APEX$^2$: Adaptive and Extreme Summarization for Personalized Knowledge Graphs

Dongqi Fu; Jingrui He; Mengting Ai; Zihao Li

REVIEW 3 major objections 5 minor 1 cited by

APEX$^2$: Adaptive and Extreme Summarization for Personalized Knowledge Graphs

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Personalized knowledge graphs can be summarized to 0.1% of their original size and still track a user's shifting interests, because APEX2 models interest as decaying heat and updates only the local neighborhood touched by each new query.

desk verdict A genuinely new adaptive PKG summarization framework with shipped code, but the best variant can exceed its own triple budget and the main comparison gives it an unfair update-frequency advantage. read the letter →

arxiv 2412.17336 v2 pith:C6EEHSOL submitted 2024-12-23 cs.LG cs.AIcs.DBcs.SC

Zihao Li , Dongqi Fu , Mengting Ai , Jingrui He This is my paper · ORCID

classification cs.LGcs.AIcs.DBcs.SC

keywords adaptivesummarizationpersonalizedknowledgegraphheatdiffusionincrementalsortingextremecompressionevolvinguserinterestsqueryanswering

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that personalized knowledge graphs can be kept extremely small—down to 0.1% or less of the full graph—while still answering a user's next query accurately, even as the user's interests drift from one topic to another. It proposes APEX2, a framework that models user interest as heat on the knowledge graph, lets that heat decay and diffuse after each query, and incrementally updates a ranked summary so re-summarization does not require scanning the whole graph. The reported experiments on graphs with up to 12 million triples show APEX2 and its entity-focused variant APEX2-N beating the existing baselines on next-query F1 while running tens of times faster than the strongest baseline. If the claims hold, storing a personalized copy of a knowledge graph on a device becomes practical for very large graphs and for users whose interests change during the day.

What carries the argument

The load-bearing object is the sparse heat tensor $\boldsymbol{H}$, defined entrywise as $\boldsymbol{H}^{(T)}[i][j][k] = \boldsymbol{e}^{(T)}[i]\,\boldsymbol{r}^{(T)}[j]\,\boldsymbol{e}^{(T)}[k]$, where $\boldsymbol{e}$ is the diffused entity-interest vector and $\boldsymbol{r}$ is the relation-frequency vector. The framework applies a decay-inject-diffuse cycle: at each timestamp all nonzero entries are multiplied by $\gamma^3$, new query heat is injected, and only entries whose entities or relation changed are recalculated. A second mechanism, incremental binary insertion sort, reuses the previous sorted order and inserts the few changed entries in $O(k \log n)$ comparisons, which Theorem 3.1 proves optimal. Together these make the per-timestamp update cost $O(c \cdot |\mathcal{Q}|^2 \log(c|\mathcal{Q}|))$, with $c$ the average number of neighbors within $d$ hops, independent of the number of entities in the full KG; an elimination threshold can reduce this to $O(c \log c)$.

What would settle it

Run the adaptation experiment with two topics deliberately chosen to have very different average connectivity or very different sizes (e.g., a dense hub topic and a sparse peripheral topic) and record the number of queries until the summary's F1 on the new topic surpasses that on the old topic. If that measured count systematically falls outside the bound given by Theorem 4.1, or if replacing the synthetic query logs with a real anonymized SPARQL log changes the F1 ranking of APEX2 versus the baselines, the central claim would be weakened.

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Extended reading notes

Core claim

APEX2 is presented as the first adaptive personalized knowledge-graph summarization framework that keeps the summary useful under extreme compression (budgets at or below 0.1% of the full graph). The core mechanism is a heat-based model of user interest: each query injects heat at the queried entity and its neighbors, heat decays by a factor $\gamma$ each timestamp, and the summary is simply the $K$ triples with highest heat. Because decay only scales all scores and new queries affect only a small neighborhood, the ordering of triples can be maintained by incremental binary insertion sort, making the per-query update cost depend on the local connectivity of the queried area rather than on the size of the whole knowledge graph. The paper also proves an adaptation bound: after a user switches from topic $U$ to topic $V$, APEX2 needs at most about $\log_\gamma \frac{1}{\frac{A}{B}(1-\gamma^a)}+1$ queries to re-adapt, where $A$ and $B$ are derived from the average connectivity of the two topics.

Load-bearing premise

The adaptation bound assumes the two topics have similar size and a well-defined average connectivity, and that products of user preferences can be replaced by products of their expectations; these approximations are not validated on the benchmark graphs, and the experiments further assume that synthetic topic-block query logs mimic real user behavior.

