Adaptive Cannistraci-Hebb Network Automata Modelling of Complex Networks for Path-based Link Prediction
Authors: Jialin Zhao
Alessandro Muscoloni
Umberto Michieli
Yingtao Zhang
Carlo Vittorio Cannistraci
Venue: NeurIPS 2025 poster ()
Keywords: Complex networks
network models
link prediction
automata theory
network automata
Cannistraci-Hebb theory
Primary area: machine_learning_for_sciences

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.

Adaptive Cannistraci-Hebb Network Automata Modelling of Complex Networks for Path-based Link Prediction
Jialin Zhao
Alessandro Muscoloni
Umberto Michielis.6
Yingtao Zhang 2.2
Carlo Vittorio Cannistraci 1,,,3,4
Center for Complex Network Intelligence (CCNI) 1
Dept. of Computer Science & Technology, School of Biomedical Engineering, Tsinghua University Biomedicaa Cybernetics Group, Technische Universität Dresden, Germany University of Padova, Italy, CCanva Research
Abstract
Many complex networks have partially observed or evolving connectivity, making link prediction fundamental task. Topological link prediction infers missing links using only network topology, with applications in social, biological, and technological systems. The Cannistraci-Hod (CH) theory provides 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 Similaritt-driveen interactions via paths of length 2 and synergetic (diversity -rivee)) interactions via paths of length 3. Building on this, we introduce the Cannistraci-Hebb Adaptive (CHA) network automata, adaptive learning machine that automatically selects the optimal CH rule and path length to model each network. CHA unifies theoretical interpretabilitty and data-driven adaptivity. bridging physics-inspired network science and machine intelligence. Across 1,269 networks from 14 domains, CHA consistently surpasses sate-offthe-art methods--nclldimn SPM. SBM, graph embedding methods, and message-passing graph neural networks-ve while revealing the mechanistic princi- ples governing link formation. Our code is available at automataaatttaaaaaaaaaaaaaaaaaaa biomedical- htt.aaaaaaayypeeettteccaaaaaaaaa
com/
Introduction
Many complex networks have connectivity that might be only partially detected or that tends to grow over time, hence the prediction of non-observed links is a fundamental problem in network science. The aim of topological link prediction is forecast these non-observed links by only exploiting features intrinsic to the network topology. It has a wide range of real applications, like suggesting friendships in social networks or predicting interactions in biological networks [13] A plethora of methods based on different methodological principles have been developed in recent years, and in this
Corresponding author, kalokagathos.agun @ gmail..om
Research center in Tsinghua Laboratory of Brain and Intelligence (THBI), Department of Psychological and Cognitive Sciences
Past affiliation of the research group in Biotechnology Center (BIOTEC), Center for Molecular and Cellular Bioengineering (CMCB), Center for Systems Biology Dresden (CSBD), Cluster of Excellence Physics of Life (PoL), Department of Physics, Technische Universität Dresden, Tatzberg 47/49, 01307 Dresden, Germany
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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study we will consider for reference the state-of-the-ar algorithms, The first is Structural Perturbation Method (SPM), a model-free global approach that relies on a theory derived from the first-order perturbation in quantum mechanics [ 'he second represents a class of generative models named Stochastic Block Models (SBM), whose general idea is that the nodes are partitioned into groups and the probability that two nodes are connected depends on the groups to which they belong [5]. The third class of methods is model-free and includes machine learning algorithms for graph embedding. Such methods convert the graph data into a low dimensional space in which certain graph structural information and properties are preserved. Different graph embedding variants have been developed aiming to preserve different information in the embedded space, some examples are: HOPE [6], node2vee [7]. NetSMF [8, ProNE (9.. A fourth category comprises neural message-passingg models tailored for link prediction. Neural Common Neighbor with Completion (NCNC) [TO] integrates structural features with message passing under an MPNN-then-SF architecture and corrects for graph incompleteness by completing missing common-neighhor structures. Finally, the Message Passing Link Predictor (MPLP) 11 approximates structural heuristics such as Common Neighbor via quasi-orthogonal vector propagation within pure pessage-passsing framework.
While the aforementioned approaches have been shown to be competitive link predictors, they do not offer clear interpretability of the mechanisms behind the network growth. This property. instead, can be fulfilled for example by mechanistic models based on the Cannistraci-Hoon theory [ 12 since each of them is based on an explicit deterministic mathematical formulation. Each model in the Cannistraci- Hebb theory represents a specific rule of self- organizaanz which is associated to explicit principles that drive the growth's 's dynamics of the underlying complex networked physical system.
The concept of network automata, like other forms of automata, rooted Al research [17- 19, where they model adaptive, decentralized, and emergent intelligence mechanisms in complex networks. The Cannistraci-Hebb theory is a recent achievement in network science [16 20 that includes theoretical framework to understand local-base link prediction on paths of length under the lens of predictive network automata theory. CH theory goes beyond any type of classical local link predictor heuristic paths of length two such common neighbors (CN) 21, resource allocation (RA) [22]. Jaccard (23) and preferential attachment (PA) 21];: and link predictor on paths of length three of Kovács et al. [24], which triggered a fundamental discovery
the organization of protein interaction networks (PPI). Following the discovery of a previous article of Daminelli et al. [12] that stressed the importance of paths of length three for link prediction in bipartite networks, on the same line Kováes et al. suggested that proteins interact according to an underlying bipartite scheme. Indeed, proteins interact not if they are similar each other, but if one of them is similar to the other's partners. This principle, such as the one proposed by Daminelli et al. [12], mathematically relies on network paths of length three (L3) [24]. whereas most of the deterministic local based models previously developed were based on paths of length two (L2) 3 These findings lead to a change of perspective in the field, highlighting the existence of different classes of networks whose patterns of interactions are organized either as L2 or L3. However, conceptual limitation of the studies of Daminelli al. [12) and Kovács et al. is that the L3-based link predictors developed [24] were not properly connected to already known principles of modelling, which prompted us to formulate and introduce a generalized theory.
In this study, we introduce four key innovations in the field of topological link prediction:
Minimization of external connectivity (CH paradigm). We formalize new principle within the Cannistraci-Hedt (CH) framework that emphasizes minimizing external local- community links (eLCCL), leading to the introduction of two new models, CH3 and CH3.1.
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- -drveen), while L3-based rules capture synergistic interactions (diversity -driven cooperation). Thus, CHA 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.
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CH3.1_ 1B8.1_L2/3
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CH3 CH3_12/3
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Liklihood scores of
Topology
CH2_ L2/3
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LB
AU
CH1 CH1_L2/3
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RA_L2//
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Figure I: Illustration of the Cannistraci-Hebb Adaptive (CHA) network automaton framework. CHA is designed to automatically adapt to the structural characteristics of the input network by selecting the most suitable Cannistraci-Hebh (CH) model and path length for link prediction. For given network, multiple CH models (e.g.. CH2, CH3) and path lengths (e.g., L2, L3) are applied independently assign likelihood scores both observed and non-observed links. For each model- path combination, performance metric, such as Area Under the Precision Recall Curve (AUPR), Precision, or NDCG, computed using observed links as positives. The combination achieving highest value under the selected metric is chosen, and the corresponding scores for the non-observed links are used as the final prediction. To further refine the ranking among node pairs receiving identical CH scores, sub-ranking strategy based on Spearman correlation of shortest-pon poortest--ath profiles is applied. This adaptive procedure ensures that CHA selects the rule best aligned with the underlying connectivity pattern of the network under analysis.
Empirically, on a benchmark of over 1000 networks, CHA achieves more than twice the win rate the best-performing baseline.
Comprehensive static and temporal benchmark. We construct a large-scale benchmark ATLAS,, consisting 1269 real-world networks (ATIAAS-static) and time-evolving networks (ATLASS-temporal)
Multi-metric evaluation. We adopt three complementary evaluation metries, Precision, NDCG, and AUPR, to capture diverse aspects of link prediction performance. Across all three metrics, our adaptive model consistently outperforms all baselines, demonstrating its robustness and general superiority under different evaluation criteria.
2 Preliminaries and Methods
2.1 Physical Modelling
2.1.1 Network Automata
CH and CHA are network automata rules for approximating the likelihood of a non-observed link appear in the network. These rules categorized network automata because they adopt only local information to infer the score of a link the network without need of pre-training of the rule. Note that CH and CHA are predictive network automata that differ from generative network automata which are rules created to generate artificial networks (25 Network automata as for any type of automata are part of AI research [17H19]. Network automata were originally introduced by Wolfram 28 and later formally defined by Smith et al. [29] as general framework for modeling the evolution of network topology. Given an unweighted and undirected adjacency matrix (t) time t, in network automaton the states of links evolve over time according to rule that depends only on local topological properties computable from a portion of the adjacency matrix (() C X
(t (()
(1)
The ruleset may depend on any property of the nodes or links and might be deterministic or stochastic. In contrast to cellular automata network (28 30), in which the states of nodes evolve and

