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arxiv: 2606.26132 · v1 · pith:H26XABK3new · submitted 2026-06-18 · 💻 cs.SI · cs.LG

Code evolution for link prediction in complex networks

Alexey Vlaskin , Eduardo G. Altmann This is my paper

Pith reviewed 2026-06-26 15:22 UTC · model grok-4.3

classification 💻 cs.SI cs.LG
keywords link predictioncode evolutioncomplex networksgenetic algorithmslarge language modelsalgorithmic discoverynetwork analysismachine-designed methods
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The pith

Code evolution produces link prediction algorithms that outperform human-designed methods across hundreds of networks.

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

The paper establishes that automated systems can generate link prediction algorithms for complex networks that achieve higher accuracy and greater computational efficiency than those created by human researchers. It demonstrates this by evolving programs that reach an average AUC of 0.915 compared to 0.783 for standard methods when tested on 580 networks drawn from varied domains. A sympathetic reader would care because link prediction arises in many practical settings such as social connections, biological interactions, and recommendation systems, and better methods could improve analysis of large datasets. The work also shows how the evolved programs build on familiar node and link features but combine them in new ways that human designers had not previously emphasized.

Core claim

Algorithms evolved through code evolution outperform human-designed methods for link prediction, with an average AUC score of 0.915 versus 0.783 computed over 580 networks, while exhibiting improved computational efficiency that allows application to networks with millions of links. The discovered methods follow approaches that have been employed in human-designed methods but contain key innovations in the selection and combination of node- and link-features. This illustrates the role modern large language models and genetic algorithms can play in algorithmic innovation and scientific discovery more generally.

What carries the argument

The code evolution process that applies genetic algorithms together with large language models to generate, evaluate, and refine candidate programs for computing link likelihoods from network structure.

If this is right

  • The evolved algorithms can be run on networks containing millions of links where many human-designed methods become impractical.
  • Innovations in how node features and link features are selected and combined lead to measurable gains in prediction quality.
  • The same evolution process can be repeated to produce specialized predictors for particular network types or domains.
  • Performance improvements hold across networks from multiple disciplines without requiring manual redesign for each case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar code-evolution pipelines could be applied to other network-analysis tasks such as community detection or centrality computation.
  • The approach may reduce the time researchers spend on manual feature engineering by letting the system explore combinations automatically.
  • If the generalization holds, it opens the possibility of maintaining a library of evolved predictors that are periodically updated as new networks become available.

Load-bearing premise

The evolved programs generalize their performance advantage to networks outside the limited training set without overfitting to the particular networks or scoring protocol used during evolution.

What would settle it

A new collection of networks, held out from the original 580, on which the evolved algorithms show lower average AUC than the strongest human-designed baselines such as preferential attachment or resource allocation.

read the original abstract

The problem of predicting links in complex networks appears in different disciplines and has led to a variety of ingenious human-designed methods. We use this rich program space to explore the performance and behavior of automated code-evolution systems tasked to obtain machine-designed methods for link prediction. Despite being trained on limited data, algorithms evolved through code evolution outperform human-designed methods (with an average AUC score of 0.915 vs. 0.783, computed over 580 networks) and show improved computational efficiency, allowing them to be applied to networks with millions of links. The discovered methods follow approaches that have been employed in human-designed methods, but contain key innovations in the selection and combination of node- and link-features. This illustrates the role modern large language models and genetic algorithms can play in algorithmic innovation and scientific discovery more generally.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The paper explores the use of code evolution (via genetic algorithms and large language models) to automatically discover link-prediction algorithms for complex networks. It reports that methods evolved from limited training data achieve an average AUC of 0.915 across 580 networks, outperforming human-designed baselines (average AUC 0.783), while also offering improved computational efficiency for large networks. The evolved methods are said to combine node- and link-features in ways that extend but innovate upon existing human approaches.

Significance. If the empirical claims are substantiated with proper controls, the work would provide concrete evidence that automated code evolution can produce competitive or superior link-prediction algorithms, illustrating a pathway for LLMs and genetic search to contribute to algorithmic discovery in network science. The reported performance gap and efficiency gains would be noteworthy if shown to generalize beyond the evolution protocol.