Editorial extensions

If this is right

Personalized knowledge graphs can be stored at compression ratios below 0.1% (as small as one triple per million) while still answering next queries with competitive F1, making on-device PKGs feasible for graphs with hundreds of millions of facts.
Users' shifting interests can be tracked without re-summarizing the whole graph: each new query triggers a local heat update and an incremental re-sort, so the cost per adapting phase is independent of the KG's total size.
APEX2-N, which ignores relations and tracks only entity heat, achieves higher next-query F1 than the full APEX2 in the experiments, suggesting that for short-horizon interest tracking entities matter more than relations.
The decay factor $\gamma$ directly controls the trade-off between adapting to new interests and retaining useful old facts; setting $\gamma$ near 1 (no forgetting) causes F1 to collapse under extreme compression.
Baseline methods that re-summarize periodically every 9 timestamps are dominated in both accuracy and speed, so existing static PKG summarizers are not a viable fallback for evolving interest.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the heat model is right, the same decay-inject-diffuse mechanism could be applied to other evolving personalization tasks, such as adaptive retrieval-augmented generation contexts or personalized recommendation, where the 'graph' is a user-specific interaction graph rather than a knowledge graph.
The paper leaves the entity-relation weight trade-off unresolved; a natural testable extension is to learn the weight per user or per query type from the query log instead of fixing it to 0 or 1.
The synthetic query logs are constructed as blocks of 10 same-topic queries; real interest shifts are likely more gradual and interleaved, so a stress test with probabilistic topic mixtures would show whether the adaptation bound degrades gracefully.
Because the update cost is independent of KG size once a threshold zeroes out decayed heat, the framework could in principle scale to graphs far beyond the 12-million-triple benchmarks tested, provided the local connectivity $c$ stays bounded.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Zihao Li and coauthors have written a paper worth a real referee, but it needs major surgery before it can be trusted as a published record. The core contribution—adaptive personalized KG summarization under extreme compression—is genuinely new, and they back it with shipped code and large-scale experiments on YAGO, DBpedia, MetaQA, and Freebase at compression ratios around 0.01% and below. The heat-decay model, incremental binary insertion sort, and the decay ablation are all reasonable engineering. The optimality proof for the incremental sort (Theorem 3.1) is clean.

The problems start with the definition of the problem. Equation 2 imposes a budget K on the number of triples, and the experiments report triple-count compression ratios. But Algorithm 5, APEX2-N, selects the top-K entities by heat and keeps all triples induced on those entities. The induced subgraph on K entities can contain far more than K triples—a clique would give ~K^2/2. The while loop in lines 13–17 adds an entity until T_p exceeds K, then exits, so the output can be arbitrarily over budget. Figure 2 shows APEX2-N as the best method, so the headline 'state-of-the-art at extreme compression' is not established unless the code enforces a cap that the pseudocode omits. The authors need to fix the algorithm (e.g., cap by triples, not entities) and re-run.

Second, the empirical comparison is not apples-to-apples. Baselines re-summarize every 9 timestamps (R=9), while APEX2 updates every timestamp (R_APEX=1). The paper justifies R=9 with a pre-experiment, but this still gives APEX2 more adaptation chances. The case study that gives baselines per-timestamp re-summarization is a good partial mitigation, but the main figure remains unfair.

Third, the 'robust theoretical guarantees' are not rigorous. The effectiveness theorems (4.1, 4.3) replace products of preferences with products of expectations and assume similar-size topic areas. That turns the bound into a heuristic. The proof sketch in Appendix E shows the approximation steps explicitly. In the current form, the theory should be labeled as an analysis of the model, not a guarantee.

Minor but annoying: the appendix theorem numbers don't match the main text. E.1 is labeled 'Theorem 4.1' but proves the sorting optimality (Theorem 3.1); E.4/E.5 prove GLIMPSE/PEGASUS non-adaptability but are labeled Theorem 3.1/3.2; E.6/E.7 are labeled 4.2/4.4 but prove the effectiveness results. This needs fixing.

Who should read this? Researchers working on on-device KG summarization, evolving user interests, or incremental top-K maintenance. The paper deserves peer review, but I'd ask for a corrected algorithm for APEX2-N, an enforced-budget re-run, a same-update-frequency comparison (or a stronger justification), and a rewritten theory section that says what's heuristic and what's exact.