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whose neighborhoods are defined by the network, in network automata the states of links evolve, and therefore the topology itself changes over time.
Smith et al. 29 provide an example in which the state update of a link (t + 1) is determined by simple topological property such as the sum of the node degrees:
(t,e (t) + de)
If the link exists, (t) 1, and the property exceeds a survival threshold (t, > Jss. then the link survives; otherwise, it is removed. If the link does not exist, (t) 0, and the property exceeds birth threshold fa.,(( (t) > zg, then the link is created; otherwise, it remains absent. In this example, the computation of the topological property and the link update based on the survival and birth thresholds constitute the operation. This basic ruleset fully describes the evolution of a network automaton.
We note that if we focus on the topological property fu, (t), we could replace the sum of node degrees with any of several mathematical models developed for link prediction, such as common neighbors (CN) [21), resource allocation (RA) [22]. Jaccard [23], and preferential attachment (PA) [21]. Al- though these models are often referred to as heuristics, the definition and example above clearly show that such local and deterministic models are in fact network automata.
As a final remark, in this link prediction study we specifically use these algorithms to predict the non- observed links that are more likely be created the next step of network evolution. Therefore, we do not consider the survival and birth thresholds in further detail, and focus solely on the topological property (t). For simplicity, we will also omit the time variable in the following discussion.
2.1.2 Network Automata on Paths of Length n2
After having recalled the framework of network automata defined by Smith et al. 29 we now introduce particular subclass named network automata on paths of length n. These automata evaluate the topological property between two nodes based on the topological information contained along the paths of length n between them. In mathematical terms, we can express the topological property as follows:
f(u,u)=
Σ M'(,,,..,22n1 Zn-1)
where and v are the two seed nodes of the candidate interaction; the summation executed over all paths of length n; 2,,..., 2n are the intermediate nodes on each path of length n; and f' 11) is some function dependent on the intermediate nodes.
simple example is represented by the resource allocation (RA) model developed by Zhou et al. 22 which is network automaton on paths of length two (L2), using as function (=( the inverse of the degree of the intermediate node (common neighbour in the L2 case). The mathematical formula is:
1
RA-L2(u, v) = Σ
2ET,2 de
where the summation is over all paths of length two; = is the intermediate node each path; and d, is the degree of node 2.
To generalize this paths of length 2, we need operator that merges the individual topological contributions of the intermediate nodes on a path of length n. Without loss of generality, if we use the geometric mean the merging operator, we derive the following generalized formula for RA paths length n:
RA-Ln(u,o) = Σ
A LeLn (dz, do, ... dzn-1)
where the summation is executed over all paths of length nt; 21.... Zn-1 are the intermediate nodes; and dz,....,d..... are their respective degrees.
We note that for paths of length three (L3), the above formula becomes equivalent to the one proposed by Kovács et al. (24), which extends the resource allocation principle to paths of length three- althuuatthouhh this connection was not properly clarified in their study, but was subsequently explained in f16] supporting the present one. From here onward, we will refer to this method as RA-L3.

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2.1.3 Sinnistraci-Hebb Network Automata on Paths of Length
In this section, we introduce a new rule of self oraanorganiztiion that can be modeled using different network automata on paths of length n. Let us first recall some theoretical background.
In 1949, Donald Olding Hebb proposed a local learning rule in neuronal networks, often summarized as: neurons that fire together wire logether (3I. However, the concept of "wiring together" was not fully specified and can be interpreted in two ways. The first interpretation is that existing connectivity between co- co-firing neurons is reinforced. The second is that new connections form between co-firing neurons not yet directly connected but already integrated within the same interacting cohort.
In 2013, Cannistraci et al. 3 termed the second interpretation epitopological learning. noting that it could be formalized as a topological link prediction problem in complex networks. The rationale is that, in networks with local-community organization, cohorts of neurons tend to be co-activated (fire together) and to learn by forming new connections (wire together), because they are topologically isolated within the same local community.
Cannistraci et al. 3 postulated that epitopological learning in neuronal networks is a special case of more general rule of local learning, valid for topological link prediction in any complex network exhibiting a local-come caadd-commnityppaii paradiigm (LCP) architecture, Based on this idea, they introduced a new class of link predictors which outperformed state-of- of-the-art predictors in both monopartite B 32 and bipartite topologies [22 13,, across various domains such as brain networks, social networks, biological systems, and economic structures.
study by Narula et al. understanding of local brain connectivity in processing, learning, and memorizing chronic pain.
B9 also highlighted that LCP and epitopological learning enhance the
Previous formulations of the LCP theory emphasized the contribution of common neighbor nodes complemented by the interactions among them, termed internal local-community links (iL.C... This was a limitation, as shown by Cannistraci [14]. who demonstrated that the local isolation of common neighbors, minimizing their interactions external to the local community (external local- community links, eLCL), is equally important to carve the LCP architecture. This minimization forms topological energy barrier that confines information processing within the community.
Here, we introduce the Cannistraci-Hebb (CH) theory, revised mathematical formalization of the LCP theory. The Cannistraci- Hebb (CH) theory provides a topological formulation of Hebbian learn- ing, grounded on two pillars: (1) the minimization of external links (eL.CL) 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. For any network automata rule, the necessary condition to be a CH rule is that explicitly incorporates the minimization of eLCL. We define Cannistraci- Hebb network automata on paths of length n as any network automaton in which the function f' (2(2,..., 2n-1) follows the CH rule.
The first CH model, introduced by Cannistraci et al. (3] and originally named Cannistraci Resoucee- Allocation (CRA), is renamed here as CH1. Its formula for L2 is:
di Σ/
CHI-L2(u,
Σ
(6)
sELs d2
where di, is the internal degree (number of iLCL) of the intermediate node 2, and d2 = 2 + +LCCL eLCL is the total degree. This model encourages minimization of eLCL, but only when dig- > otherwise the node does not contribute to the sum.
To address this, Cannistraci et al. [66] proposed the second CH model:
1 1+ dig
CH2-L2(u, = Σ
1 + de,
ZEL,2
where di, and de, the internal and external degrees (iL.CL and eLCL.) of node z, respectively. The unitary terms in the numerator and denominator prevent saturation when either value is zero
Next, we introduce CH3, novel model proposed for the first time in this study, which mathematically represents the basic principle of being a CH rule. Indeed, CH3 is based solely on eLCL minimization:
CH3-L2(u,
Σ =EL2
1
(8)
+ de;
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CH3 departs from earlier formulations by fully discarding the internal degree component and focusing purely on penalizing external connectivity, providing clean and principled expression of the CH paradigm.
Finally, we propose CH3.1, which embodies an adaptive mechanism: it accounts for the reward of internal links di, when the node follows the CH principle (because its number of external links de, is low), and progressively neglects the reward dig for larger values of external links deN, which indicates a violation of the CH principle:
CH = E
1 + diz
zELa (1 + dee) + dea 1+dex
We note that the RA model is also CH network automaton, while CN is not. In this study, we focus on five CH models: RA, CHI, CH2, CH3 and CH3.1. We do not consider non-CH models, including those studied by Kováes et al. [24], as they were shown to underperform.
Similar to RA, the CH models can be generalized paths of length n (Ln). Their formulas for L2, L3, and Ln are summarized in Figure 4 In the generalized case, the local community is the set of all intermediate nodes in any path of length n between the seed nodes. iLCL are the internal links among those nodes; eLCL are the links between any intermediate node and external nodes (excluding the seed nodes).
2.1.4
CH Model Sub-Ranking Strategy
Here describe the sub-ranking strategy adopted by the CH model to internally sub-rank all node pairs that receive the same CH score. The goal is to refine link prediction by reducing the ranking uncertainty among node pairs that are tied-ranked.
Given network and set of CH scores CH.,, computed for all node pairs (i,j) according to a given CH model, the sub-ranking procedure proceeds follows:
1. Assign to each link (i,j) in the network weight wij = I/CH, to transform similarity into dissimilarity.
2. Compute the shortest paths (SP) between all node pairs in the resulting weighted network.
3. For each node pair (i. compute the prediction score SPcorr,, as the Spearman's 's rank correlation between the two vectors of all shortest paths from node i and from node j to every other node the network.
4. Generate a final ranking of node pairs such that pairs are first ranked by CH. and any ties are sub-ranked using Sp If both scores are tied, then the node pairs receive the same final rank.
5. (Optional) Map the final ranking back to likelihood score if numerical prediction score is required by downstream applications (see details in Appendix
Although the SPcorr score could be replaced by other link predictors, we chose this approach for its neurobiological grounding, which aligns with the CH model's conceptual framework. Specifically, based on one interpretation of Peters' rule, the probability of two neurons being connected is proportional to the spatial apposition of their respective axonal and dendritic arbors [40] other words, connectivity depends on the geometrical proximity of neurons.
This biological principle resonates with the SPcorr score within the CH modelling framework: high correlation implies that two nodes share similar shortest-par path distances to all other nodes, which suggests, within the network topology, that they are spatially proximate due to their network- -egometce closeness.
2.2 Engineering the Adaptive Network Automata Machine
Different types of complex networks exhibit distinct structural patterns, some are better captured by L2 connectivity, others by L33, making it unlikely that single network automaton rule can perform optimally across all domains (see Figure F.. For instance, 1.2-based rules may suit social networks, whereas protein-protein interaction (PPI) networks often favor L3- L3-based approaches.

