{"id":"97248547-2928-4726-afdf-ae4c7c84a530","arxiv_id":"2505.00983","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":6,"one_line_summary":"EDEN uses structural entropy to build a hierarchical knowledge tree over digraphs and distills parent-to-child knowledge to boost graph-neural-network accuracy, though the mutual-information lower bound it relies on is invalid.","lead":"A new framework, EDEN, builds a hierarchical tree over a directed graph using structural entropy, then distills information from tree parents to child nodes to improve graph neural network training. It reports accuracy gains on node and link prediction tasks, but its core mutual-information derivation is mathematically flawed.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The MI lower bound in Theorems 3.1–3.3 is invalid: Eqs. (5)–(6) split a single supremum into two independent suprema, so Eq. (6) is not a valid MI lower bound and the affinity scores in Eq. (9) lose their claimed information-theoretic grounding.","rationale":"The reader's weakest assumption identifies exactly the load-bearing flaw: the variational MI bound in Theorems 3.1–3.3 is invalid because Eqs. (5)–(6) use two independent suprema/maximizations instead of one shared scoring function. I agree, and this concern is decisive for the paper's central claim. Equation (6) is the official criterion C(Ω) (Theorem 3.3), directly used to compute affinity scores S in Eq. (9), to form parent representations X_p, and to drive the KD loss in Eq. (10). If C(Ω) is not a lower bound on MI, then the 'profile-aware' refinement of the HKT has no information-theoretic justification, and the claimed distinction between EDEN and a learned hierarchical clustering plus self-distillation heuristic is unsupported. The empirical results in Tables 1–3 and 6–9 are broad and internally consistent, and the plug-in gains may survive as an engineering contribution, but they cannot validate the theoretical mechanism without a correct bound; no formal verification or released code is provided. The paper could be repaired by using one shared critic in a standard f-GAN/DV objective or by reframing EDEN as a data-level regularizer without MI claims, but as written the central claim is not established. I would therefore keep the reader's REJECT verdict, with the same recommended revision path.","tokens_in":36413,"tokens_out":6803,"duration_ms":76166,"concrete_test":"Re-derive Eqs. (A.17)–(A.19) with a(F) = E_P log σ(F) and b(F) = E_Q log(1−σ(F)); verify that sup_F(a+b) ≤ sup_F a + sup_F b, so the inequality direction in Eq. (19) is reversed. Then run a numerical check on two small distributions (e.g., product of two-point marginals): evaluate the RHS of Eq. (6) with two independently optimized networks and compare it with the same objective using one shared network; if the independent-maximum value exceeds the shared-network value or exceeds a Monte Carlo estimate of the f-divergence, the claimed lower bound in Theorem 3.3 fails. Also recompute Eq. (4) with a single F in both expectation terms, using sup_F[E_P F − E_Q e^{F−1}] rather than two separate suprema.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.1 (Eqs. (4)–(6)) and Appendix A.6–A.8 derive a GAN-style MI estimator by starting from the valid f-divergence variational form D_f ≥ sup_{F∈F} [E_P F(X, X_N) − E_Q f^*(F(X, X_N̄))] (Eq. A.17), then replacing the single supremum with two independent suprema: sup_F E_P log σ(F(·)) + sup_F E_Q log(1−σ(F(·))) (Eqs. A.18–A.19; main text Eqs. (5)–(6)). This is not a lower bound: for any a(F) and b(F), sup_F(a+b) ≤ sup_F a + sup_F b, so the split expression is an upper bound on the valid lower bound and can exceed the true divergence. The two maximizers need not be the same network; Eq. (6)'s max_w + max_w is therefore not a variational MI estimate. Since the criterion C(Ω) in Theorem 3.3 is exactly this invalid objective, the affinity scores in Eq. (9) and the refined HKT used to build parent representations for KD are not MI-grounded. Theorems 3.1–3.3 are unproven as stated; the central claim that EDEN's knowledge flow is guided by mutual information collapses, leaving an empirically motivated heuristic with an unsupported theoretical core.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes EDEN, a data-centric framework for directed graph learning. It first constructs a coarse hierarchical knowledge tree (HKT) by minimizing a directed structural-entropy measure, then refines the tree using a claimed mutual-information neural estimator between node profiles and generalized neighborhoods, and finally performs parent-to-child knowledge distillation within the tree, followed by a random-walk-based leaf prediction. The paper claims state-of-the-art results on node- and link-level tasks over 14 (di)graph datasets and reports consistent improvements when EDEN is plugged into existing (Di)GNNs.","tokens_in":36781,"tokens_out":10859,"duration_ms":117059,"significance":"The proposed direction—hierarchical data-level knowledge distillation for digraphs—is timely, and the experimental campaign is broad: 14 datasets, multiple tasks, ablations, and sparsity and sensitivity analyses. If the theoretical underpinning were sound, EDEN would be a notable contribution to data-centric graph