Referee Report

3 major / 5 minor

Summary. The paper proposes APEX2 and a variant APEX2-N for adaptive, extremely compressed personalized knowledge graph (PKG) summarization under evolving user interests. The method maintains a heat-based interest model with decay, incrementally updates entity/relation/triple preferences, and uses incremental sorting to select top-heat triples under a storage budget. The authors provide theoretical claims on adaptation speed and incremental time complexity, and report experiments on YAGO, DBPedia, MetaQA, and Freebase showing that APEX2 and APEX2-N outperform GLIMPSE, PEGASUS, iSummary, and personalized PageRank in next-query F1 at compression ratios below 0.1%.

Significance. If the claimed results hold, the paper would make an important step toward practical on-device PKGs: it targets the regime of extreme compression (≤0.1%) and continuous interest shift, which prior work (e.g., GLIMPSE, PEGASUS) was not designed for. The paper ships code and provides a clean problem formulation with a heat-decay mechanism that is intuitive and likely extensible. The theoretical results, though conditional on several assumptions, offer a starting point for reasoning about adaptation speed. The main empirical claim hinges on the correctness of the experiments, and two load-bearing issues currently prevent acceptance: APEX2-N may violate the stated storage budget, and the baselines are evaluated at a disadvantageous update frequency. The central idea is promising, but the current evidence is not sufficient to substantiate the headline claims.

major comments (3)

[§5.2, Algorithm 5 (Appendix A.5), Eq. (2)] APEX2-N does not enforce the size budget K on triples. The algorithm selects top-K entities in line 6 and then in lines 7 and 15 constructs T_p as all triples induced by those entities, with no final trimming. Because an induced subgraph on K entities can contain substantially more than K triples, the resulting PKG can violate the constraint |P| ≤ K from the problem definition in Eq. (2). The compression ratios reported in Section 5.2 are triple-based (e.g., 0.01% of MetaQA’s 231,103 triples corresponds to about 23 triples), so APEX2-N’s summaries may be much larger than the claimed budget. The F1 results for APEX2-N in Figure 2 and Table 3 are therefore not valid evidence for effectiveness at the stated compression ratios unless the code enforces a triple-level cap that is missing from Algorithm 5. Please revise the algorithm to guarantee |T_p| ≤ K (for example, by keeping only the top-K triples from the induced set) and rerun the experiments.
[§5.1.3 and §5.3] The main experimental comparison is not at equal update opportunity. APEX2 and APEX2-N update the summary every timestamp (R_APEX = 1), while the baselines GLIMPSE, PEGASUS, iSummary, and PageRank re-summarize only every R = 9 timestamps, as described in Section 5.1.3. Thus APEX receives nine times more update chances than the baselines. The additional experiments in Section 5.6 show that APEX2-N’s F1 drops from 0.858 to 0.680 when R_APEX increases from 1 to 6 on MetaQA, which suggests that a significant part of the advantage may be due to the asymmetric update schedule. To support the claim that APEX2 and APEX2-N outperform baselines, please report results at matched update frequencies (e.g., R_APEX = 9 for APEX methods or R = 1 for baselines, if computationally feasible).
[§4.1, Theorem 4.1 (proof in Appendix E.6) and Theorem 4.3 (proof in Appendix E.7)] The query-count adaptation bounds in Theorems 4.1 and 4.3 rely on assumptions that are not stated in the theorem and are not validated empirically. The proofs in Appendices E.6 and E.7 replace products of entity/relation preferences by products of their expectations and assume the topics U and V have similar sizes, i.e., |E_u| ≈ |E_v| and |R_u| ≈ |R_v|, in order to obtain the closed-form bound b > log_gamma(...). Without these assumptions, the derivation does not produce the stated bound. Please state these assumptions explicitly in the theorems and either validate them on the benchmark KGs or discuss the sensitivity of the bound when they are violated.

minor comments (5)