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Here, we aim to go step further from the engineering perspective and design a computational machine that adaptive the network under investigation and is capable of automatically selecting the model that is most likely to provide the best prediction.
To achieve this, exploit partieular property of the network automata models discussed in Section 2.1 These deterministic rules for link prediction assign scores to both observed and non-observed links in a way that is directly comparable; that is, the scores of observed links are not inherently biased to be higher or lower than those of non-observed links. This is because the mathematical formulation used to compute the score between two nodes independent of the existence of the link in the current topology.
Specifically, given set of candidate models and network, compute metric (c.g., AUPR, Precision, or NDCG) based on how well each model separates observed from non-observed links. The assumption is that, if the model tends to assign higher scores to observed links than non- observed links, is likely more effective at predicting missing or future links.
This mechanism enables CHA to self-adapt to wide range of network topologies. The time complexity and runtime details are provided in Appendix B
3 Experiments
3.1 Datasets and Baselines
Datasets. We evaluate our meth- ods on comprehensive collection of real-world networks drawn from two benchmark sets:
Table 1: Number of real-world networks tested for the link prediction task in related literature.
Algorithm
Field
Year
Networks
Ref.
SBM
Statisticul Physics
2014
ATLAS-static includes 1269 undirected static networks from 14 domains such as biological, social, and economic systems (see Appendix for full details).
SBM-DC
Statisticul Physics
2014
Ab]
SBM-N, SBM-DC-N
Statisticul Physics
2014
5
SPM
Quantum Physies
2015
13
HОPE
Computer Science
2016
mode2vec
Computer Science
2016
ATLAS-temporal consists 14 real-world networks with temporal snapshots rep- resenting dynamic evolution across time (see Supplemen- tary material for details).
ProNE, PrONE-SMF
Computer Science
2019
NaSMF
Computer Science
2019
MPLP, MPLP+
Computer Science
2024
15
|||
CHA
Physics CS
2025
For static networks, we adopt a 10%
link removal evaluation protocol: 10% of existing edges are randomly removed from the original network and used positives in a held-out test set, while the remaining links form the input to the prediction algorithm. The evaluation is repeated 10 times with different random splits, and the average metric is reported. For temporal networks, we evaluate predictions across successive snapshots: links that appear at future time steps are used as positives, while links absent at prediction time are ranked and scored. The average performance across all time pairs is reported. Full evaluation protocols are described in Appendix B
Baselines. We compare the proposed CHA framework against four categories of state-of-the-art link prediction methods:
SPM (Structural Perturbation Method) 4 model-free global approach based on spectral perturbation.
SBM variants
5 including SBM, degree-corrected SBM, and nested extensions.
Graph Embedding Methods: including HOPE node2vee NetSMF g and ProNE [9].
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Message-Passing Graph Neural Networks: including NCNC [TO] MPLP M. MPLP+, and MPLP+A. NCNC combines message passing with structural features under the MPNN- then-SF architecture and performs graph completion to mitigate incompleteness. MPLP and MPLP+ approximate classical heuristics such as Common Neighbor through quasi- orthogonal message propagation. The new variant MPLP+A adaptively selects between the L2-based (homophilic) and L3-based (synergetic) versions of MPLP+ according to validation performance.
Detailed descriptions of all baseline methods, including their underlying principles, implementation details, and hyperparameter settings, as well as complete evaluation procedures, are provided in the Appendix A
Scale of Evaluation. Unlike prior work that typically evaluates on a limited number of networks (often fewer than 20), our study conducts large-scale benchmarking on total of 1283 real-world networks (1269 static and 14 temporal), This represents the most extensive evaluation to date for the link prediction task. Table summarizes the number of networks used in related literature, highlighting the comprehensiveness of our experimental setup.
3.2 Link Prediction on ATLAS-static
To evaluate robustness across network scales, we define three nested subsets of the ATLAAS-tatic
dataset: ATLAS-small (N < 100, 900 networks), ATLAS-medium (N < 2500, 11 1126 networks), and ATLAS-large (N < 10000, 1269 networks). As shown in Figures 2 CHA consistently achieves both the highest win rate and the best average rank across all three settings, outperforming SPM, all SBM variants, graph embedding methods (HOPE, node2vec, NetSMF, ProNE), and message-passing models (NCNC, MPLP and MPLP+). Notably, CHA achieves more than twice the win rate of other baselines under AUPR, underscoring the strength of its adaptive mechanism. These results highlight CHA's superiority not only in winning the top position more frequently but also maintaining consistently strong performance across networks of varying size and structure.
Additional mean-rank and win-rate comparisons based on NDCG and Precision also reported in Figures 6 7 confirming the consistency of CHA's superiority under multiple evaluation metrics.
Notably, the adaptive variant MPLP+A achieved better average rank and higher win rate than MPLP+, confirming that adaptivity between L2 (homophilic) and L3 (synergetic) path rules can also benefit message-passing models. This observation supports the generality of the Cannistraci-HHe Hebb theory beyond the CHA framework.
3.3 Temporal Link Prediction
We further evaluate the generalization of CHA in time-cvolving settings using 14 roal-world temporal networks from the ATLAS-temporal collection. Figures 2g and 2h report the performance of CHA compared to baseline methods in terms of mean rank and win rate based on AUPR, respectively. CHA ranks among the pop-performing methods and achieves the best mean rank and highest win rate. These results indicate that CHA is well-suited for modeling dynamic connectivity in temporal networks, offering stable and competitive predictions across diverse time-evolving settings.
3.4 Path Length Preference Across Network Classes
To better understand the need for adaptivity in CHA, analyze how different network classes prefer different path lengths. For each network in the ATLAS-stati dataset (N < 10000), and for each CH model (RA, CHI, CH2, CH3, CH3.I), we apply the 10% link removal protocol and compute AUPR on both L2 and L3 paths. For each network and path length, we record the maximum AUPR across all CH models.
Figure reports, for each network class, the win rate of L2 versus L3, i.e. how often one path length outperforms the other within the class. The results clearly show that single path length dominates across all classes: some network types (e.g., coauthorship or connectome) are better captured by L.2 structures, while others (e.g., PPI or transcription) favor L3.
8

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(a) ATI.AS-smal rank
(b) ATLAAS-mall win rate
R000000 DRR -ben
0.53
3.62
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Noya
1460
HDPE
500
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(c) ATLAS-medium mean rank
(d) ATLAS-medium win rate
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SPM
NCWC
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(e) ATLAS-larg large mean rank
(I) ATLAS--Saaa win rate
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ATLAS-temporal rank
(h) ATLAS- win rate
1.50
6.8 6.7
2.89
Whrse
6.8
7elCard
HBP2
Prett
Bulk
Protd
RetMar
Figure 2: Comparison of CHA with baseline methods across network categories using AUPR. Each pair of subplots compares the performance of CHA and baselines in terms (left) mean rank and (right) win rate across: (a, b) ATLASSsmall (N < 100, 900 networks), (c, d) ATLAS- --ee-m (N < 2500, 1126 networks), (c, f) ATLAS-large (N S 10000, 1269 networks), and (g, h) ATLAS- temporal (14 time-evolving networks). CHA consistently achieves the best mean rank and highest win rate across all categories.
This observation provides empirical motivation for using an adaptive mechanism that automatically selects the most suitable path length and CH model for each network, rather than relying on fixed configuration. Similar trends are observed when using Precision and NDCG evaluation metrics (see Appendix Figures 8 and 19). A related experiment on ogbl-collab (Appendix G further confirms that other message-pan passing models such as MPLP+ exhibit similar L.2-L3 preference pattern, consistent with the adaptive principles of CH theory.
3.5
Validation of the CH Adaptive Strategy
The CHA framework operates by adaptively selecting, for each network, the CH model and path length combination that achieves the best predictive performance observed links. In this study, we instantiate CHA using the CH3 and CH3.1 models, as they best capture the core CH principle of minimizing external links (eLCL).
To directly assess the value of the adaptive mechanism CHA, we compare its performance with all individual CH variants (RA, CHI, CH2, CH3, CH3.1) under the same 10% link removal evaluation

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transcription (8)
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software (9)
foodweb (?1)
trade (200)
flightmap (36)
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Figure 3: For each network of the ATLAS (considering N < 10000, 1269 networks) and for each CH model (RA, CHI. CH2, CH3. CH3.1) of path lengths L2-L3, we applied the 10% link removal evaluation, obtaining the AUPR as measure of performance. For each path length L2-L3, we assigned as performance on network the maximum AUPR over the CH models. The barplots report for each network class and for each path length the win rate over the networks of that class. For each class, the number of networks is shown in brackets.
the full ATLAS-static dataset (N < 10000, 1269 networks). For each network, identify the winning method per metric, and report the average win rate across the 14 network classes.
Figure 5 shows results under three metrics: AUPR, Precision, and NDCG. In all cases, CHA outperforms every individual CH variant, consistently achieving the highest win rate. This confirms the effectiveness of the adaptive design in CHA, which dynamically selects the best CH model and path length combination for each network rather than relying on fixed configuration.
We also compare this choice against other combinations of CH models and report the win rate of each configuration on the ATLAS-static dataset. As shown Table 2 the (CH3, CH3. 1 setting achieves the highest win rate, confirming its effectiveness as the optimal rule set for CHA.
4 Conclusion and Discussion
We proposed Cannistraci-Hebl Adaptive (CHA), an adaptive network automata machine that exploits principles of network topological self-organization for link prediction. CHA based on the CH theory, 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- interactions via paths of length 2 and synergetic (diversity-driven) interactions via paths of length CHA leverages two models, CH3 and CH3.1, and adaptively selects the optimal path length network to capture local community Moanistrai-Hebbbane driven topological dynamics and organization. Experiments on more than 1000 real networks, static and temporal, show that CHA consistently outperforms state-of-the-arr baselines across multiple metrics. Bridging the physics of complex networks and artificial intelligence via adaptive network automata, this study confirms the effectiveness of combining theoretical grounding with adaptive engineering design.
While CHA is deterministic and interpretable, it does not leverage node attributes, which may be crucial some domains. Future extensions could integrate topological and feature-based signals. CHA can support applications such recommender systems biological discovery, but its use sensitive data should carefully monitored to avoid unintended inferences.
Acknowledgements
This work was supported by: The Zhou Yahui Chair Professorship award of Tsinghua University (to CVC); The National High-Level Talent Program of the Ministry of Science and Technology of China (grant number 20241710001. to CVC). The Center for Information Services and High Performance Computing (ZIH) of the TU Dresden and AC for providing HPC resources.
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Figure 4: menistrrcc-Hebbe epitopological rationale. The figure shows an explanatory example for the topological link prediction performed using the L2 or L3 Cannistraci-Hoe -ebb epitopological rationale. The two black nodes represent the seed nodes whose non-observed interaction should be scored with a likelihood. The white nodes are the L.2 or L.3 nommon-neighbone nodes, further neighbours are not shown for simplicity. The cohort of common- -einonn-eighhburs and the iLCL form the local community. The different types of links are reported with different colours: non-LCL (green), external-LCI. (red), internal-LCL. (white). The set of L.2 and L3 paths related to the given examples local communities are shown. the bottom, the mathematical description of 12, L.3 and Ln methods considered in this study are reported. Notation: u,y are the seed nodes; is the intermediate node (CN) in the L2 path; de is the degree of z; di, is the internal degree (number of iLCL) of z; deg is the external degree (number of cLCL) of z. For any degree it is valid the following: d' = 1 For L3 and Ln paths the definitions are analogous.
(CNs) of the seed
21