learning. However, the central theoretical result, the mutual-information neural estimator in Theorems 3.1–3.3, is not a valid variational MI bound as stated; since the affinity scores and the fine-grained HKT refinement in Eqs. (7)–(9) rest on this result, the main conceptual claim is not supported. The method may still work as a heuristic self-distillation regularizer, but that is a substantially weaker claim than the one made in the paper.","major_comments":[{"comment":"The derivation of the GAN-based MI neural estimator is mathematically invalid. The f-divergence variational form in Eq. (17) of Appendix A.7 is D_f(P||Q) ≥ sup_F [E_P T(F) + E_Q U(F)] with a single function F appearing in both the positive and negative expectation terms. In Eqs. (5) and (6) (and Eq. (21) of Appendix A.8), the authors replace this single supremum by two independent suprema: max_w over the positive term plus max_w over the negative term. This is a different optimization problem, since sup_F a(F) + sup_F b(F) is an upper bound on sup_F(a(F)+b(F)) and the two maximizing functions need not coincide. Consequently, C(Ω) in Eq. (6) is not the standard variational lower bound of MI. In fact, with the sigmoid parameterization both terms are non-positive and can be driven toward 0 separately on any finite sample when the function class is rich enough, making C(Ω) degenerate and largely data-independent. Theorems 3.2 and 3.3 therefore do not establish that C(Ω) estimates MI, and the affinity scores in Eq. (9) lose their claimed information-theoretic grounding.","section":"Section 3.1, Eqs. (5)–(6); Appendix A.7–A.8"},{"comment":"The paper's data-centric framing is weakened by the model dependence of the knowledge extraction. The topology-based HKT construction in Sec. 3.1 and Appendix A.5 is independent of the trained model, but the fine-grained refinement explicitly uses the trainable model M: F_intra and F_inter in Eqs. (7)–(8) are computed from M(X_v), and the knowledge generation in Eq. (9) and the KD loss in Eq. (10) are functions of the same trainable encoder. EDEN is therefore closer to a self-distillation and regularization scheme than to the extraction of intrinsic data knowledge. The authors should either provide a data-only variant of the profile refinement or substantially temper the claim that the method is data-centric.","section":"Section 3.1, Eqs. (7)–(10); Section 2.2"},{"comment":"Even if the split-supremum expression is treated as a heuristic score rather than an MI lower bound, the criterion C(Ω) as written is degenerate: both terms are non-positive (log σ ≤ 0 and log(1−σ) ≤ 0), and with independent maximization over w each term can approach its trivial optimum separately. Thus C(Ω) does not quantify the dependency between nodes and their generalized neighborhoods, and it cannot rank nodes by information content in the way claimed. A valid criterion would need to use the same network in both expectations, as in the standard f-divergence variational form, or be re-derived explicitly for the proposed architecture.","section":"Theorem 3.3 and Eq. (6)"}],"minor_comments":[{"comment":"The phrase 'hot-and-plug' appears repeatedly and should likely be 'plug-and-play'.","section":"Abstract and Section 1"},{"comment":"The definition of U_{X_p}^p and the expression X_p^{X_p}/U_{X_p}^p are unclear; the notation mixes partition indices, node indices, and class sums, and should be rewritten with explicit dimensionalities and a precise definition of each symbol.","section":"Eq. (10)"},{"comment":"The figures are referenced as 'Fig. ??' in the text; the figure numbers are missing.","section":"Appendix A.4"},{"comment":"The symbol σ is used both for the activation mapping into the conjugate domain and for the sigmoid function; this collision makes the derivation of Eq. (19) difficult to follow.","section":"Appendix A.7"},{"comment":"The transition probabilities 1/p_rw, 1/s_rw, and 1/c_rw are not normalized as written; the authors should clarify whether these are weights, probabilities, or unnormalized scores.","section":"Eq. (11)"}],"recommendation":"reject","confidential_remarks":"The principal weakness is the invalid MI estimation in Theorems 3.1–3.3, which is load-bearing for the paper's central claim. The empirical study is extensive, but without a valid theoretical grounding the contribution reduces to a heuristic regularizer. I would not rule out a future submission if the authors rework the theoretical section and reposition the method accordingly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: EDEN is a clever, well-engineered framework for directed-graph knowledge distillation, and the empirical work is more thorough than most. But the theoretical engine — the MI lower bound in Theorems 3.1–3.3 — is not a valid lower bound as written. That is a load-bearing flaw, because the paper leans on it to claim information-theoretic grounding for the affinity scores.\n\nWhat is new and good: the pipeline itself — directed structural entropy to build a coarse hierarchical tree, then a learned MI-style refinement, then parent-to-child knowledge distillation — is genuinely novel as far as I know. The experiments are extensive: 14 datasets, node and link tasks, plug-in gains on several DiGNNs (up to ~4.96%). The ablations and efficiency analysis are solid, and the gains look consistent rather than cherry-picked. If the method works as a heuristic, it is a useful one.