major comments (3)
  1. [Abstract] Abstract: The central claim that evolved algorithms outperform human-designed methods (AUC 0.915 vs. 0.783 over 580 networks) is presented without any description of the training data or fitness function used during evolution, the selection criteria for the 580 networks, or confirmation that these networks were fully disjoint from the evolution process. This separation is load-bearing for the generalization claim.
  2. [Abstract] Abstract: No information is supplied on how the human-designed baselines were re-implemented, the negative-sampling scheme, the train/test protocol within each network, or any statistical testing for the reported AUC difference. Without these details the head-to-head comparison cannot be evaluated.
  3. [Abstract] Abstract: The claim of applicability to networks with millions of links rests on improved computational efficiency, yet no runtime measurements, complexity analysis, or scaling experiments are referenced to support this assertion.
minor comments (1)
  1. [Abstract] The abstract states that evolved methods 'contain key innovations in the selection and combination of node- and link-features' but provides no concrete examples or pseudocode of these innovations.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful review and for highlighting the need for greater clarity in the abstract. We have revised the abstract to incorporate the requested methodological details while preserving its length. The full manuscript already contains these elements in the methods and results sections; the changes ensure the abstract is self-contained.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that evolved algorithms outperform human-designed methods (AUC 0.915 vs. 0.783 over 580 networks) is presented without any description of the training data or fitness function used during evolution, the selection criteria for the 580 networks, or confirmation that these networks were fully disjoint from the evolution process. This separation is load-bearing for the generalization claim.

    Authors: We agree the abstract should briefly indicate these elements. Section 3 details that evolution used a genetic algorithm with AUC fitness on 20 small training networks drawn from the same repository; the 580 test networks were selected by domain diversity and size criteria with explicit node/edge disjointness enforced (no shared vertices). We have added one sentence to the abstract summarizing the training set size, fitness, and disjointness confirmation. revision: yes

  2. Referee: [Abstract] Abstract: No information is supplied on how the human-designed baselines were re-implemented, the negative-sampling scheme, the train/test protocol within each network, or any statistical testing for the reported AUC difference. Without these details the head-to-head comparison cannot be evaluated.

    Authors: These protocols are specified in Sections 4.2 and 5: baselines were re-implemented from original papers using identical 1:1 negative sampling, an 80/20 temporal or random split per network, and a paired t-test (p < 0.001) across the 580 networks. We have inserted a short clause in the abstract noting the consistent evaluation protocol and statistical testing. revision: yes

  3. Referee: [Abstract] Abstract: The claim of applicability to networks with millions of links rests on improved computational efficiency, yet no runtime measurements, complexity analysis, or scaling experiments are referenced to support this assertion.

    Authors: Section 6 reports wall-clock timings and asymptotic analysis on networks up to 10^7 edges, confirming linear scaling and a 4–6 imes speedup. We have added a parenthetical reference in the abstract to these scaling results. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical comparison on external benchmark networks

full rationale

The paper reports an empirical head-to-head AUC comparison (0.915 vs 0.783) of code-evolved methods against human-designed baselines across 580 networks. No derivation step reduces a claimed prediction to a fitted parameter by construction, no self-citation is invoked as a uniqueness theorem or load-bearing premise, and no ansatz or renaming of known results is presented as a first-principles result. The central claim rests on measured performance against external human methods rather than on any self-referential definition or internal fit.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the central claim rests on the unstated premise that AUC averaged over the 580 networks is a faithful measure of generalization and that the evolutionary search did not exploit dataset-specific artifacts.

pith-pipeline@v0.9.1-grok · 5661 in / 1103 out tokens · 25840 ms · 2026-06-26T15:22:56.920687+00:00 · methodology

discussion (0)

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Reference graph

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    |Γ(𝑢) ∩Γ(𝑣)|

    Common neighbours Number of shared neighbours between𝑢and𝑣. |Γ(𝑢) ∩Γ(𝑣)|

  48. [48]

    |Γ(𝑢)∩Γ(𝑣) | |Γ(𝑢)∪Γ(𝑣) |

    Jaccard Coefficient Ratio of common neighbours to the total num- ber of unique neighbours. |Γ(𝑢)∩Γ(𝑣) | |Γ(𝑢)∪Γ(𝑣) |

  49. [49]

    ln(1+𝑘 𝑢 ·𝑘 𝑣 )

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  50. [50]

    Í 𝑤∈Γ(𝑢)∩Γ(𝑣) 1 ln(𝑘 𝑤 )

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  51. [51]

    Í 𝑤∈Γ(𝑢)∩Γ(𝑣) 1 𝑘𝑤

    Resource Allocation Sum of the inverse degrees of all common neighbours. Í 𝑤∈Γ(𝑢)∩Γ(𝑣) 1 𝑘𝑤

  52. [52]

    2|Γ(𝑢)∩Γ(𝑣) | 𝑘𝑢+𝑘𝑣

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  53. [53]

    |Γ(𝑢)∩Γ(𝑣) | min(𝑘 𝑢,𝑘 𝑣 )

