Adaptive Cannistraci-Hebb Network Automata Modelling of Complex Networks for Path-based Link Prediction

We appreciate the reviewer’s insightful and encouraging assessment. We will integrate the suggested clarifications and additional analyses in the revision to further strengthen the paper’s presentation and scope.

Weakness 1: The main weakness of the method/paper is the lack of clarity in motivation behind the use of the CH framework and an explicit distinction between methods under the CH framework and other probabilistic frameworks such as the SBM or spectral methods such as the SPM. In lines 56-57, the authors note the clear interpretability of the growth mechanisms under the CH framework, but this is not explained in detail, e.g. with reference to one of the alternatives. Or in the discussion of results.

Reply: Our goal is to capture local, mechanistic drivers of link formation, how distinct structural patterns affect a missing edge, within an interpretable, predictive network‑automata framework. CHA consists of local, training‑free rules driven by local‑community patterns and provides per‑edge mechanistic explanations. By contrast, SBM (probabilistic) and SPM (spectral) are global models that prioritize macro‑structure (block partitions, global eigenstructure) rather than edge‑level local mechanisms. A formal side‑by‑side distinction between CH and these alternatives is provided in [1]. We have revised the manuscript to make this motivation and reference the relevant comparison.

Weakness 2: Another weakness in my view is the lack of references and/or comparisons with more latest contributions on link prediction in networks, such as (1) Gaucher, Solenne, and Olga Klopp. "Optimality of variational inference for stochastic block model with missing links." Advances in Neural Information Processing Systems 34 (2021): 19947-19959. (2) Zhao, Yunpeng, et al. "Link prediction for partially observed networks." Journal of Computational and Graphical Statistics 26.3 (2017): 725-733.

Reply: Thank you for the suggestion. We have updated the manuscript and included these articles in the related work section.

Question 1: Can the practical importance of the interpretability of the mechanism behind network growth as offered under the CH framework in contrast to link prediction under SBMs be illustrated via a simple example?

Reply: CH/CHA offers per‑edge, mechanistic explanations grounded in local structure: each score decomposes into internal community vs. external leakage (e.g., iLCL/eLCL). By contrast, SBMs explain links via global group memberships and block‑level probabilities inferred from the entire graph.

Question 2: I found the paper to be completely lacking in intuition. For example, what is the idea behind the definition of RA-Ln in equation (5)? What does a local-community paradigm (LCP) architecture refer to in simple words? Why is minimization of external connectivity key to this framework?

Reply:

• RA‑Ln (Eq. 5). We introduce RA‑Ln as an example of a network automata on paths: it extends the classic resource‑allocation idea from common neighbors (L2) to Ln.

• What is the LCP (local‑community paradigm) architecture, in simple words? In the Fig. 3 in appendix we show that a local community is a subnetwork formed by the common-neighbors (defined on a certain path of length n) of two seed nodes and by the interactions between such common-neighbors. The local-community architecture is the fact that the network is organized according to such local community organization and it is measured by the LCP-correlation see Fig. 4, 5,6 of the reference: From link-prediction in brain connectomes and protein interactomes to the local-community-paradigm in complex networks. CV Cannistraci, G Alanis-Lobato, T Ravasi. Scientific reports 3 (1), 1613. In Fig.5 and 6 of this reference you will find examples of LCP or non-LCP real and artificial complex networks.

• Why minimize external connectivity? This is widely discussed in the reference “Local-community network automata modelling based on length-three-paths for prediction of complex network structures in protein interactomes, food webs and more. A Muscoloni, I Abdelhamid, CV Cannistraci. BioRxiv, 346916”. The fact that a common-neighbor node has a low or absent number of external (to the local community) links ensures from the physical point of view its participation in local processing units that perform a function in the complex systems. For instance: a protein complex in biological pathways, a neural circuit associated with an engram in the nervous system, or a club of associates in a social network.

Question 3: Can the method be compared to Gaucher and Klopp 2021 - which seems to be a more recent work on prediction of missing links under SBM?

Reply: Thanks for this suggestion. We are conducting experiments to compare with it.

Question 4: lines 98-99: the likelihood is mentioned here. What is the model under which the likelihood is evaluated? Please state this clearly.