[Appendix E: theorem numbering] The appendix proof labels do not match the main-text theorem numbers: E.2 proves the APEX2 time complexity (Theorem 4.2) but is titled Theorem 4.3; E.3 proves APEX2-N time complexity (Theorem 4.4) but is titled Theorem 4.5; E.6 proves APEX2 effectiveness (Theorem 4.1) but is titled Theorem 4.2; E.7 proves APEX2-N effectiveness (Theorem 4.3) but is titled Theorem 4.4. Please renumber for consistency.
[Algorithm 5, line 7] The condition in line 7, "v in e", is unclear and appears to contain a typo; it should presumably refer to entities being endpoints in E_p^(0) (or to the relation k). Please correct.
[Eq. (5) and Eq. (6)] Equation (5) writes Pr(e|Q) with an unweighted indicator 1(e_o in q), whereas the vector q_total in Eq. (3) weights answer entities by 1/|A_i|. If Eq. (6) is the actual computation, please clarify that Eq. (5) is an informal illustration and point to Eq. (3)–(4) for the exact weighting.
[Table 3] The PageRank row for YAGO reports a mean of 22.81 with standard deviation 259.7, which is implausibly large relative to the mean; please check whether this is a typo or an artifact of a small number of outlier runs.
[Section 5.2 and Table 3] The notation "APEX-N2" in Table 3 is inconsistent with the rest of the paper, which uses "APEX2-N"; please unify.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the APEX2 objective and experiments are grounded in external baselines and held-out next-query evaluation, and the effectiveness theorems are self-consistency statements rather than input-equivalent reductions.

full rationale

The paper's derivation chain is self-contained. The triple-preference objective in Eq. 14-15 is explicitly inherited from the external GLIMPSE baseline ('we stick to GLIMPSE's choice for triple preference'), not from a fitted constant or self-citation. The experimental F1 comparisons are measured against GLIMPSE, PEGASUS, iSummary, and personalized PageRank on 'the very next query' after adaptation, and no parameter is fitted to the F1 numbers, so the reported comparisons are not forced by construction. The effectiveness theorems (4.1 and 4.3) analyze the heat-decay model's own scoring function: 'adaptation' is formalized as the point where the model's preference for the new topic exceeds that for the old topic, and the bounds in Appendix E.6/E.7 follow from the model equations under explicit assumptions (|E_u|≈|E_v|, |R_u|≈|R_v|, and expectation-of-products approximations). This is an internal consistency analysis rather than a circular prediction, because the theorems do not assume their conclusions and are not used to define the input quantities. Self-citations such as [34] for the Neumann-series closed form and the PPR baseline are not load-bearing: the closed form is a standard identity and PPR is an external benchmark. Section 8's limitation statement ('there may be no significant advantage to summarizing a PKG rather than querying the KG directly') is candid and does not conceal a circular step. Two non-circular concerns fall outside this pass: Algorithm 5 can select an induced triple set exceeding the budget K, and the Appendix E proofs rely on unvalidated homogeneity approximations; these affect correctness and rigor, not circularity.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central claims rest on several modeling choices: conditional independence of triple preference, heat diffusion as a proxy for user interest, topic-as-entity, plus unquantified approximations in the adaptation proofs. The three hyperparameters gamma, alpha, and d are hand set, and the heat-elimination threshold is left unspecified. No new entities are introduced.

free parameters (5)

gamma (decay factor) = 0.5 (default; ablated 0.1-1.0 in Fig. 3)
Hand-set hyperparameter controlling how quickly old interests fade; the adaptation bound in Theorem 4.1 depends on it.
alpha (damping factor of neighbor) = 0.3
Hand-set neighbor diffusion weight; used in Eq. 6 and in the theoretical adaptation analysis.
d (diffusing diameter) = 1
Number of diffusion hops; with d>1 the step from Eq. 5 to Eq. 6 is not exact because A^l counts walks rather than distance-l neighbors.
epsilon threshold for heat elimination = unspecified ("small enough value")
Used in Section 4.1 to zero out decayed heat and make incremental complexity independent of graph size; never defined or reported in experiments.
entity/relation weight ratio in APEX2-N = entities: 1, relations: 0
Deliberate modeling choice that makes APEX2-N ignore relation interest; the paper leaves tuning of this trade-off to future work.

assumptions (5)

standard math Convergence and invertibility of (I - alpha A)^-1 in Eq. 7
Used to write the infinite-diffusion closed form; with unnormalized adjacency A and alpha=0.3 it need not hold for high-degree KG nodes, though experiments use finite d=1.
domain assumption Triple preference factorizes as Pr(e_i) Pr(r_k) Pr(e_j)
Eq. 14; inherited from GLIMPSE (Eq. 22), not derived from query logs.
domain assumption User interests follow heat diffusion and topics correspond to entities/nodes
Section 3.1 and Appendix D; used to justify query entity as topic label and the entire adaptation analysis.
ad hoc to paper Topic areas have similar sizes and expectation products can be replaced by products of expectations
Appendix E.4, E.6, and E.7 derive the adaptation bounds only after assuming |E_u| approximately equals |E_v|, |R_u| approximately equals |R_v|, and approximating products by expectations; no error bound is given.
domain assumption Queries are simple one-hop queries with known answers
Section 2 defines the query log this way and Section 5.2 generates only 1-hop simple queries, so claims do not extend to complex multi-hop queries.