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high-level description of the procedure as (a) Mean rank (AUPR)
(b) Mean rank (Precision)
(c) Mean rank (NDCG)
(d) Win rate (AUPR)
(c) Win rate (Precision)
(f) Win rate (NDCG)
Figure 5: Comparison of CHA and CH variants across evaluation metrics. For each network in ATLAS-stati (N v 10000, 1269 networks), we apply the 10% link removal evaluation and compute both mean rank (top row) and win rate (bottom row) across three metries: (a, d) AUPR, (b, e) Precision, and (c, f) NDCG. Bars report average values over the 14 network classes. In all cases, CHA achieves the best mean rank and highest win rate, consistently outperforming all individual CH variants.
A Link Prediction Methods
A.1 Structural Perturbation Method (SPM)
The Structural Perturbation Method (SPM) is based on theory analogous to the first-order per- turbation technique in quantum mechanies follows:
1. Randomly remove 10% of the links from the network adjacency matrix X, obtaining reduced network X' X R, where is the set of removed links.
2. Compute the cigenvalues and cigenvectors of X'.
3. Considering the set of links R as a perturbation to X', construct the perturbed matrix Xp via first-order approximation that allows the eigenvalues to change while keeping the eigenvectors fixed.
4. Repeat steps 1-3 for 10 independent iterations and take the average of the resulting perturbed matrices Xp.
The link prediction result given by the values the averaged perturbed matrix, which represent the scores assigned to the non-observed links. A higher score indicates a higher likelihood that the corresponding interaction exists.
The rationale behind the method is that a missing portion of the network is predictable if its absence does significantly alter the structural characteristics of the observable part, captured by the cigenvectors of the adjacency matrix. If this condition holds, the perturbed matrices should serve as good approximations of the original network (4.
A.2 Stochastic Block Model (SBM)
The general idea behind the Stochastic Block Model (SBM) is that the nodes of network partitioned into B blocks, and a B x matrix specifies the probabilities of links existing between nodes belonging to each pair of blocks. SBM provides a general framework for statistical analysis and inference networks, particularly for tasks such as community detection and link prediction [41].
The concept of degree-corrected (DC) SBM was introduced for community detection predicting spurious and missing links [42, order to keep into account the variations node degree typically observed in real networks. A nested (N) version of SBM has been introduced ( to overcome two major limitations: the inability to separate true structures from noise, and the difficulty in detecting smaller yet well-defined clusters network size increases.
4 and for
22

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ATT.AS-small rank (NDCG)
(b) Amal win (NDCG)
300000 ORDD -be0
2.61
3,75
5.84
619
N0y
HDPE
MPcre
500
mokeve
Fdd
Poat 5td
Recter
Scat
Hard
ac
sese
HPLF
aPtea
un montee poll sas
Wetse
(c) ATLAS-medium mean rank (NDCG)
(d) ATLAS -LLAA-m win rate (NDCG)
10.56
3.48
4.02
Wivete
0.31
-01%
Proft Mr
RC
HoP
MPLP
HPLP
HeLP
notehee
Preid
Pottr Paa
(c) ATT.AS-large mean rank (NDCG)
(f) ATT-AS-lare win rate (NDCG)
Wirre
2.
LS0
2,22
1PM
KOP
(g)
ATLAS-temporal mean rank (NDCG)
(h) ATLAS ATLAS-temporal win rate (NDCG)
6/1
EB
235
65
Whate
CHA
SP8
HDPS
Rode
Pott
Restar
Prett
Figure 6: Comparison of CHA with baseline methods across network categories using NDCG.
All four tested variants (SBM, SBM-DC, SBM-N, SBM-DC-N) require finding an appropri- partitioning of the network perform inference. We used the implementation provided in Graph-tool [44] which uses an optimized Markov Chain Monte Carlo (MCMC) algorithm sample the space of possible partitions
41 Graph-tool is a Python module available at:
httt:://rraph-tool. skewed.ded.. de/
As suggested in 45 predictive performance is generally higher when averaging partitions rather than relying on a single most plausible partition, since the latter approach can lead to overfitting. Therefore, for each network, we sampled P partitions, computed the likelihood scores for the non-observed links in each partition, and averaged the scores across all partitions to obtain the final link prediction result. We set P = 100 for ATLAS networks with N < 100, and P 50 for connectomes with N > 100.
multiple
A.3 HOPE
Ffigh-Order Proximity preserved Embedding (HOPE) is graph embedding algorithm designed to preserve high-order proximities in graphs and to capture asymmetric transitivity [6]. Asymmetric transitivity depicts the correlation among directed edges, making HOPE particularly suitable for embedding directed networks, although it can also be used for undirected networks.
Many high-order proximity measures can reflect asymmetric transitivity in graphs, such the Katz index [46). Many of these measures share a common algebraic structure. Instead computing the proximity matrix and then applying singular value decomposition (SVD), HOPE leverages this shared
23

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(a) ATL.AS-smal rank (Precision)
(b) ATL.AS-small win (Precision)
0.34034
3000000 ORO -bco
2.87
3.61
619
1860
HOPE
50t
mokeee
HCaC
Hong
sawa
sata
SBM oB
at
Areur
wnza MP
Pont swr
(c) ATLAS-medium mean rank (Precision)
(d) ATLAS-medium win rate (Precision)
Wirrlt
+1248
1013
CHA
SPM
NCBC
Holt
MPLPal
Hesto
HRW
Prenc sMr
(e)
ATLAS large mean rank (Precision)
(I) ATLAS-large win rate (Precision)
2012
08
whreee
#DPS
SPX
(g) ATLAS-temporal mean rank (Precision)
(h) ATLAS--eemme -eempol-t win rate (Precision)
6/1 68
Whnge
SPH
HDPS
Rodeceer
Pose
RedSMr
Figure 7: Comparison of CHA with baseline methods across network categories using Precision.
structure transform the standard SVD problem into generalized SVD problem. This formulation allows for the direct computation of embedding vectors, thereby avoiding explicit construction of the proximity matrix 6
However, for the purpose link prediction, an approximation of the proximity matrix can be reconstructed from the learned embedding. In this context, HOPE provides scalable solution for approximating the Katz index through graph embedding. The entries of the approximated Katz proximity matrix represent the link prediction scores: the higher the proximity, the more likely the existence of the interaction.
The implementation of HOPE is available at: https://githi: com ZW ZHANG/ HOPE For experiments, we set the embedding dimension to min(128, N), where the number of nodes, and used the default values for all other parameters.
A.4 node2vec
node2vec is graph embedding algorithm that maps nodes to a low-dimensional feature space by maximizing the likelihood of preserving the network neighborhoods of nodes [7]. The maximization performed on a custom graph-based objective function using stochastic gradient descent, inspired by prior work in natural language processing and related to the Skip-gram model (7).
To define node neighborhoods flexibly, node2vec employs second-order random walk strategy to sample node neighborhoods. The behavior of the random walk is governed by two parameters;

---

coauthorship (2))nntcoo (529)
contact (32)
friendship (16)
socialnetwork (68)
covert (86)
collaboration (18)
0.8
Winrat
0.6
0.2
L.
L3
L22
L.3
L2
L3
L2
L3
L.2
L3
PPI (14)
transcriptinn (8)
software (9)
foodweeb (71)
internet (162)
trade (200)
fiigh (36)
Winrate
0.6
0.4
L2
L3
L2
L3
L2
L3
12
L3
L2
L3
L3
Figure 8: Path length preference across network classes based on Precision. For each network in ATttAS-static (N I 10000) and each CH model (RA, CH1, CH2, CH3, CH3.1), 1), we compute Precision under 10% link removal. For each path length (L2, L3), retain the best-performing CH model per network. Bar plots report the win rate L2 versus L3 across networks each class (class size in brackets).
coauthorship (20) connectome (529) friendship (16)
collaboration (18)
contact (32)
covert (86)
socialnetwork (68)
0.6
L2
L3
L2
L3
L2
L3
L2
L3
L2
L3
L2
L3
L2
L3
PPI transcription
software (9)
foodweb (71)
Internet (162)
trade (200)
flightmap (36)
Winrats
0.4
L2
L3
L2
L3
L2
L3
L2
L2
L3
L2
L3
Figure 9: Path length preference across network classes based on NDCG. For each network in ATLAAS-tatic (N S 10000) and each CH model (RA, CHI, CH2, CH3, CH3.1), we compute NDCG under 10% link removal. For each path length (L2, L3), we retain the best-performing CH model per network. Bar plots report the win rate of L2 versus L3 across networks each class (class size in brackets).
(return parameter) and y (in-out parameter), which bias the walk towards different network exploration strategies (e.g., breadth-firs vs. depth-first search).
After computing the node embeddings, we generate feature vectors for node pairs by applying the Hadamard (element-wise) product to the embedding vectors of each node in the pair, suggested the original node2vec study [7]. These node-pair feature vectors are then used to train a logistic regression classifier, which outputs likelihood scores for the non-observed links in the network.
The implementation of the node2vec embedding method available at: https://gitte We set the embedding dimension to min 128, N 1), where N is the
com/
snap-stanfor stan-staffoo wawaannordd//aaaa number nodes, and discarded node features that were constant across all nodes.
We tested the parameters and using three configurations: (p = 0.5, = 2); (p= = 1,4 = 1): and (p = q 0.5). The best configuration was selected via tross-validation. All other parameters were kept at their default values.
A.5 ProNE and ProNE-SMF
ProNE been proposed as fast and scalable graph embedding algorithm that maps nodes to low-dimensional feature space using two-step procedure [9]
25