Reply: In our work, “likelihood” is used in the predictive sense: the CH/CHA score $s(u,v)$ is a local‑based measurement whose ordering correlates with the probability that a non‑observed link $(u,v)$ exists. Formally, we assume that there exists a monotone link $g$ such that $p(u,v) ≈ g(s(u,v))$. This explains why using the CHA score for link prediction is appropriate, as the evaluation is ranking‑based (Precision/NDCG/AUPR).

Question 5: Lines 114-115: are x_S and x_B function of \tilde{X}(t) or independently chosen?

Reply: In [2], x_S and x_B are independently chosen.

Question 6: Can iLCL and eLCL be mathematically defined?

Reply: Thank you for the suggestion. For L2 paths, we define iLCL and eLCL for an intermediate node $z$ between seeds $(u,v)$ as

• iLCL of node z given seed node (u, v): $di_z = | \mathcal{N}_z \cap \mathcal{N}_u \cap \mathcal{N}_v |$
• eLCL of node z given seed node (u, v): $de_z = | \mathcal{N}_z \setminus ((\mathcal{N}_u \cap \mathcal{N}_v) \cup \{u, v\}) |$

where $\mathcal{N}_v$ denotes the neighborhood of a node v. Intuitively, iLCL counts $z$’s links inside the local community of common neighbors of $u$ and $v$, while eLCL counts those outside it. We have added these formulas in the revised manuscript, together with the corresponding L3 definitions.

Question 7: Lines 252-253: How/why is this expected to work in the case of static networks where for example, a high proportion of non-observed links are simply missing links?

Reply: We are running a sensitivity analysis that increases the edge‑removal rate. We will report the result as soon as possible.

Question 8: Appendix A.2 (final paragraph): are each of the P sampled partitions of the same size?

Reply: No. We do not fix the size of partitions.

Formatting 1: NDCG -abbreviation used without being defined first

Reply: Thank you for pointing this out. We will define NDCG at first use as Normalized Discounted Cumulative Gain (NDCG) and add the formula.

Formatting 2: How exactly are 'mean rank' and 'win rate' as used for evaluation defined?

Reply:

Mean rank (lower is better): For each network realization and each metric (AUPR, Precision, NDCG), we rank all methods (1 = best). A method’s mean rank is the average of its ranks over all realizations.

Win rate (higher is better): For each realization and metric, we check which method ranks 1st; win rate is the fraction of realizations where a method is 1st (ties counted as both wins).

We prefer mean rank / win rate over averaging raw metric values because networks differ widely in size and class imbalance, making absolute scores non‑comparable; rank‑based summaries capture relative performance fairly across heterogeneous networks.

[1] Muscoloni, Alessandro, Umberto Michieli, and Carlo Vittorio Cannistraci. "Local-ring network automata and the impact of hyperbolic geometry in complex network link-prediction." arXiv preprint arXiv:1707.09496 (2017).

[2] Smith, David MD, et al. "Network automata: Coupling structure and function in dynamic networks." Advances in Complex Systems 14.03 (2011): 317-339.

We sincerely thank the reviewer for the thorough and highly positive evaluation. Below we offer brief clarifications and note the additional analyses and experiments we will incorporate in the revision to address the remaining questions.

Weakness 1: Limited Theoretical Analysis: While the mathematical formulations are clear, the paper lacks theoretical analysis of when and why the adaptive mechanism should work. There's no convergence analysis for the selection procedure or theoretical guarantees about optimality.

Reply: A convergence analysis is not applicable because our adaptive mechanism is non‑iterative: CHA computes closed‑form scores, and the “adaptation” is a finite model‑selection over a small set of CH variants and path lengths (no gradient updates).

Why the adaptive mechanism is expected to work. For each network, we create random train/test splits of the same graph; thus, the train‑ and test‑positive links are exchangeable samples from the same underlying distribution. Under exchangeability, the metric computed on the observed (training) links is an unbiased estimator of the test metric; therefore, selecting the rule that maximizes this metric on the observed data is statistically justified.

Weakness 2: Computational Complexity: The paper mentions time complexity details are in supplementary material but doesn't discuss scalability in the main text. With multiple models tested per network, computational overhead could be significant for large networks.

Reply: Thank you for the helpful suggestion. As noted on line 65 of the Supplementary, although CHA internally evaluates multiple CH models and path lengths, all candidates operate on the same intermediate statistics (iLCL and eLCL), which are computed once per graph. Consequently, evaluating multiple candidates introduces only a small overhead relative to running a single model.