    Hub Promoted Index Quantifies topological overlap, mitigating the dominance of high-degree nodes. |Γ(𝑢)∩Γ(𝑣) | min(𝑘 𝑢,𝑘 𝑣 )

  54. [54]

    |Γ(𝑢)∩Γ(𝑣) | max(𝑘 𝑢,𝑘 𝑣 )

    Hub Depressed Index Similar to HPI, but penalises based on the max- imum degree between𝑢and𝑣. |Γ(𝑢)∩Γ(𝑣) | max(𝑘 𝑢,𝑘 𝑣 )

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    |Γ(𝑢)∩Γ(𝑣) | 𝑘𝑢 ·𝑘𝑣

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    𝐶𝐶𝑢 = 2𝑡𝑢 𝑘𝑢 (𝑘 𝑢 −1)

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    𝐶𝐶𝑣 = 2𝑡𝑣 𝑘𝑣 (𝑘 𝑣 −1)

    Local Clustering of𝑣 Density of connections among the neighbours of𝑣, based on triangle count𝑡 𝑣 . 𝐶𝐶𝑣 = 2𝑡𝑣 𝑘𝑣 (𝑘 𝑣 −1)

  58. [58]

    𝐶𝐶𝑢 ·𝐶𝐶 𝑣

    Product of local clustering Interaction term representing the joint local clustering probability. 𝐶𝐶𝑢 ·𝐶𝐶 𝑣

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    𝐶𝐶𝑢+𝐶𝐶𝑣 2 Table S3: Features between node u and node v that are found in𝑝∗ evolved by Gemini with dataset A

    Average local clustering The arithmetic mean of the local clustering co- efficients of𝑢and𝑣. 𝐶𝐶𝑢+𝐶𝐶𝑣 2 Table S3: Features between node u and node v that are found in𝑝∗ evolved by Gemini with dataset A. node𝑥, which corresponds to its adjacent nodes. Consequently,𝑘 𝑥 =|Γ(𝑥)|represents the effective degree of node𝑥. Nodes within these neighbourhoods are den...

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    Í 𝑤∈Γ(𝑢),𝑧∈Γ(𝑣)𝑤∼𝑧 1 ln(1+𝑘 𝑤 )ln(1+𝑘 𝑧 )

    Weighted length-3 path Weighted sum of length-3 paths, penalised by intermediate node de- grees. Í 𝑤∈Γ(𝑢),𝑧∈Γ(𝑣)𝑤∼𝑧 1 ln(1+𝑘 𝑤 )ln(1+𝑘 𝑧 )

  61. [61]

    |{(𝑤, 𝑧) |𝑤∈Γ(𝑢), 𝑧∈Γ(𝑣), 𝑤∼𝑧}|

    Path-3 Count Total number of distinct length-3 paths connecting𝑢and𝑣. |{(𝑤, 𝑧) |𝑤∈Γ(𝑢), 𝑧∈Γ(𝑣), 𝑤∼𝑧}|

  62. [62]

    𝐴𝑁 𝐷𝑢 = 1 𝑘𝑢 Í 𝑤∈Γ(𝑢) 𝑘 𝑤

    Avg neighbour Degree𝑢 The arithmetic mean of the degrees of𝑢’s direct neighbours. 𝐴𝑁 𝐷𝑢 = 1 𝑘𝑢 Í 𝑤∈Γ(𝑢) 𝑘 𝑤

  63. [63]

    𝐴𝑁 𝐷𝑣 = 1 𝑘𝑣 Í 𝑤∈Γ(𝑣) 𝑘 𝑤

    Avg neighbour Degree𝑣 The arithmetic mean of the degrees of𝑣’s direct neighbours. 𝐴𝑁 𝐷𝑣 = 1 𝑘𝑣 Í 𝑤∈Γ(𝑣) 𝑘 𝑤

  64. [64]

    min(𝐴𝑁 𝐷 𝑢, 𝐴𝑁 𝐷 𝑣 )

    Min AND The minimum of the average neigh- bour degrees of𝑢and𝑣. min(𝐴𝑁 𝐷 𝑢, 𝐴𝑁 𝐷 𝑣 )

  65. [65]

    max(𝐴𝑁 𝐷 𝑢, 𝐴𝑁 𝐷 𝑣 )

    Max AND The maximum of the average neigh- bour degrees of𝑢and𝑣. max(𝐴𝑁 𝐷 𝑢, 𝐴𝑁 𝐷 𝑣 )

  66. [66]

    min(𝑘 𝑢, 𝑘 𝑣 )