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Table Results of CHA variants on ATLAS-static. We evaluate several CHA variants on path lengths 13,L3, each using different combination of CH models. For every network in ATLAS- static, we apply the 10% link removal evaluation and compute AUPR the performance metric. For each network class, we report the win rate and mean AUPR of each algorithm across the networks that class. The table shows the average these values over all classes. Algorithms are sorted by decreasing win rate. The upper bound indicates the performance that would be achieved by selecting the best CH model and path length per network.
Method
Win rate
AUPR
upper bound
1.00
0,30
CH3-CHB,1
0.29
0.72
0.29
RA-CH2-CHN-CHM
RA-CH3-CH3.1
0.69
0,29
CH2-CH3.1
0.67
0.28
CHB.1
0.65
0,28
RA-CH2-CH3.1
0.62
0,28
RA-CHB31
CHI-CHB-CH3I
0.56
0.27
CHI-CH2-CHR-CHB.I
0.56
CHI-CH2-CH3.1
0.54
0,26
RA-CHI AA-CHI-CH3-CHII
0.53
0.27
RA-CHI-CH2-CHS-CCBII
0.53
0.27
CHI-CH3.1
0.52
0,26
CA CI2
RA-CHI-CHB.I
0.47
0,26
CH3
0.28
CH2-CHB
0.41
0,28
RA-CH3
0.38
0.28
RA-CH2-CHB
CH2
0.26
0.27
CHI-CH2-CHB
0.26
0,26
RA
0.26
CHI-CHB
0.25
0,26
RA CIII CH2 CHB-CIIB CIB
0.23
0.25
RA-CHI-CH3
CIIl CH1
0.16
0,25
RA-CHI-CH2
RA-CHI
0.12
0,24
CHI
0.10
0.23
The first step initializes the network embedding using sparse matrix factorization (SMF), which effi- ciently provides an initial node representation via randomized truncated singular value decomposition. The second step, inspired by the higher-order Cheeger's inequality, performs spectral propagation to enhance the initial embedding
In our analysis, we considered both the embeddings obtained after the first step (ProNE-SMP and those obtained after the second step (ProNE).
After computing the embeddings, we generated feature vectors for node pairs by applying the Hadamard product to the corresponding node embeddings. These node-pair features were then used train a logistic regression classifier produce likelihood scores for the non-observed links in the network.
The implementation of the ProNE and ProNE-SMF embedding methods is available at: httpp //github. Conn//HUUM/PPOOO the number of nodes, and discarded node features that were constant across all nodes. Default values were used for all other parameters.
We set the embedding dimension to min(128, N N-1), where N is
A.6 NetSMF
NerSMF is graph embedding algorithm that maps nodes to low-dimensional feature space [8. It based on the observation that several network embedding algorithms implicitly factorize a specific closed-form matrix, and that explicitly factorizing this matrix can lead to improved performance. However, the matrix in question typically dense, making it computationally expensive handle for large networks.
26

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NetSMF addresses this limitation by proposing a scalable solution that first applies spectral graph sparsification techniques to construct a sparse matrix that is spectrally close the original dense matrix. It then performs randomized singular value decomposition (SVD) on the sparse matrix to efficiently obtaín the node embeddings [8]
After generating the embeddings, we compute feature vectors for node pairs using the Hadamard product of their corresponding node embeddings. These pairwise feature vectors are then used to train a logistic regression classifier that produces likelihood scores for the non-observed links in the network.
The implementation of the NetSMF method available at: https:/ www.cooo......e / NetSMF/ We set the embedding dimension to min(128, N - 1), where N is the number of nodes, and discarded node features with constant values across all nodes. We set rounds = 10000 and used default values for all other parameters.
A.7 Logistic Regression Classifier
After obtaining feature vectors for each node pair from the network embeddings generated by node2vec, ProNE, ProNE-SMF, and NerSMF, we trained a logistic regression classifier to compute likelihood scores for the non-observed links in the network.
In particular, we performed a repeated 5-fold cross-validation 10 times. For each repetition E [1, 10), the following steps were executed:
1. Create learning set consisting of all the observed links and an equal number of non- observed links (if available; otherwise, include all non-observed links).
2. Split the learning set into 5 folds for cross-validation.
3. For each cross-validation iteration j E 5]:
(a) Train: Train logistic regression classifier using 4 folds and obtain the coefficient
estimates B.y-
(b) Validation: Using the coefficients Bıj, obtain the likelihood scores for the remaining fold and compute the prediction performance using AUPR,
After completing the repetitions, we compute the mean coefficient estimates B across all (i, j) pairs. These coefficients B are then used compute the final likelihood scores for the non-observed links, which constitute the link prediction result.
In the case of node2vec, where multiple parameter configurations are tested, we also compute the mean validation performance AUPR over the repetitions and cross-validation iterations for each configuration. The final link prediction result corresponds to the configuration with the highest AUPR.
In contrast, for ProNE, ProNE-SMF, and NerSMF, which use a single parameter configuration, step 3.(b) (validation) is not necessary.
We used the MATLAB implementation of the logistic regression classifier, specifically the mnrf it and mnrval functions, to perform model training and scoring.
A.8 MPLP and MPLP+
Message Passing Link Predictor (MPLP) is a graph neural model specifically designed for the link prediction task [T]] Unlike general-purpose graph embedding methods that focus on node- level representations, MPLP explicitly estimates link-level structural features such as the Common Neighbor (CN) score. It achieves this by propagating quasi-orthogonal vectors through message- passing layers and leveraging their inner products to approximate structural similarities between node pairs.
To improve scalability, MPLP+ introduces more efficient variant that avoids expensive multi-hop preprocessing. Instead, it computes approximated structural signals using only one-hop neighbor- hoods, making it suitable for large-scale graphs with limited memory time constraints.
The official implementation is available at: htttps://iithbb con//arcavin/ efficient -oweiieeetnnn----eeeeee labelling In this study, followed the original MPLP experimental
27

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settings, with the exception of the early stopping strategy. The original implementation used Hits@ 100 on validation with 1:1 Desitiv-to-negativ ratio, which is not consistent with our evaluation metric, AUPR, where positives are ranked among all missing links. This misalignment could result in suboptimal model selection. To address this, we modified MPLP's early stopping criterion to monitor the AUPR on the validation set and stop training when it no longer improves, ensuring better alignment with our evaluation framework.
Hyperparameter search for MPLP/ / MPLP+. To ensure a fair comparison, we performed a targeted hyperparameter search on MPLP and MPLP+ varying the batch size E {128, 256, 512] (other settings kept as in the official code). This follows the hyperparameter sensitivity emphasized in the original study.
MPLP+A. To further align with CHA's adaptive design, we introduce an additional baseline, MPLP+A, which automatically selects between the 1.2-only and L3-only versions of MPLP+ based on validation performance.
A.9 NCNC
Neural Common Neighbor with Completion (NCNC) [10) is recent neural link predictor that combines message passing with structural features under the MPNN-then architecture. It enhances the original Neural Common Neighbor (NCN) model by completing missing common-neighbon structures to mitigate graph incompleteness before applying NCN on the completed graph. In our experiments NCNC failed to run on small subset of networks (about 10%), and repeated runs consistently crashed; these failed cases were assigned the lowest rank in evaluation. For fairness, we also conducted a hyperparameter search over the batch size B E {128, 256, 512), selecting the best configuration for each dataset.
B Link Prediction Evaluation
B.1
10%
Link Removal Evaluation
The 10% link removal evaluation framework is employed when there is information available about missing links or links that may appear in the future relative to the current state of the network.
Given network X, 10% of its links are randomly removed, resulting in reduced network X' X R, where R is the set of removed links. To evaluate given algorithm, the reduced network. X' provided as input, and the algorithm outputs likelihood scores for the non-observed links in X'.
These non-observed links are ranked in descending order of their predicted scores. The removed links R are treated as positives, and the remaining non-links as negatives. Evaluation metrics such as area under the precision-recall curve (AUPR), Precision, and normalized discounted cumulative gain (NDCG) are computed to assess the ranking quality.
Because the link removal is random, the procedure repeated 10 times with different train/test splits. The final performance on network X is reported as the average metric over these repetitions.
B.2 Temporal Evaluation
The temporal evaluation framework is employed when information is available regarding links that will appear in the future relative to the current time point of the network under consideration.
For given network, a sequence of T snapshots is available, each corresponding to a different time point. For each snapshot at time E [I, 1]. the snapshot is provided as input the algorithm being evaluated, which outputs likelihood scores for the non-observed links at time i.
For each pair of time points (i,j) with E [ - 1] and [i + 1,T the non-observed links at time are ranked by their predicted scores, and the links that actually appear at time j are treated as positives. Non-obxerved links time i involving nodes that no longer exist at time j are excluded from the evaluation.
Multiple ranking-based metrics are computed for each (i.j) pair, including area under the precision- recall curve (AUPR), Precision, and normalized discounted cumulative gain (NDCG). The final performance for the network is reported as the average of each metric over all valid time pairs.
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C
Datasets
C.1 ATLAS
have collected a dataset 1269 real-world networks, either downloaded from publicly available online sources or provided directly by the authors of prior scientific studies, The networks have been categorized into distinct classes, with the number of networks in each class
Table 3: Number of networks per class.
Class
Count
Collaboration
18
Contact
32
Covert
86
Friendship
16
PPI
14
Connectome
529
Foodweh
71
Trade
200
Transcription
8
Coauthorship
20
Flightmap
36
Internet
162
Socialnetwork
68
Software
9
Total
1269
For complete list of the networks along with their basic properties (such as number of nodes and edges), references, data sources, and descriptions, please refer Supplementary Material.
C.2 Temporal Networks
We have collected dataset of 14 real networks with temporal information, downloaded from publicly available online sources. For each network, certain number of snapshots are available, corresponding to different time points.
For complete list of the networks along with their basic properties (such number of nodes and edges), references, data sources, and descriptions, please refer to Supplementary Material.
D Compute Resources
All experiments conducted high-performance computing server equipped with 256 logical CPUs and 2 TB of RAM. The machine supports 64-bit architecture with 256 MiB of L3 cache and base frequency of 1.5 GHz (boost up to 2.6 GHz). No GPUs were used, as CHA is CPU-based.
We estimate that running all 1269 networks required approximately 72 CPU hours.
E Time Complexity and Runtime Analysis
E.1 Time Complexity of CHA
The time complexity of CHA is determined by the number of length-L paths in the network and the cost of computing iLCL and eLCL statistics for the intermediate nodes along those paths.
Let and m denote the number of nodes and edges, respectively. d = 2m/n h is the average degree.
For L = 2.
29

