Title: Adaptive Cannistraci-Hebb Network Automata Modelling of Complex Networks for Path-based Link Prediction
Venue: NeurIPS 2025
Accepted: yes
Keywords: Complex networks

network models

link prediction

automata theory

network automata

Cannistraci-Hebb theory
Abstract: Many complex networks have partially observed or evolving connectivity, making link prediction a fundamental task. 
Topological link prediction infers missing links using only network topology, with applications in social, biological, and technological systems. 
The Cannistraci-Hebb (CH) theory provides a topological formulation of Hebbian learning, grounded on two pillars: 
(1) the **minimization of external links** within local communities, and 
(2) the **path-based definition of local communities** that capture homophilic (similarity-driven) interactions via paths of length 2 and synergetic (diversity-driven) interactions via paths of length 3. 
Building on this, we introduce the Cannistraci-Hebb Adaptive (CHA) network automata, an adaptive learning machine that automatically selects the optimal CH rule and path length to model each network. 
CHA unifies theoretical interpretability and data-driven adaptivity, bridging physics-inspired network science and machine intelligence. 
Across 1,269 networks from 14 domains, CHA consistently surpasses state-of-the-art methods—including SPM, SBM, graph embedding methods, and message-passing graph neural networks—while revealing the mechanistic principles governing link formation. Our code is available at https://github.com/biomedical-cybernetics/Cannistraci_Hebb_network_automata.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a novel methodology for prediction of missing/unobserved links in static networks and future links in temporal networks. The method relies on a novel network automaton following the Cannistraci-Hebb rule. The proposed method is compared to a range of link prediction techniques - from the statistical blockmodel (SBM) based methods to message passing link predictor. The method is validated using a large collection of datasets from different sources and the average performance for small, medium and large sized networks is reported.\n\nSTRENGTHS AND WEAKNESSES:\nThe 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. 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. Another weakness in my view is the lack of references and/or comparisons with more latest contributions on link prediction in networks, such as \n1. 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.\n\n2. Zhao, Yunpeng, et al. \"Link prediction for partially observed networks.\" Journal of Computational and Graphical Statistics 26.3 (2017): 725-733.\n\nQUESTIONS:\nKEY QUESTIONS:\n1. 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?\n2. 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?\n3. 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?\n4. lines 98-99: the likelihood is mentioned here. What is the model under which the likelihood is evaluated? Please state this clearly.\n\nMINOR:\n5. Lines 114-115: are x_S and x_B function of \\tilde{X}(t) or independently chosen?\n6. Can iLCL and eLCL be mathematically defined?\n7. 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? \n\nAppendix \nA.2 (final paragraph): are each of the P sampled partitions of the same size?\n\nLIMITATIONS:\nYes. Included in Conclusions and Discussion. Could be expanded further for clarity.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces the Cannistraci-Hebb Adaptive (CHA) network automaton for link prediction in complex networks. The authors present two main theoretical contributions: (1) novel Cannistraci-Hebb (CH) models (CH3 and CH3.1) that formalize the minimization of external local-community links (eLCL) as a core principle, and (2) an adaptive mechanism that automatically selects the optimal CH model and path length (L2 vs L3) for each network. The CHA framework is evaluated on ATLAS, a comprehensive benchmark of 1269 static networks and 14 temporal networks across 14 domains, showing superior performance compared to state-of-the-art methods including SPM, SBM variants, graph embeddings, and message-passing models across multiple evaluation metrics (AUPR, Precision, NDCG).\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**:\n\n***Quality***: The experimental evaluation is exceptionally comprehensive, testing on over 1000 networks - significantly larger than previous studies (typically <20 networks). The adaptive selection mechanism is well-motivated by empirical evidence showing different network types prefer different path lengths. The mathematical formulations are clearly presented, and the connection to network automata theory provides solid theoretical grounding.\n\n***Clarity***: The paper is generally well-written with clear motivation and methodology. Figure 1 effectively illustrates the adaptive framework, and the mathematical notation is consistent throughout. The distinction between generative and predictive network automata is helpful for positioning the work.\nSignificance: The work addresses a fundamental problem in network science with broad applications. The adaptive mechanism represents a meaningful advance over fixed-rule approaches, and the large-scale benchmark (ATLAS) will likely be valuable for future research. The interpretability advantage over black-box methods is practically important.\n\n***Originality***: The CH3 and CH3.1 models are novel formalizations of the eLCL minimization principle. The adaptive selection framework and the scale of empirical evaluation represent clear contributions. However, the core Cannistraci-Hebb theory builds incrementally on prior work by the same research group.\n\n**Weaknesses**:\n\n***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.\n\n***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.\n\n***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.\n\n***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.\n\nQUESTIONS:\n***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?\n\n***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?\n\n***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?\n\n***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?\n\n***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?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-07-02"}

comments: SUMMARY:
This paper introduces Cannistraci-Hebb Adaptive (CHA) based on topological self-organization and external connectivity. CHA adaptively selects optimal path lengths to capture local community dynamics by an adaptive strategy. The core contribution of this paper lies in proposing two heuristic indexes (CH3, CH3.1). The authors apply these indexes with a sub-ranking strategy to a large number of networks.

STRENGTHS AND WEAKNESSES:
Strength:
1. The paper reviews relevant definitions in the research field, logically and coherently introducing the proposed heuristic metrics with a well-structured approach.

2. The experimental setup of the paper is well-designed, with validation conducted across multiple large-scale datasets, effectively mitigating potential biases from dataset selection and demonstrating rigorous experimental design.

3. The new index demonstrates good performance compared with the baselines selected.

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

2. The baselines are relatively old.

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

QUESTIONS:
1. I did not see any machine learning tools in the method proposed. The references are mostly physics papers. Why do the authors believe the work fits the scope of the conference?

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

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

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 3
IMPACT: 2
ORIGINALITY: 1
DATE: 2025-07-02

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a new series of heuristics for temporal link prediction. The difference of the proposed method and previous path-based heuristics is dividing the degree of nodes in the path into internal degree and external degree. The authors construct a set of link prediction datasets and compare the performance of the proposed method with baselines on them.\n\nSTRENGTHS AND WEAKNESSES:\nStrength\n\n1. The proposed method seems to achieve better performance than the SOTA neural link prediction method.\n\nWeaknesses\n\n1. From the perspective of the reviewer, the main contribution of this paper lies in proposing a new heuristic for link prediction. However, the design of the heuristic appears somewhat ad hoc and lacks solid theoretical justification. The authors briefly mention the importance of distinguishing between internal and external structures, which seems to be the underlying intuition, but this idea is not formally developed or analyzed. As a result, it remains unclear under what structural conditions the proposed method is expected to outperform existing heuristics. To address this, it would be valuable for the authors to construct illustrative examples or controlled settings that highlight the strengths and limitations of their approach. Such examples could help clarify the mechanism behind the heuristic and provide concrete insights into when and why it works better than others.\n2. The computational complexity of the proposed method seems high. The authors are encouraged to compare its computational complexity with the baselines.\n3. The paper would be more convincing if the authors could report results of the proposed method on standard link prediction datasets such as Collab, PPA, and Citation2. In addition, the reviewer is concerned about whether the hyperparameters of MPLP and MPLP+ have been properly tuned to ensure a fair comparison.\n\nQUESTIONS:\nPlease see the weaknesses part.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}