In the revised manuscript, we have summarized this scalability point in the main text.

Weakness 3: Baseline Limitations: Some baseline implementations may not be optimal - for example, the modification of MPLP's early stopping criterion, while reasonable, makes direct comparison less straightforward. The SBM results are described as "irrelevant and unsatisfactory" which seems dismissive without deeper analysis.

Reply: We appreciate this concern and have taken steps to ensure fair, transparent comparisons.

• MPLP/MPLP+ early stopping. We used the official hyperparameters and code for MPLP/MPLP+, with a single change to early stopping to match our evaluation protocol. The original implementation monitors Hits@100 on a 1:1 negative‑sampling validation set, whereas our test metric is AUPR with all‑missing‑links ranking. Using original implementation systematically favors models that excel on easy negatives and can lead to premature stopping. To align model selection with the test objective, we monitor validation AUPR where validation positives are also ranked among all non‑observed links (Appendix Sec. A.8). Importantly, this modification only affects the stopping time, mostly increasing the number of training epochs, but does not change the model architecture, loss, optimizer, data pipeline, or any other training setting. For the harder all‑missing‑links task, allowing additional epochs is typically beneficial, as the model requires more optimization steps to learn to separate positives from hard negatives.

• Action item for the revision. To address the reviewer’s point directly, we will report both variants side‑by‑side: (i) the original MPLP/MPLP+ early‑stopping protocol and (ii) the aligned AUPR‑based early‑stopping protocol. Our preliminary runs indicate that the original protocol yields substantially lower performance under the all‑missing‑links test (consistent with under‑learning for the harder task). We will report the result as soon as possible.

We have removed the “irrelevant and unsatisfactory” phrasing in the abstract. We hope these changes address the concern and improve the clarity and fairness of the comparisons.

Weakness 4: Statistical Rigor: The authors acknowledge not providing error bars, arguing that networks aren't i.i.d. samples. However, some form of statistical analysis (e.g., bootstrap confidence intervals for win rates) would strengthen the claims.

Reply: Thank you for the suggestion. We have added a stratified bootstrap (by network category) to quantify standard deviations for average ranks. Concretely, we run 10,000 stratified resamples; in each resample we draw, within each category and without replacement, the same number of networks as in the original split, compute the metrics, and aggregate with the original category weights. We report mean and std in bellow. The conclusions are unchanged: CHA maintains the best average rank with small standard deviations. We have included this in the revised manuscript.

ATLAS-small mean rank AUPR

Question 1: Theoretical Justification: Can you provide theoretical insight into why the adaptive mechanism works? Under what conditions might it fail, and are there networks where a fixed rule would be preferable?

Reply: As noted in our response to Weakness 1, the adaptive mechanism’s effectiveness follows from train‑ and test‑positive links being sampled from the same underlying distribution. Regarding cases where a fixed rule may be preferable, Fig. 4 shows that many network categories exhibit stable preferences for L2 or L3.

Question 2: Computational Scalability: What is the computational complexity of CHA compared to baselines? How does runtime scale with network size, and at what point does the adaptive overhead become prohibitive?

Reply: Thank you for the suggestion. A comprehensive analysis is provided in the Supplementary (“supplementary.pdf”). Below we summarize the key results for sparse networks.

CHA complexity（sparse regimes）

As shown in Fig. 1 of the Supplementary, most real-world networks—especially large ones—are very sparse.

Baselines（sparse regimes, training + scoring over all missing links

Here, $r\ge 1$ is the hop count, $F$ the node-signature dimension, and $K$ the embedding dimension. Note that these complexities include both training and scoring over all non-observed links (for several methods, the scoring step dominates). $M$ is the number of missing links in the network, and $B$ represents the number of blocks in the SBM family of models. For SBM-based models, the primary computational cost arises from fitting the network using MCMC equilibration and the posterior edge probability estimation. The listed complexity corresponds to the sparse version; for denser networks, the complexity increases to $O(M\cdot(n^2 + B^2))$.

Adaptive overhead. Although CHA evaluates multiple CH variants/path lengths, all candidates reuse the same precomputed intermediate statistics (iLCL/eLCL), computed once per graph.

Thanks reviewer for this suggestion. We have added a short scalability summary and a table of baseline complexities in the revised manuscript.