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Path count. Each length-2 path u - - is defined by an intermediate node 2 connected both u and v. The total number of such paths is given by:
d=
1
(d. - 1)
#L2_path
Σ
2
==1
where d2 is the degree of node 2. This represents the number of unique unordered two-hop
paths in the network.
Computation per path. For each 1ength-2 path, CHA computes a score based on the iLCL and eLCCI of the intermediate node z. This requires checking the neighbors of against the local community associated with the pair (u, v), which takes (dz) time per path.
Overall time complexity. Multiplying the path count and per-path cost gives the total time complexity:
O
Σ
d² dg
007
Σ
We now analyze this quantity under three typical network regimes:
Sparse, degree-homogeneous. If the graph is Sparse (i.e. d = 2-m2mmmn---- O(1)) with relatively uniform degrees (i.e., d= = O(1) for 2), then:
Σ d = (())
So the overall time complexity of O(n).
Sparse, degree-heterogeneeou::: If the graph is sparse (i.e., d = 0(1)), but has skewed degree distribution (c.g.. power law), we can no longer assume dg = O(1) for all nodes. To handle this case, we apply relaxation via Hölder's ' inequality 44 to upper-bound the
root-mean-cube degree ! Σ d9)
1/3 in terms of the average degree:
1/3
a
S 2/3
Σ
d=
2/3 -= On213)
12
This relaxation allows us to express the cubic-degree term in the overall complexity as:
1 Σ d )) =0 (n )
8) = On33)
Σ d = O n
Thus, the overall time complexity in this case is O(z²).
Dense graphs: In the worst-case scenario of dense graphs, where d, O(n) for all nodes. we obtain:
Σ d = 0(n))
leading to an overall time complexity of O(n
For f /= 3.
Path count. Each length-3 path u + + passes through a central edge (i,j) E. The number of such paths using (i, as the central segment is (dę - 1) - -1), where d, and d, the degrees of and j, respectively. The total number of such paths is:
#L3 path =
Σ (d; - 1)(d, - = (LJ)EE
Σ d,dy (i.J)EE
30

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Computation per path. For each length-3 path - CHA computes the iLCL and eLCL of intermediate nodes and with respect to the seed pair (u, v).
Each such computation, i.e., evaluating the iLCL/eLCL of node with respect to (u, v), requires scanning the neighborhood of and takes ((d,) time. However, this computation is performed only once for each triplet (u, v, i), and the result is reused across all paths in which (u, i) appears.
Since each such triplet (u, v, 2) is associated with (d.) paths on average, the total cost is distributed across multiple paths. Thus, the amortized cost per path remains (l).
Overall time complexity. For compact notation, we define the RMS degree product over edges:
dedge
Σ d,dz
(FJ]EE
and upper bound the total complexity as:
O m desge
We now analyze this quantity under three typical network regimes:
Sparse, degree-homogeneone If the graph sparse (i.e., d= O()) with relatively uniform degrees (i.e., d, = (()) for all nodes), then d,dj = (1) for all edges and m (n).
This yields:
O m date)) (n)
So the overall time complexity of O(n).
Sparse, degree-heterogenneon If the graph is sparse (i.e., = O(1)). but has a skewed degree distribution (e.g., power law), we upper bound:
L edge
Σ (KJ)EE
d,dy I max d,d
(f.5)
In the worst case, this maximum can scale as O(n)). Since the total number of edges is n.d it follows that m = O(n). This leads to an overall complexity:
m 2
O detged O(n n²) Onn)
Dense networks: the network dense (m (n and degrees are O(n)). then dedgo (n) and:
O(m dedge) O(n*)
Summary. The overall time complexity of CHA depends on the path length (only when 4) and the structural characteristics of the network. For both & and l. = 3, we observe the following regimes:
Sparse, degree-homogenneuu:: When the average degree is 0(1) and degree distribution is uniform, the complexity is: O(n)
Sparse, degree-heterogeneous: When the average degree is O(1) but degree distribution is skewed (e.g., power-law), the complexity is higher due to hubs: O(n²)
Dense networks: When the average degree (nn), the worst-case complexity becomes On+*)
Although CHA internally evaluates multiple CH models and path lengths, all models operate on the same set intermediatt--oode statistics (iLCL and eLCL), which are computed only once. Therefore, evaluating multiple models does not increase the overall asymptotic complexity.
Subranking Complexity. The subranking step based on Spearman correlation requires computing all-pairs shortest paths, which has worst-case time complexity of (n²²).
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0.200
0.1946
0.175
0.150
Average density
0.125
0.100
0.075
0.050
0.0391
0.025
0.0058
0.000
All networks (# 1269)
N 100 (# = 3369)
N > 2500 (# 143)
Figure 10: Average edge density across different network sizes in ATLAS-static. We report the average edge density, defined as the ratio between average degree and N - 1, for three subsets of the ATLAAS-tatic dataset: all networks (N 1269, density = 0. 1946), networks with N > 100 (n 369, density = 0.0391), and networks with N > 2500 (n = 143, density = 0.0058). The results show that large networks in ATLAS-statie are extremely sparse, which makes CHA extremely fast to compute on such networks. (see Figure 11).
E.2 Running Time of CHA
We empirically evaluate the running time of CHA under different graph conditions to validate its efficiency in practice.
Figure FigureI0] reports the average edge density across networks of increasing size in the ATLAS-static dataset. While the overall dataset has moderately low density (0.194)), the average density for larger networks (N > 2500) drops sharply 0.0058. This confirms that large real-world networks in ATLAS are typically sparse.
To further explore CHA's computational behavior under varying sparsity levels, Figure II shows its running time on synthetic networks with 500 nodes and increasing edge densities. The edge density defined the average degree divided by N As predicted by theoretical analysis, computation time increases steeply with density. For sparse graphs, CHA is significantly faster.
Together, these results indicate that CHA benefits from the inherent sparsity of real-world networks and achieves high scalability across the ATLAS benchmark.
E.3 Time Complexity Comparison
We summarize here the computational complexity CHA compared with major baselines, consider- ing both all-missing- -inks and subset prediction settings.
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CHA 500
70
60
2300 (# = 1431
4 100 (# [366
All networks 1269)
50
Runtime (s)
30
20
10
0.0
0.2
0.4
0.6
0.8
1.0
Edge density p
Figure 11: Runtime CHA on artificial networks with increasing density. We generate synthetic networks with N = 500 nodes and vary the density, defined average degree divided by (N 1). The plot shows that CHA runs extremely fast on sparse networks, while its runtime increases rapidly density grows. This highlights the efficiency of CHA in low-density regimes, which common in many real-world networks.
All-missing-links evaluation. In this setting, positives are ranked against all non-observed links (O(n))). CHA requires only one pass to compute its local community statistics (iLCL/eLCL) and reuse them across all CH variants, giving the following asymptotic costs:
Table 4: Asymptotic time complexity of CHA and baselines for all-missing-links evaluation. All costs refer the sparse-network regime and include both training and scoring over all non-observed links.
Method
Time Complexity
CHA
(
CHA (dggree-hetroogeeeous))
O
SBM variants
O(M + B²)) ((
SPM
Graph embedding methods
O(K n²)
Message passing methods
O(Fn2t))
Here, 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 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 ( + B²)).
Subset prediction. When only subset of missing links evaluated < n²)), the complexity of MPLP follows the same analysis as in the original paper, O(td" F), where d is the maximum degree,
33