\n\nNow the soft spot, and it is not minor. The standard variational f-divergence lower bound is a single supremum: sup_F [E_P F - E_Q f*(F)]. In Eqs. (4)–(6) the paper splits this into two independent suprema, and Appendix A.7 repeats the move. That is not a lower bound — for any a(F) and b(F), sup_F(a-b) is not bounded below by sup_F a + sup_F b. The two network parameterizations in the two max_w terms need not coincide. So Theorem 3.3's criterion is not an MI estimate, and the affinity scores in Eq. (9) lose their claimed information-theoretic meaning. This is not a stylistic issue; it is the formal justification for the whole refinement step.\n\nA secondary concern: the \"data-centric\" framing is weaker than the text implies. The teacher representations are computed from the trainable model M itself (Eqs. 7–9), so this is closer to self-distillation with tree structure than to extracting model-independent data knowledge. The \"data knowledge K\" in Assumption 2.3 is never defined. These are fixable in a revision, though.\n\nIf the authors correct the bound — for instance, by using the same network for both terms and re-deriving, or by dropping the MI claim and presenting the method as an empirically motivated regularizer — the core contribution would survive. As written, the central theoretical claim collapses.\n\nThis paper deserves a serious referee: the empirical contribution is substantial, and the fix may be straightforward. I would send it back for major revision rather than reject outright.","headline":"EDEN has a novel and empirically strong hierarchical KD pipeline for digraphs, but the core MI lower-bound theorems are invalid as written, so the paper's theoretical grounding collapses.","tokens_in":37335,"tokens_out":2598,"would_cite":false,"duration_ms":28144,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"EDEN claims that an entropy-built hierarchical knowledge tree, refined by mutual-information scores, distills data-level digraph knowledge into better node and link predictions.","keywords":["directed graphs","knowledge distillation","structural entropy","mutual information","hierarchical knowledge tree","data-centric learning","graph neural networks","heterophily"],"falsifier":"Take a small digraph with known node features, compute the exact mutual information for every node, and compare it against the trained affinity scores produced by Eq. (9). If the ordering of the scores does not match the ordering of the true mutual-information values, or if the right-hand side of Eq. (6) exceeds the true mutual information, then the information-theoretic refinement step is not doing what the theorems claim.","tokens_in":36128,"feed_emoji":"🌳","tokens_out":8056,"duration_ms":78491,"temperature":0.7,"pith_summary":"This paper proposes EDEN, a data-centric framework that treats the knowledge hidden in a directed graph's topology and node profiles as something to be distilled, rather than only processed by a message-passing model. The claim is that constructing a coarse hierarchical knowledge tree from directed structural entropy, refining it with neural mutual-information estimates, and then distilling parent knowledge to child nodes improves downstream digraph learning. If correct, EDEN provides both a standalone digraph learner and a model-agnostic plug-in that improves existing directed and undirected GNNs; experiments report gains up to 3.12% over baselines and up to 4.96% as a plug-in across 14 (di)graph datasets and 4 tasks. The broader significance is that data-level knowledge extraction, captured in a tree, can substitute for or complement model-level knowledge distillation.","feed_headline":"Entropy tree distills digraph knowledge into better GNNs","feed_subtitle":"A hierarchical knowledge tree refined by mutual information lifts node and link prediction on 14 graph datasets.","key_machinery":"The central object is the Hierarchical Knowledge Tree (HKT), a height-h partition tree over the digraph's nodes built by minimizing directed structural entropy; it carries the argument by supplying the teacher-student structure in which parent nodes act as teachers and child nodes as students. The mechanism that refines the tree is the mutual-information neural estimator of Theorems 3.1-3.3, which scores nodes by the dependency between a node and its generalized neighbourhood, where the neighbourhood is defined by HKT partitions rather than by the adjacency matrix. Those scores set the affinity weights in Eq. (9), the distillation loss in Eq. (10) transfers refined parent knowledge to child nodes, and the tree-based random walk in Eq. (11) aggregates multi-level representations for leaf prediction.","core_discovery":"On the paper's own terms, the discovery is that digraph data knowledge can be organised as a Hierarchical Knowledge Tree and then distilled. EDEN first computes directed structural entropy from random walks that include reverse probability and sink self-loops, and minimises this entropy greedily to build a coarse tree. It then uses a GAN-style mutual-information neural estimator, justified by Theorems 3.1-3.3, to score how well each node