Question 3: Path Length Selection: The analysis focuses on L2 vs L3 paths. Have you considered extending to longer path lengths (L4, L5)? What theoretical or empirical evidence suggests L3 is sufficient?

Reply: Thanks for the suggestion. We additionally evaluated CHA-L2…L6 on ATLAS and summarize the average AUPR win rate:

These results show that L2/L3 substantially outperform L4+. Accordingly, we focus the main analysis on L2/L3 and include the L4–L6 results in the appendix for completeness.

Question 4: Sub-ranking Strategy: The SPcorr sub-ranking seems ad-hoc despite the neurobiological motivation. Have you compared against other tie-breaking strategies? How sensitive are the results to this choice?

Reply: Thank you for the question. We are currently running an ablation on tie‑breaking that compares SPcorr with other tie-breaking strategies. We will report the result as soon as possible.

Question 5: Cross-domain Generalization: The adaptive mechanism selects models based on observed links in the same network. How would CHA perform when applying models learned from one domain to predict links in a different domain?

Reply: “Cross‑domain transfer” would mean fixing the rule selected on a source domain and applying it to a different domain. We do not expect this to work reliably unless the domains are structurally similar (e.g., comparable local‑community patterns such as L2 vs. L3 preference). By contrast, within the same domain, applying the same fixed rule across different networks is often reasonable—indeed, Fig. 4 shows stable L2/L3 preferences at the category level.

We sincerely thank the Reviewer for the insightful and constructive comments.

Weakness 1: The work seems not relevant to the topic of the conference.

Reply: We sincerely thank the reviewer that gives us the chance to clarify why our study represents a novel and timely contribution to the field of AI for link prediction. The reply is based on three points

1.1 Why the work goes far beyond the introduction of two heuristics and is relevant to the conference

The Reviewer’s comment indicates that we must more clearly emphasize our main innovation in the Introduction and strengthen its explanation throughout the Results and Discussion. As stated at the end of the Introduction Section 1, L79: “In this study, we introduce four key innovations in the field of topological link prediction.” These are then briefly described: Minimization of external connectivity (the Cannistraci-Hebb, or CH, paradigm) — which is the specific point the Reviewer seems to highlight; Adaptive model selection; Comprehensive static and temporal benchmarking; Multi-metric evaluation. This clearly demonstrates that our contribution is not limited to two heuristic indexes (CH3, CH3.1), but spans two core theoretical innovations and two robust experimental advances. Importantly, this perspective is independently supported by two other reviewers (rqqa and AZk2) in the NeurIPs review process. For instance, Reviewer AZk2 writes: “The strength of the method lies in it being adaptive as it can automatically learn the rule that better explains the pattern of connectivity in the network under study. Their method leads to an overall higher link prediction performance in comparison to a wide range of techniques.”

To address the Reviewer's concern, we propose the following revision to Section 1, L83: “Engineering the adaptive network automata learning machine CHA. We design an adaptive intelligent machine CHA, that automatically learns from the network topology the most suitable CH rule and path length to model each network, using internal validation to guide selection. This adaptive modeling is the central innovation of the study. Crucially, our framework infers the physical principle that governs link formation: L2-based rules reflect homophilic interactions (similarity-driven), while L3-based rules capture synergistic interactions (diversity-driven cooperation). Thus, CHA is not a black‑box scorer but an interpretable, mechanistic machine that recovers the effective rule explaining the prediction and governing the topological evolution directly from data. This bridges AI and network science, enabling both predictive power and scientific insights across physics domains. Empirically, on a benchmark of over 1000 networks, CHA achieves more than twice the win rate of the best-performing baseline.” Due to space limitations, we omit the corresponding edits in the Results and Discussion, but we confirm they directly address the reviewer's concern.

1.2 Network Automata are part of AI

The Reviewer was right to express this concern, because we did not clarify this point enough in the section 1 Introduction and in the section 2.2.1 Network Automata. To address the comment, we revised the text as follow:

L62: “The concept of network automata, like other forms of automata, is rooted in AI research [1,2,3], where they model adaptive, decentralized, and emergent intelligence mechanisms in complex networks. The Cannistraci-Hebb theory is a recent achievement in network science that includes a theoretical framework to understand local-based link prediction on paths of length n under the lens of predictive network automata theory. Cannistraci-Hebb theory goes beyond any type of classical local link predictor heuristic on paths of length two such common neighbors (CN), …”

L102: “Network automata as for any type of automata are part of AI research [1,2,3]. Network automata were originally introduced by Wolfram and later formally defined by Smith et al. as a general framework for modeling the evolution of network topology.”