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r the hop count, and F the node-signature dimension. For CHA, the per-link complexity can be derived explicitly as follows.
L2 paths.
(1) Compute the common-neighbor set (u) (o(()) in O(d).
(2) For each of the (dd) common neighbors scan M²)) once to accumulate its iLCL/eLCL counts, each scan being O(d).
Total work per link: ((d) x O(d) = O(d)).
I.3 paths.
(1) Enumerate length-3 paths i - by intersecting N(u) with the two-hop neighborhood v, costing (dd²).
(2) Each discovered path contributes one iLCL for and j; while enumerating, these counters are updated in place without an additional pass.
Total work per link: still (dd)).
Therefore, CHA (without sub-ranking) runs in overall Ottd²), which is asymptotically lower than MPLP+'s Otdd" F) with
(as used the MPLP article). Empirically, this theoretical difference is reflected in practice: on ogbl-collab (Table 5 CHA completes inference in about 6 seconds, whereas MPLP+ requires roughly minutes per run on a single A100 GPU.
Adaptive overhead. Although CHA adaptively evaluates multiple CH variants and path lengths, all candidates reuse the same precomputed iLCLLeLCL statistics, computed once per graph, so the adaptive overhead negligible practice.
F
Mapping Subranking to Likelihood Score
The CH sub-ranking mechanism refines the ordering of node pairs with tied CH scores by leveraging secondary scores (c.g., SPcorr). In some applications, is desirable map this refined ranking back to continuous likelihood scores.
To achieve this, we provide two interpolation-based strategies that preserve the original CH ranking while assigning distinct values to previously tied scores:
Score-guided interpolation. Tied scores are adjusted based on the actual SPcorr values, preserving their relative magnitudes within the group. This results in smooth, value-aware distribution of scores.
Rank-based interpolation. Tied scores are redistributed uniformly according to their sub-rank positions, regardless of the SPcorr values. This maintains only the order but not the magnitude.
Both strategies are optional and used during the main CHA evaluation, but are supported for downstream scenarios requiring continuous-valuee outputs.
G Experiment on ogbl-collab
The ogbl-collab dataset 48 is large-scale author collaboration graph with about 200K nodes, where each node represents an author and edges indicate co-authorships. The task to predict future collaborations given the past, evaluated by Hits @50, ranking positive edges among randomly- sampled negative edges, following the standard metric used in previous fnk-prediction literature and the official OGB leaderboard protocol.
MPLP+ results are averaged over 10 runs, whereas CHA is deterministic and reported from a single run. For efficiency, we use MPLP+, the faster variant of MPLP, and run all models on single A100 GPU. For fairness, CHA is evaluated without sub-ranking step, mirroring the approximation trade-off used by MPLP+
34

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We also introduce MPLP+L2 and MPLP+L3, which restrict structural feature to paths of length and 3, respectively, corresponding the homophilic and synergetic regimes described by Cannistraci- Hebb theory. To account for potential hyperparameter sensitivity, we conducted small search on batch size {8192, 16384, 32768) and report the best configuration for each variant.
Table 5: Results ogbl-collab. CHA outperforms all MPLP+ variants in accuracy while being much faster. Each MPLP+ run takes about minutes on a single A100 GPU, whereas CHA completes inference in approximately 6 seconds.
Method
Test Hits@ 50 (%)
MPLP+ (with feature)
66.47 + 0.94
MPLP+ (without feature)
64.84 + 1.21
MPLP+L2 (without feature)
63.97 1.10
MPLP+L3 (without feature)
65.01 + 0.43
CHA (CH3 L3)
66.85
As shown in 5 CHA achieves higher accuracy than all MPLP+ variants, even surpasses MPL.+ using node features, despite relying solely on topology. Each MPLP+ run takes about 6 minutes on single A100 GPU, whereas CHA completes inference in approximately 6 seconds. The superior performance of MPLP+L3 over MPLP+L2 further mirrors CHA's adaptive preference for L3-based (synergetic) rules, confirming the broader relevance of the Comnistraci-Hebe theory beyond CHA.
H Evaluation on Classical Non-Attributed Networks
We further evaluate CHA on the eight non-attributed networks used in the original MPLP paper [1] USAir, NS, PB, Yeast, Celegans, Power, Router, and Ecoli. For a fair comparison, both MPLP and MPLP+ are run using the official hyperparameters provided for each network in the authors' repository, All results are reported under the all-missing-link: evaluation protocol, where positives ranked against all non-observed links rather than 1:1 sampled subset. This protocol offers more realistic assessment, since in woal-world applications (e.g., protein-protedo interactions) the true fraction of missing links is unknown, and 1:1 sampling tends to simplify the task artificially.
Table Comparison on the 8 additional non-attributed networks. Evaluation metric is AUPR (higher is better).
Network
MPLP+
MPLP
CHA
USAir
0.4948
0.4708
0.4795
NS
0.5259
0.3363
0.6823
PB
0.2093
0.1723
0.1984
Yeast
0.5105
0.4251
0.4652
Celegans
0.1315
0.1350
0.1285
Power
0.0069
0.0055
0.0092
Router
0,0673
0.0573
0,0960
Ecoli
0.5615
0.5603
0.5876
Average 0.3135
0.2703
0.3308
As shown in Table 6 CHA achieves the highest average AUPR, demonstrating better performance than both MPLP and MPLP+ across the classical non-attributed benchmarks.
I CH Theory for Prediction of Complex Network Connectivity
The Cannistraci-Hebb (CH) framework represents theory, not a heuristic. Heuristics provide practical shorteuts without explanatory rigor, whereas a theory offers systematic, empirically grounded framework capable of explaining and predicting phenomena. Over more than decade of
35

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studies, CH has demonstrated both predictive power and mechanistic interpretability in modeling link formation across complex systems. It unifies homophilic (similarity-driver and synergetic (diversity-driven principles of connectivity, forming coherent explanation for self-organization in networks.
On the Importance of Distinguishing Internal and External Connectivity in CH Theory
CH theory distinguishes between internal and external connectivity, describing how local commu- nities emerge from the balance between intra- and inter-community links. Our experiments reveal that external-link minimization alone often achieves comparable or superior performance to full internal-extente formulations. As shown in Figure CH3 and CH3.1- solety on external-link minimization-mall or exceed CH2, suggesting that "external-line minimization is all you need" to drive the emergence of local-community structures essential for link prediction.
K Path-Length Extension Study
To examine whether longer paths provide additional predictive benefit, we extended CHA to evaluate path lengths L.2L6 on the ATLAS dataset. Table reports the average AUPR win rates. Results show that performance peaks at L3, while longer paths (L4-66) contribute negligibly, confirming that L2/L3 capture the essential topological scales for link prediction.
Table 7: Average AUPR win rate of CHA with different path lengths on ATLAS.
Path length
Avg. AUPR win rate
L2
0.48
L3
0.53
0.02
L.5
0.05
L6
0.03
Accordingly, the main analysis focuses L2 and L3, they substantially outperform longer path configurations.
L Comparison with Variants of SBM
To assess the relationship between CHA and recent statistical models for missing-link prediction under the stochastic block model (SBM), directly compare CHA with three SBM estimators [49] use the authors' official R implementation on the coauthorship network (Section 4.2.2 of the original paper). We then evaluate AUPR, NDCG, and Precision by ranking test positives among all missing links. As shown Table 8 CHA substantially outperforms all SBM estimators across metrics.
Table 8: Comparison with SBM estimators from Gaucher & Klopp (2021) on the coauthorship network. Evaluation metries: AUPR, NDCG, and Precision (higher is better).
Method
AUPR
NDCG
Precision
SBM-VAR -AR
0.1102
0.6910
0.2128
SBM-missSBM
0.1215
0.7046
0.2126
SBM-softlmpute
0.1766
0.7030
0.2797
CHA (CH3_L2)
0.4567
0.8481
0.5073
36