represents its generalized neighbourhood; these scores refine the tree by keeping or reassigning nodes across partitions. Finally, parent nodes act as teachers and child nodes as students in an online knowledge-distillation loss, and a tree-based random walk gathers multi-level representations for leaf prediction. The paper reports that this pipeline attains better accuracy than the compared baselines on node and link tasks across homophilic and heterophilic digraphs, and that it improves existing directed and undirected GNNs when plugged in.","pith_inferences":["The parent-child distillation view suggests a practical recipe for semi-supervised graphs: leaf nodes with labels can refine their parents' soft targets, which then regularize unlabeled children; the paper's label-sparsity experiments are consistent with this, though they do not isolate it.","The same two-stage construction of an entropy tree followed by mutual-information refinement could be applied to graph-level tasks, where the tree would summarise collections of graphs rather than nodes; that is a plausible extension the paper does not pursue.","Because the coarse tree is built without labels, EDEN could be adapted to unsupervised representation learning by using the tree-based random walk as a graph-equivariant augmentation; the paper only evaluates supervised tasks."],"forward_implications":["A digraph learner built on EDEN should beat existing DiGNNs on node classification and on link existence, direction, and three-class link classification; the paper reports consistent wins across 10 digraph datasets.","Existing (Di)GNNs can be upgraded by attaching EDEN as a plug-in module, with reported accuracy gains up to 4.96% over the same model without EDEN.","EDEN transfers to undirected graphs and to heterophilic settings where connected nodes do not share labels; the experiments include both homophilic and heterophilic benchmarks.","Under feature, edge, or label sparsity, EDEN degrades more gracefully than baselines because tree-based multi-level representations compensate for missing input signals.","The weighting of the distillation loss matters by task: edge-existence prediction favours small weights, while node-level and finer link tasks favour larger weights, indicating the distilled knowledge is most useful where supervision is fine-grained."],"supporting_citations":[{"why":"Supplies the structural-information and partition-tree formulation that EDEN's coarse HKT construction minimizes.","marker":"Li & Pan (2016)"},{"why":"Supplies the Donsker-Varadhan neural mutual-information estimator that Theorems 3.1-3.3 adapt to node-neighbourhood scoring.","marker":"Belghazi et al. (2018)"},{"why":"Supplies the f-divergence and GAN-style lower bound used to turn mutual-information estimation into the objective in Eq. (6).","marker":"Nowozin et al. (2016)"},{"why":"Provides the GAN discriminator-style objective that Theorem 3.2 and Theorem 3.3 repurpose for node selection.","marker":"Goodfellow et al. (2014)"},{"why":"Provides the personalised-PageRank-style propagation used to build generalized neighbourhood representations in Eq. (20).","marker":"Klicpera et al. (2019)"},{"why":"Grounds the premise that hierarchical community structure, encoded as a tree, predicts unseen links and reveals latent organization.","marker":"Clauset et al. (2008)"},{"why":"Gives the entropy measure underlying the directed structural measurement used for topology-aware knowledge discovery.","marker":"Shannon (1948)"}],"fun_headline_variants":["EDEN: entropy tree distills digraph knowledge for GNNs","Entropy tree distillation improves directed GNNs","Data-centric digraph learning via entropy tree KD","Entropy-guided tree distillation for digraph node and link tasks"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on the premise that the GAN-style objective in Eq. (6) is a valid lower bound on the mutual information between a node and its generalized neighbourhood; if that bound is not valid, the affinity scores that refine the HKT are not information-theoretically grounded.","fun_headline_variants_meta":{"raw":{"variants":["EDEN: entropy tree distills digraph knowledge for GNNs","Entropy tree distillation improves directed GNNs","Data-centric digraph learning via entropy tree KD","Entropy-guided tree distillation for digraph node and link tasks"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001353,"raw_usage":{"total_tokens":5547,"prompt_tokens":1053,"completion_tokens":4494,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":669,"completion_tokens_details":{"reasoning_tokens":4427}},"tokens_in":669,"tokens_out":4494,"duration_ms":29300,"temperature":1.0,"reasoning_tokens":4427,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:30:59.061741+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a small digraph with known node features, compute the exact mutual information for every node, and compare it against the trained affinity scores produced by Eq. (9). If the ordering of the scores does not match the ordering of the true mutual-information values, or if the right-hand side of Eq. (6) exceeds the true mutual information, then the information-theoretic refinement step is not doing what the theorems claim.","supporting_citations":[],"review_version":1}