1.3 Cannistraci-Hebb is not a heuristic but a theory

We also thank the Reviewer for giving us the opportunity to clarify a deeper conceptual point: Cannistraci-Hebb is not a heuristic but a theory. For example, Zhou et al. [4] write: “Cannistraci–Hebb theory plays an important role in link prediction in most considered networks.” and “We implement extensive experimental comparisons between 2-hop-based and 3-hop-based similarity indices on 137 real networks. The class of Cannistraci–Hebb indices performs the best among all considered candidates.”

To make this distinction explicit, we have added a new appendix section titled: “Cannistraci-Hebb Theory for Prediction of Complex Network Connectivity”. In this section, we delineate why CH constitutes a theory, not a heuristic, and under which network science conditions this framework applies. First, we clarify the definitional boundary: Heuristics are practical shortcuts offering approximate solutions based on experience, often without rigor. Theories, in contrast, are frameworks grounded in systematic empirical evidence, capable of explaining and predicting phenomena—even when not fully captured by formal mathematics. A theory often starts with observations and empirical evidence. It provides a framework to explain and predict phenomena. Mathematical proof is not always required, especially in fields where direct mathematical derivation isn't possible. For instance, theories like Newton's law of gravity are based on empirical evidence and have been tested extensively, even without a mathematical proof. The Cannistraci-Hebb framework has evolved over a decade of empirical validation and is supported by a substantial body of work across disciplines. It forms a predictive, mechanistic explanation for link formation in complex systems. Given length constraints, we summarize rather than reproduce the full appendix here. However, the final version of the paper includes a curated review of independent studies—collectively approaching 1000 citations—that justify the scientific maturity of CH as a theory rather than a heuristic.

Weakness 2: The baselines are relatively old.

Reply: Thank you for the comment. Our baseline suite includes MPLP, a recent NeurIPS 2024 method that, per its original study, outperforms classic heuristics (CN, AA, RA), node‑level GNNs (GCN, GraphSAGE), and link‑level GNNs (SEAL (NeurIPS 2018)), Neo-GNN (NeurIPS 2021), ELPH (ICLR 2023), NCNC (ICLR 2024). If there are specific recent methods we have missed, please let us know and we are happy to include them in the revision.

Weakness 3: The novelty is unclear. It seems to be a simple modification of the old index.

Reply: This point was already addressed in the reply above: 1.1 Why the work goes far beyond the introduction of two heuristics and is relevant to the conference.

Moreover, two independent reviewers explicitly recognized the novelty and significance of our contributions. Reviewer rqqa stated that “CH3 and CH3.1 are novel formalizations of the eLCL minimization principle” and that “the adaptive mechanism represents a meaningful advance over fixed‑rule approaches,” further noting that ATLAS “will likely be valuable for future research.” Reviewer AZk2 also emphasized that “the strength of the method lies in it being adaptive as it can automatically learn the rule that better explains the pattern of connectivity.” These independent assessments validate that our contributions go well beyond a simple modification of a prior index.

Question 1: No machine learning tools in the method proposed. The references are mostly physics papers. Why the work fits the scope of the conference?

Reply: Link prediction is a core machine‑learning on graphs problem. Concretely, we cast it as a predictive network automaton: a local rule assigns scores to non‑observed links and we adapt the rule to each network via data‑driven model selection over CH variants and path length. This is a non‑parametric, training‑free, self‑supervised ML approach—the model learns solely from the observed topology, with no external labels or features required. The model is interpretable (the learned rule reveals whether L2 vs. L3 connectivity dominates), and we evaluate it as an ML system—against strong modern GNN baselines, across multiple metrics and large‑scale benchmarks (ATLAS).

To address this, we have already reported our modification to the study in the reply above: 1.1 Why the work goes far beyond the introduction of two heuristics and is relevant to the conference

Question 2: Why not consider more recent methods with deep learning as the baseline?

Reply: We actually do it. See reply to point above: The baselines are relatively old.

Question 3: Why do the authors not use the commonly adopted Hits and MRR metrics as evaluation metrics for the experimental results?