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Cite this review

Pith. "Pith review of APEX$^2$: Adaptive and Extreme Summarization for Personalized Knowledge Graphs." pith.science (2026). https://pith.science/paper/C6EEHSOL

@misc{pith2026241217336,
  author       = {Pith},
  title        = {Pith review of: APEX$^2$: Adaptive and Extreme Summarization for Personalized Knowledge Graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C6EEHSOL}},
  note         = {Machine review of arXiv:2412.17336}
}

abstract

Knowledge graphs (KGs), which store an extensive number of relational facts, serve various applications. Recently, personalized knowledge graphs (PKGs) have emerged as a solution to optimize storage costs by customizing their content to align with users' specific interests within particular domains. In the real world, on one hand, user queries and their underlying interests are inherently evolving, requiring PKGs to adapt continuously; on the other hand, the summarization is constantly expected to be as small as possible in terms of storage cost. However, the existing PKG summarization methods implicitly assume that the user's interests are constant and do not shift. Furthermore, when the size constraint of PKG is extremely small, the existing methods cannot distinguish which facts are more of immediate interest and guarantee the utility of the summarized PKG. To address these limitations, we propose APEX$^2$, a highly scalable PKG summarization framework designed with robust theoretical guarantees to excel in adaptive summarization tasks with extremely small size constraints. To be specific, after constructing an initial PKG, APEX$^2$ continuously tracks the interest shift and adjusts the previous summary. We evaluate APEX$^2$ under an evolving query setting on benchmark KGs containing up to 12 million triples, summarizing with compression ratios $\leq 0.1\%$. The experiments show that APEX outperforms state-of-the-art baselines in terms of both query-answering accuracy and efficiency. Code is available at https://github.com/iDEA-iSAIL-Lab-UIUC/APEX.

Figures

Figures reproduced from arXiv: 2412.17336 by the authors.

**Figure 2.** Effectiveness Comparison Under Querying Scenario [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Querying Effectiveness in Multiple Decay Levels [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (12 more)

**Figure 4.** Figure 4: Parameter Study. From left to right: compression ratio [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: Cosine similarity between consecutive query topics in different datasets for one user. The cosine similarity is computed [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

**Figure 6.** Figure 6: The distribution of cosine similarities between consecutive query topics in our synthetic queries. From left to right: [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]

**Figure 7.** Figure 7: Our pre-experiment to determine that 𝑅 = 9 is a good trade-off for accuracy and efficiency. C.4.5 Freebase. https://paperswithcode.com/dataset/f b15k237. The entire FB15K237 dataset can be downloaded by clicking "Homepage". C.4.6 Environments. We run all our experimen…

**Figure 8.** Figure 8: Querying Effectiveness in Multiple Decay Levels for different [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]

**Figure 9.** Figure 9: Parameter Study adjusting different 𝑅𝐴𝑃𝐸𝑋 . From left to right: compression ratio 𝜅, damping factor of neighbor 𝛼, diffusing diameter 𝑑. we adopt a heat decay-inject-diffuse framework for heat diffusion, as shown in [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]

**Figure 10.** Figure 10: We adopt a heat decay-inject-diffuse framework for heat diffusion. For each timestamp, first, all the heat (i.e., interest) [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]

**Figure 11.** Figure 11: The specific part of MetaQA Knowledge Graph within the interest of our case study. The summarization operates on [PITH_FULL_IMAGE:figures/full_fig_p021_11.png]

**Figure 12.** Figure 12: Case Study on APEX2 [PITH_FULL_IMAGE:figures/full_fig_p022_12.png]

**Figure 13.** Figure 13: Case Study on APEX2 -N [PITH_FULL_IMAGE:figures/full_fig_p023_13.png]

**Figure 14.** Figure 14: Case Study on GLIMPSE [PITH_FULL_IMAGE:figures/full_fig_p024_14.png]

**Figure 15.** Figure 15: Case Study on PageRank [PITH_FULL_IMAGE:figures/full_fig_p025_15.png]

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Homepage

The entire FB15K237 dataset can be downloaded by clicking "Homepage". C.4.6 Environments. We run all our experiment on a Windows 10 machine with Intel(R) Core(TM) i7-10750H CPU @ 2.60GHz and 32GB RAM. For other different platforms such as Linux, you may need to reset the path ...

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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