecting training are applied in training and evaluation, followed by retraining from initialization.Source: §F.3 · pp. 57–58

F.4 TRANSFERRING SNAM TO OTHER MODELS

  • SNAM transfer freezes both the backbone and activity parameters, selecting only β on validation for each dataset, backbone, prediction setting, and negative-sampling protocol.The selected coefficient is applied unchanged to the corresponding test split.Source: §F.4 · pp. 58–59

F.5 CODE RELEASE AND REPRODUCIBILITY CHECKS

  • Reproducibility checks found at most 3 × 10^-5 change in selected test AP across GPUs, and seed-0 UCI AP measurements reproduced within 0.35 percentage points.The transfer pipeline also reproduces backbone results when β = 0.Source: §F.5 · p. 59

G ADDITIONAL EXPERIMENTAL RESULTS AND ANALYSIS

  • Table 19 reports inductive link-prediction AUC across random, historical, and inductive negative sampling, with results averaged over seeds.DyGFormer uses three seeds and LastFM one; the table marks corrected retraining with †.Source: Table 19 · p. 60

G.1 COMPLETE MAIN RESULTS

Across the reported tables, the full model ranks first under each of three evaluation protocols for both AP and AUC. Its advantages are strongest under historical and inductive negative sampling, including in inductive prediction.

G.1 COMPLETE MAIN RESULTS

  • 95.54% AP on MOOC under historical negatives exceeds TPNet, the strongest baseline, by 13.69 points in the inductive setting.Adding SNAM improves both inductive-setting average ranks under historical and inductive negatives to 2.35, with the highest AP on 9 of 13 datasets under each protocol.Source: §G.1 · p. 60
  • 2.44, 3.18, 2.46, and 3.33 are the full model’s overall average ranks for transductive AP, inductive AP, transductive AUC, and inductive AUC, respectively, ranking first in every table.Under historical negatives, it has the highest AUC on 11 of 13 datasets; on LastFM it reaches 94.74%, exceeding the strongest baseline by 10.10 points.Source: §G.1 · p. 60

G.2 COMPLETE TRANSFER RESULTS

SNAM improves AP across the reported backbone-transfer comparisons, while module ablations identify benefits from multiple timescales and within-block updates. The appendix also quantifies efficiency trade-offs across datasets and configurations.

G.2 COMPLETE TRANSFER RESULTS

  • All 40 comparisons across four backbones and five datasets improve AP in the two reported settings.Across the remaining eight datasets, 19 of 24 hard-negative comparisons improve, four are unchanged, and the sole decrease is 2.42 points on UN Vote; that extended study uses one seed per dataset.Source: §G.2 · p. 61

G.3 ROLES OF THE INDIVIDUAL MODULES

  • On UCI, transductive historical AP rises from 79.48% at K = 1 to 95.43% at K = 8 and 96.12% at K = 28; the authors find covering temporal ranges more important than continually adding timescales.The trend is not universal: inductive historical performance on Wikipedia is better at K = 1.Source: §G.3 · p. 61
  • Disabling within-block updates weakens the activity-term benefit; at C = 192, transductive historical AP falls 2.13–5.30 points across five datasets.The exact-to-query-time update claim applies to exponential counts, not to recomputing every higher-order representation at every query.Source: §G.3 · p. 61
  • Removing all three relation-input groups on UCI reduces transductive random AP from 95.81% to 83.09%.The results support complementary roles for multi-timescale historical representations and candidate-specific relation features; “no projections” also removes the dependent recent-interaction encoding.Source: §G.3 · p. 61
  • The count-based MLP exceeds SNAM by 0.17–0.35 points under transductive historical evaluation, but is not consistently better in other settings.The authors choose SNAM for its K + 1 parameters, event-intensity interpretation, shared historical statistics, and point-process likelihood without additional negative sampling or numerical integration.Source: §G.3 · pp. 61–62
  • For a fixed source and query time, adding β log λu(t) leaves candidates’ relative ranking unchanged, although mixed-query batch AP or AUC can change.Under hard negatives, a moderate activity correction generally improves AP before performance saturates or declines as β increases.Source: §G.3 · p. 62

G.4 EFFICIENCY AND COST TRADE-OFFS

  • On Flights, TPNet and DyGFormer take approximately 22× and 100× our per-epoch training time, respectively, while test scoring takes approximately 6× and 84× our time.On smaller streams such as UCI, fixed per-block overhead means the method is not fastest; the advantage is more pronounced on larger streams.Source: §G.4 · p. 62
  • Raising C from 48 to 192 reduces peak inference memory on LastFM to about 30% and on Reddit from about 3.8 GB to 2.6 GB, but block-start reconstruction can increase training time.Increasing timescales from 14–16 to 27–32 raises average training time to approximately 1.3×; measured stream costs do not grow faster than event count over the tested tenfold range.Source: §G.4 · p. 62
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