Reply:

1. Precision vs. Hits. We report Precision@|P|, where |P| is the number of test positives. This is the same metric as Hits@|P|. Because network sizes vary widely in ATLAS, using a fixed cutoff (e.g., Hits@100 or Hits@50) would be inconsistent; using @|P| adapts naturally to each network.

2. NDCG vs. MRR. Both are rank‑based. Our protocol ranks all test positives against all non‑observed links (a single global ranking), for which NDCG is appropriate. MRR is most suitable when one positive vs. multiple negatives. Hence we report NDCG under our all‑missing‑links setup.

[1] Grattarola et al. Learning graph cellular automata. NeurIPS 2021.

[2] Najarro et al. HyperNCA: Growing developmental networks with neural cellular automata. ICLR 2022.

[3] Zhang et al. Intelligence at the Edge of Chaos. ICLR 2025.

[4] Zhou, et al. Experimental analyses on 2-hop-based and 3-hop-based link prediction algorithms. Physica A: Statistical Mechanics and its Applications 2021.

We sincerely thank the reviewer for the constructive feedback.

W1: The main contribution of this paper lies in proposing a new heuristic for link prediction.

R: We respond in two parts:

• This comment indicates that we must clarify our main innovation in the Introduction and strengthen its explanation throughout the Results and Discussion. We state four key innovations (L79): Minimization of external connectivity (the Cannistraci-Hebb (CH) paradigm) — the specific point the Reviewer seems to highlight; Adaptive model selection; Comprehensive static and temporal benchmarking; Multi-metric evaluation. This shows our work goes beyond a single heuristic, spanning two theoretical and two experimental advances. Two other reviewers (rqqa, AZk2) independently support this perspective. For instance, AZk2 writes: “The strength of the method lies in it being adaptive as it can automatically learn the rule that better explains the pattern of connectivity in the network under study...”

  To address the Reviewer's concern, we propose the following revision to Sec 1, L83: “Engineering the adaptive network automata learning machine CHA. We design an adaptive intelligent machine CHA, that automatically learns from the network topology the most suitable CH rule and path length to model each network, using internal validation to guide selection. This adaptive modeling is the central innovation. Crucially, our framework infers the physical principle that governs link formation: L2-based rules reflect homophilic interactions (similarity-driven), while L3-based rules capture synergistic interactions (diversity-driven cooperation). Thus, CHA is not a black‑box scorer but an interpretable, mechanistic machine that recovers the effective rule explaining the prediction and governing the topological evolution directly from data. This bridges AI and network science, enabling both predictive power and scientific insights across physics domains. Empirically, on a benchmark of over 1000 networks, CHA achieves more than twice the win rate of the best-performing baseline.” Due to space limitations, we omit the corresponding edits in the Results and Discussion, but we confirm they directly address the reviewer's concern.

• We also thank reviewer for giving us the opportunity to clarify a deeper conceptual point: CH is not a heuristic but a theory. For example, Zhou et al. [1] write: “CH theory plays an important role on link prediction in most considered networks.” and in abstract: “We implement extensive experimental comparisons between 2-hop-based and 3-hop-based similarity indices on 137 real networks. The class of CH indices performs the best among all considered candidates.”

  To make this distinction explicit, we have added:

  • Sec 1, L62: “The concept of network automata, like other forms of automata, is rooted in AI research [5-7], where they model adaptive, decentralized, and emergent intelligence mechanisms in complex networks. The CH theory is a recent achievement in network science that includes a theoretical framework to understand local-based link prediction on paths of length n under the lens of predictive network automata theory. CH theory goes beyond any type of classical local link predictor heuristic on paths of length two such common neighbors (CN), …”
  • A dedicated appendix section titled: “CH Theory for Prediction of Complex Network Connectivity”. Here, we delineate why CH constitutes a theory, not a heuristic, and under which network science conditions this framework applies. First, we clarify the definitional boundary: Heuristics are practical shortcuts offering approximate solutions based on experience, often without rigor. Theories, in contrast, are frameworks grounded in systematic empirical evidence, capable of explaining and predicting phenomena—even when not fully captured by formal mathematics. A theory often starts with observations and empirical evidence. It provides a framework to explain and predict phenomena. Mathematical proof is not always required, especially in fields where direct mathematical derivation isn't possible. For instance, theories like Newton's law of gravity are based on empirical evidence and have been tested extensively, even without a mathematical proof. The CH framework has evolved over a decade of empirical validation and is supported by a substantial body of work across disciplines. It forms a predictive, mechanistic explanation for link formation in complex systems. Given length constraints, we summarize rather than reproduce the full appendix here. However, the final version of the paper includes a curated review of independent studies—collectively approaching 1000 citations—that justify the scientific maturity of CH as a theory rather than a heuristic.