Reviews:
[{"review_id": "0Z96POagSk", "reviewer": "Reviewer_qoLq", "summary": "This 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.", "questions": "Please see the weaknesses part.", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strength\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.", "was_revised": true, "final_justification": "The rebuttal of the authors has addressed my concerns, so I raise my score to borderline accept.", "rebuttal": "We sincerely thank the reviewer for the constructive feedback.\n\n**W1:** The main contribution of this paper lies in proposing a new heuristic for link prediction.\n\nR: We respond in two parts:\n- 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...”\n\n  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.\n\n- 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.” \n\n  To make this distinction explicit, we have added:\n    + 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), …”\n    + 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.\n\n**W2:** The heuristic ... why outperform existing heuristics.\n\nR: We respond in two parts:\n\n- 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.\n\n  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.\n\n- 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.\n\n**W3:** It would be valuable ... limitations. \n\nR: Thank you for the suggestion. We constructed comparisons on two representative regimes: an LCP-type domain (coauthorship) and a non-LCP domain (molecular).\n\nWin rates (AUPR)\n\n||CHA|SPM|\n|-|-|-|\n|Coauthorship|1.00|0|\n|Molecular|0.33|0.67|\n\nMean AUPR\n||CHA|SPM|\n|-|-|-|\n|Coauthorship|0.76|0.58|\n|Molecular|0.19|0.27|\n\nCoauthorship (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.\n\n\n**W4:** Compare its computational complexity with the baselines.\n\nR: Due to space constraints, please see our response to Reviewer rqqa — Question 2.\n\n**W5:** Results of ... a fair comparison.\n\nR: 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).\n\nWe 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).\n\nWhy 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.\n\nWe are running additional hyperparameter sweeps for MPLP.\n\n[1,5,6,7] Reply for Reviewer 6XrJ citation [4,1,2,3]\n\n[2] In article [15]\n\n[3] Vaswani, et al. Attention is all you need. NeurIPS 2017.\n\n[4] In article [3]"}, {"review_id": "3w1y6FoVbC", "reviewer": "Reviewer_6XrJ", "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.", "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?\n\n2. Why not consider more recent methods with deep learning as the baseline?\n\n3. Why do the authors not use the commonly adopted Hits and MRR metrics as evaluation metrics for the experimental results?", "limitations": "Yes", "rating": 2, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 1, "strengths_and_weaknesses": "Strength:\n1. The paper reviews relevant definitions in the research field, logically and coherently introducing the proposed heuristic metrics with a well-structured approach.\n\n2. 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.\n\n3. The new index demonstrates good performance compared with the baselines selected.\n\nWeakness:\n1. The work seems not relevant to the topic of the conference.\n\n2. The baselines are relatively old.\n\n3. The novelty is unclear. It seems to be a simple modification of the old index.", "was_revised": true, "final_justification": "My main concern is about the relevance of the work to the conference. I am not against its performance. My judgment is based on its reference, which corresponds to the knowledge it built on, and the paradigm of the work. I left the decision to the Chair. I am fine if the Chair eventually decides to accept it.", "rebuttal": "We sincerely thank the Reviewer for the insightful and constructive comments.\n\n**Weakness 1:** The work seems not relevant to the topic of the conference.\n\n**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\n\n**1.1 Why the work goes far beyond the introduction of two heuristics and is relevant to the conference**\n\nThe 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.”\n\nTo 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.\n\n**1.2 Network Automata are part of AI**\n\nThe 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:\n\nL62: “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), …”\n\nL102: “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.”\n\n**1.3 Cannistraci-Hebb is not a heuristic but a theory**\n\nWe 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.” \n\nTo 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.\n\n\n**Weakness 2:** The baselines are relatively old.\n\n**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.\n\n**Weakness 3:**  The novelty is unclear. It seems to be a simple modification of the old index.\n\n**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.\n\nMoreover, 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.\n\n**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?\n\n**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).  \n\nTo 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\n\n**Question 2:** Why not consider more recent methods with deep learning as the baseline?\n\n**Reply:** We actually do it. See reply to point above: **The baselines are relatively old.**\n\n**Question 3:** Why do the authors not use the commonly adopted Hits and MRR metrics as evaluation metrics for the experimental results?\n\n**Reply:** \n\n1) **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.\n\n2) **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.\n\n[1] Grattarola et al. Learning graph cellular automata. NeurIPS 2021.\n\n[2] Najarro et al. HyperNCA: Growing developmental networks with neural cellular automata. ICLR 2022.\n\n[3] Zhang et al. Intelligence at the Edge of Chaos. ICLR 2025.\n\n[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."}, {"review_id": "zejoPvEagW", "reviewer": "Reviewer_rqqa", "summary": "This 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).", "questions": "***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?", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 4, "originality": 4, "strengths_and_weaknesses": "**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.", "was_revised": true, "final_justification": "I have read the response and all my concerns have been addressed. Given the overall quality of the work, I recommend maintaining the current rate.", "rebuttal": "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.\n \n**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.\n\n**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).\n\n**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.\n\n**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.\n\n**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.\n\nIn the revised manuscript, we have summarized this scalability point in the main text.\n \n**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.\n\n**Reply:** We appreciate this concern and have taken steps to ensure fair, transparent comparisons.\n\n- **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**.\n\n- **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.\n\nWe 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.\n\n \n**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.\n\n**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.\n\n**ATLAS-small mean rank AUPR**\n\n| Method   | mean_rank |  std  |\n|:--|-:|--:|\n| CHA |  2.009 | 0.067 |\n| SPM |  3.177 | 0.104 |\n| SBM_N |  4.884 | 0.103 |\n| SBM |  5.152 | 0.105 |\n| HOPE |  5.449 | 0.114 |\n| SBM_DC |  6.180 | 0.133 |\n| SBM_DC_N|  6.264 | 0.137 |\n| MPLP+ |  7.490 | 0.162 |\n| MPLP |  7.750 | 0.165 |\n| node2vec | 9.376 | 0.122 |\n| prone | 10.233 | 0.124 |\n| pronesmf| 11.039 | 0.085 |\n| NetSMF | 11.996 | 0.119 |\n\n\n\n \n**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?\n\n**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**.\n\n \n**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?\n\n**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**.\n\n**CHA complexity（sparse regimes）**\n\n| Regime                      | Time complexity |\n|-|-|\n|Sparse, degree-homogeneous|$O(n)$|\n|Sparse, degree-heterogeneous|$O(n^3)$|\n\nAs shown in **Fig. 1** of the Supplementary, most real-world networks—especially large ones—are very **sparse**.\n\n**Baselines（sparse regimes, training + scoring over all missing links**\n\n| Method| Time complexity   |\n|-|-|\n| MPLP / MPLP+ | $O(F n^{2+r})$|\n| SPM| $O(n^3)$|\n| HOPE, node2vec, ProNE, ProNE-SMF, NetSMF | $O(K n^2)$|\n| SBM, SBM-DC, SBM-N, SBM-DC-N| $O(M\\cdot(n + B^2))$|\n\nHere, $r\\ge 1$ 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 $O(M\\cdot(n^2 + B^2))$.\n\n\n\n**Adaptive overhead.** Although CHA evaluates multiple CH variants/path lengths, all candidates reuse the **same precomputed intermediate statistics (iLCL/eLCL)**, computed **once** per graph. \n\nThanks reviewer for this suggestion. We have added a short **scalability summary** and a table of **baseline complexities** in the revised manuscript.\n \n**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?\n\n**Reply:** Thanks for the suggestion. We additionally evaluated **CHA-L2…L6** on **ATLAS** and summarize the **average AUPR win rate**:\n\n| Path length | Avg. AUPR win rate |\n|-------------|---------------------|\n| L2          | 0.48              |\n| L3          | 0.53              |\n| L4          | 0.02             |\n| L5          | 0.05              |\n| L6          | 0.03              |\n\nThese 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.\n \n**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?\n \n**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.\n\n**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?\n\n**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."}, {"review_id": "2r9sxC6MVN", "reviewer": "Reviewer_AZk2", "summary": "This 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.", "questions": "KEY 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?", "limitations": "Yes. Included in Conclusions and Discussion. Could be expanded further for clarity.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "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. 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.", "was_revised": true, "final_justification": "The authors have clarified most of my clarity related queries and I have increased the score accordingly.", "rebuttal": "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.\n \n**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.\n\n**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.\n\n**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.\n\n**Reply:** Thank you for the suggestion. We have updated the manuscript and included these articles in the related work section.\n \n**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?\n\n**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.\n \n**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?\n\n**Reply:**\n\n- **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**.\n\n- **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. \n\n- **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. \n \n**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?\n\n**Reply:** Thanks for this suggestion. We are conducting experiments to compare with it. \n \n**Question 4:** lines 98-99: the likelihood is mentioned here. What is the model under which the likelihood is evaluated? Please state this clearly.\n\n**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).\n \n**Question 5:** Lines 114-115: are x_S and x_B function of \\tilde{X}(t) or independently chosen?\n\n**Reply:** In [2], x_S and x_B are independently chosen.\n \n**Question 6:** Can iLCL and eLCL be mathematically defined?\n\n**Reply:** Thank you for the suggestion. For **L2** paths, we define iLCL and eLCL for an intermediate node \\(z\\) between seeds \\((u,v)\\) as\n\n- iLCL of node z given seed node (u, v): $di_z = | \\mathcal{N}_z \\cap \\mathcal{N}_u  \\cap \\mathcal{N}_v |$\n- eLCL of node z given seed node (u, v): $de_z = | \\mathcal{N}_z \\setminus ((\\mathcal{N}_u  \\cap \\mathcal{N}_v) \\cup \\{u, v\\}) |$\n\nwhere $\\mathcal{N}_v$ 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.\n \n**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?\n\n**Reply:** We are running a sensitivity analysis that increases the edge‑removal rate. We will report the result as soon as possible. \n \n**Question 8:** Appendix A.2 (final paragraph): are each of the P sampled partitions of the same size?\n\n**Reply:** No. We do not fix the size of partitions.\n \n**Formatting 1:** NDCG -abbreviation used without being defined first\n\n**Reply:** Thank you for pointing this out. We will define **NDCG** at first use as **Normalized Discounted Cumulative Gain (NDCG)** and add the formula.\n\n**Formatting 2:** How exactly are 'mean rank' and 'win rate' as used for evaluation defined?\n\n**Reply:** \n\n**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.\n\n**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).\n\nWe 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.\n \n\n[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).\n\n[2] Smith, David MD, et al. \"Network automata: Coupling structure and function in dynamic networks.\" Advances in Complex Systems 14.03 (2011): 317-339."}]

Author rebuttal:

Meta-review: This paper primarily introduces new heuristics for link prediction in complex networks. The proposed heuristic relies on topological self-organization and minimizing external connectivity. Link prediction is a fundamental problem in network analysis, and the questions they examine here are core to this line of work. The reviewers also agree that other strengths of this paper include the evaluation across a wide range of networks. 

At the same time, the reviewers highlighted several concerns. The main ones were around (1) clarity of exposition, and in particular providing better intuition for the heurstics as well as conditions under which the heurstics may perform well, (2) limitations around theoretical contributions, though this may have been a natural consequence of the question studied, and (3) empirical rigour empirics, e.g., including standard baselines and concerns around computational overhead.

We thank the authors for their active engagement throughout the rebuttal period, which addressed some of these questions. We encourage the authors to incorporate the reviewers' detailed feedback in their next revision. (As a minor additional note, the AC encourages reducing the abstract by at least half as it is far too long as it stands and a shorter abstract may also free up space to flesh out the writing and make it easier to follow in other places.)
Decision: Accept (poster)