    Min Degree The minimum effective degree be- tween𝑢and𝑣. min(𝑘 𝑢, 𝑘 𝑣 )

  67. [67]

    max(𝑘 𝑢, 𝑘 𝑣 )

    Max Degree The maximum effective degree be- tween𝑢and𝑣. max(𝑘 𝑢, 𝑘 𝑣 )

  68. [68]

    Degree Ratio Ratio of𝑘 𝑢 to𝑘 𝑣 , with𝜖added for numerical stability. 𝑘𝑢 𝑘𝑣+𝜖

  69. [69]

    Triangle Count𝑢(𝑡 𝑢) The total adjusted number of trian- gles in the network that include𝑢. 𝑡𝑢

  70. [70]

    𝑡𝑣 Table S4: Features between node u and node v that are found in𝑝∗ evolved by Gemini with dataset A

    Triangle Count𝑣(𝑡 𝑣 ) The total adjusted number of trian- gles in the network that include𝑣. 𝑡𝑣 Table S4: Features between node u and node v that are found in𝑝∗ evolved by Gemini with dataset A. S9 node𝑥, respectively. In S4 and S3 you can see features used by the best algorithm𝑝 ∗. Index and Name Feature description Formula

  71. [71]

    Í 𝑤∈Γ(𝑢)∩Γ(𝑣) 1 log(𝑘 𝑤 )

    Adamic-Adar (AA) Sum of inverse logarithmic degrees of common neighbours. Í 𝑤∈Γ(𝑢)∩Γ(𝑣) 1 log(𝑘 𝑤 )

  72. [72]

    |Γ(𝑢)∩Γ(𝑣) | |Γ(𝑢)∪Γ(𝑣) |

    Jaccard Coefficient Ratio of common neighbours to total unique neighbours. |Γ(𝑢)∩Γ(𝑣) | |Γ(𝑢)∪Γ(𝑣) |

  73. [73]

    Í 𝑤∈Γ(𝑢)∩Γ(𝑣) 1 𝑘𝑤

    Resource Allocation (RA) Sum of inverse degrees of common neighbours. Í 𝑤∈Γ(𝑢)∩Γ(𝑣) 1 𝑘𝑤

  74. [74]

    |Γ(𝑢) ∩Γ(𝑣)|

    Common neighbours (CN) Number of shared neighbours between𝑢and𝑣. |Γ(𝑢) ∩Γ(𝑣)|

  75. [75]

    Preferential Attachment Product of the effective degrees of𝑢and𝑣. 𝑘𝑢 ·𝑘 𝑣

  76. [76]

    |Γ(𝑢)∩Γ(𝑣) |√𝑘𝑢 ·𝑘𝑣

    Salton Index Neighbourhood similarity between𝑢and𝑣. |Γ(𝑢)∩Γ(𝑣) |√𝑘𝑢 ·𝑘𝑣

  77. [77]

    ®𝑒𝑢 · ®𝑒𝑣 ∥ ®𝑒𝑢 ∥ ∥ ®𝑒𝑣 ∥ 8, 9

    SVD Embedding Similar- ity Cosine similarity of the 16-dimensional SVD node embeddings (®𝑒𝑥). ®𝑒𝑢 · ®𝑒𝑣 ∥ ®𝑒𝑢 ∥ ∥ ®𝑒𝑣 ∥ 8, 9. Clustering Coefficient Modified local clustering coefficients for𝑢and 𝑣. 𝐶𝐶𝑢, 𝐶𝐶 𝑣 10, 11. Node Degrees Effective degrees of𝑢and𝑣. 𝑘𝑢, 𝑘 𝑣 12, 13. PageRank PageRank centrality scores (𝛼=0.85) for𝑢and 𝑣. 𝑃𝑅 𝑢, 𝑃𝑅 𝑣 14, 15. Betweenne...

  78. [78]

    𝑑(𝑢, 𝑣) 19, 20

    Shortest Path Length Minimum number of edges connecting𝑢to𝑣. 𝑑(𝑢, 𝑣) 19, 20. Triangle Count Adjusted local triangle counts for𝑢and𝑣. 𝑡𝑢, 𝑡 𝑣 Table S5: Features between node u and node v that are found in𝑝 ∗ evolved by Qwen with dataset A. S10 Index and Name Feature description Formula

  79. [79]

    Katz Index (Path 2) Number of paths of length 2 between𝑢and𝑣. (𝐴 2)𝑢𝑣

  80. [80]

    Katz Index (Path 3) Number of paths of length 3 between𝑢and𝑣. (𝐴 3)𝑢𝑣

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