W2: The heuristic ... why outperform existing heuristics.

R: We respond in two parts:

• Theoretical Justification: We appreciate the Reviewer’s critical observation, which helps clarify our conceptual foundations. We agree the internal/external link distinction is fundamental and merits clearer articulation in the manuscript. While our paper does not re-derive this conceptual foundation from first principles, it builds upon and operationalizes prior theoretical work—most notably [2], which introduced the CH2 rule based on maximizing internal and minimizing external connectivity. That work provides a theoretical framework justifying why external-link minimization is a necessary condition for the emergence of topological communities that support localized dynamical processes. Our contribution is not to re-justify that foundational intuition, but rather to generalize and formalize the full family of CH rules, and to design an adaptive machine (CHA) that learns, directly from data, which specific CH rule best governs link formation in a given network. Importantly, we do not claim the existence of a single universally optimal CH rule. Different systems—biological, technological, social—may obey different principles of local organization. The strength of our framework lies precisely in its capacity to let the data decide, by adaptively learning among CH2 (maximize internal, minimize external), CH3 (prioritize external minimization), and CH3.1 (refined CH3). In designing CHA, we realized that the CH theoretical space was incomplete unless we explicitly included external-link-minimization-dominant rules such as CH3 and CH3.1. This insight emerged naturally from the empirical process of building an adaptive learner capable of traversing the full CH rule space.

  To more explicitly address this point, we will add an appendix section titled: “On the Importance of Distinguishing Internal and External Connectivity in CH Theory.” In that section, we will review the theoretical origins of the internal/external link paradigm [2] and integrate it with our new empirical discovery: that external-link minimization alone is often sufficient to ensure high link prediction performance. This is reminiscent of the breakthrough in the seminal “Attention Is All You Need” paper[3], which showed that a stripped-down mechanism could outperform more complex, conventional designs. Our results show that CH3 and CH3.1—based solely on external-link minimization—can perform comparably or even better than CH2 (see Fig. 9). This leads us to the insight that: “External-link minimization is all you need” to trigger the emergence of local community structures critical for link prediction. This finding challenges the prior assumption that maximizing internal links was indispensable and refines our understanding of the governing principles behind local connectivity.

• Empirical Validation: From an empirical standpoint, we believe that Figure 9 (“Comparison of CHA and CH variants across evaluation metrics”) provides robust support for the theoretical claims. It shows that external-link-minimization-driven rules (CH3 and CH3.1) often match or surpass CH2, reinforcing the validity of our expanded CH framework and the utility of CHA in discovering structural drivers of link formation across diverse networks.

W3: It would be valuable ... limitations.

R: Thank you for the suggestion. We constructed comparisons on two representative regimes: an LCP-type domain (coauthorship) and a non-LCP domain (molecular).

Win rates (AUPR)

Mean AUPR

Coauthorship (LCP) strongly favors CHA, while molecular (non-LCP) favors SPM. Many biological and social networks exhibit strong LCP-corr (local communities), whereas road and atomic-level systems have low LCP-corr [4]. These controlled results clarify when CHA excels and when global models are preferable. We have incorporated this analysis in the revision.

W4: Compare its computational complexity with the baselines.

R: Due to space constraints, please see our response to Reviewer rqqa — Question 2.

W5: Results of ... a fair comparison.

R: Networks with the same semantics as OGB tasks are already in ATLAS: Collab → Collaboration/Coauthorship (38 networks) and PPA → PPI (14 networks); metadata is in ATLAS_static.xlsx (Suppl).

We use the official settings of MPLP, with one change to early stopping: the original 1:1 negatives sampling is misaligned with our all-missing-links evaluation, so we monitor validation AUPR under the same protocol (Appendix A.8).

Why MPLP underperform CHA. 1:1 sampling over-represents easy negatives and hides hard negatives, inflating scores. Our test ranks positives against all non-observed links, restoring the natural imbalance and exposing hard negatives; under this stricter setting, GNN methods drop while CHA remains strong.

We are running additional hyperparameter sweeps for MPLP.

[1,5,6,7] Reply for Reviewer 6XrJ citation [4,1,2,3]

[2] In article [15]

[3] Vaswani, et al. Attention is all you need. NeurIPS 2017.

[4] In article [3]

Thanks for your further response. The promising performance of the proposed rule on the standard benchmark has largely addressed my concern about the evaluation. To completely address it, the authors should promise that: 1) make the code to produce the results on ogbl-collab available for full reproducibility if the paper is accepted, including the code of your method and baselines. 2) incorporate the results on the standard benchmarks into the revised version of the paper, and it would be better to include more datasets rather than only the ogbl-collab dataset. Based on the current situation, I decide to raise my score to borderline accept and I will further adjust my score according to your next update.

Dear reviewer,

We greatly appreciate your thoughtful follow-up and encouraging feedback.

1) make the code to produce the results on ogbl-collab available for full reproducibility if the paper is accepted, including the code of your method and baselines

Reply: Certainly. Upon acceptance, we will release the full code.

2) incorporate the results on the standard benchmarks into the revised version of the paper, and it would be better to include more datasets rather than only the ogbl-collab dataset

Reply: Yes, we will add results for additional OGB benchmarks in the revised paper.

Regarding W1, which pertains to motivation: Although the authors provided controlled experimental results, their categorization into "LCP-type" and "non-LCP" domains is overly coarse and fails to offer insights that distinguish the proposed method from existing baselines. For instance, MPLP can also capture local dependencies between nodes. Why, then, does the proposed method outperform it?

Reply: Thanks for the reviewer’s question, which gives us the opportunity to further clarify the innovation of CHA. CHA differs from existing link-prediction baselines in two key aspects that are the pillars of Cannistraci-Hebb theory, both of which contribute significantly to its performance improvements.

1. Explicit minimization of the external local-community-links.

   For clarification, we report the formula of CH3 on path 2, that we use to explain the principle behind the minimization of the external local-comunity links.

   $\text{CH3-L2}(u, v) = \sum_{z \in L_2} \frac{1}{1 + {de}_z}$

   The term ${de}_z$ (the external degree of the node z) indicates external local community links (eLCL) in CH3 formula, thereby establishing a physical interpretation that minimize the eLCL is aiming to form a topological energy barrier that constrains information processing within the local community. The higher is the number of external links of a node in the local community, the less it exclusively fires together (according to Cannistraci-Hebb learning rule) with other common neighbors in the local community, which is a necessary condition to implement dynamic function in the local module of the complex systems. This means that the eLCL of the common neighbors should be minimized to form a local community. The fact that two nodes establish an interaction between them in a networked complex system depends on the likelihood that they participate to implement a function in the same operational module (local community) of the complex system. Having many common neighbors with few external links is a topological condition that ensures the presence of a functional active local community which increases the likelihood of two nodes to connect each other under a certain function in the complex systems.

   To experimentally demonstrate the effectiveness of minimizing eLCLs, we evaluate two non-eLCL baselines, Common Neighbors (CN) and the internal local-community-links approach (iLCL method, $\sum_{z \in L_3} \frac{di_{z}}{1+di_{z}}$), which by design does not use eLCL minimization, and compare them with methods that explicitly minimize external links to the local community (CH2/CH3/CH3.1).

   non-external link based methods vs external link based methods on Collab

   	Hit@50
   CN_L3	61.61
   iLCL_L3	61.82
   CH2_L3	66.13
   CH3_L3 (selected by CHA)	66.66
   CH3.1_L3	66.27

   The table presents the results of the mentioned link predictors using the L3 version (consistently outperform L2) on the Collab dataset evaluated by Hit@50. As shown, eLCL-minimizing methods consistently outperform CN_L3 and iLCL_L3.

   To the best of our knowledge, we are the first to explicitly highlight the importance of minimizing eLCLs for link prediction. In contrast, most GNN-based methods—e.g., MPLP—aggregate local information via message passing but do not penalize edges that leak outside the common-neighbor set. While we would like to test an eLCL-augmented version of MPLP, there is no straightforward way to incorporate this term at present. In future work, we plan to integrate eLCL minimization into the MPLP framework; we expect that combining these two strong link-prediction strategies will further improve performance. However, the result presented in the table above should be already convincing.