Pith. sign in

REVIEW 4 major objections 7 minor 1 cited by

Correlated Growth of Causal Networks

T0 review · 4 major / 7 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper claims that degree correlations in causal networks emerge from two microscopic correlations—memory in individual degree growth and state transmission from parent to child—and that a single stationary master equation predicts…

desk verdict A clean mean-field theory for correlations in growing DAGs; the math is solid, but the empirical validation is in-sample and needs an out-of-sample test. read the letter →

arxiv 2412.16647 v2 pith:T2TBB5V3 submitted 2024-12-21 physics.soc-ph cond-mat.stat-mechnlin.AOphysics.data-an

Jiazhen Liu , Kunal Tamang , Dashun Wang , Chaoming Song This is my paper · ORCID
classification physics.soc-phcond-mat.stat-mechnlin.AOphysics.data-an MSC 05C8205C20 PACS 89.75.Fb89.75.Hc
keywords causalnetworksdegreecorrelationsdirectedacyclicgraphsnetworkgrowthcitationassortativityreinforcedPoissonprocesslatentfitness
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that the degree correlations observed in causal networks are not imposed by rewiring rules but emerge from marginalizing over two underlying correlations: dynamic correlation in how a single node's degree grows over time, and causal correlation in how new events inherit state from the nodes they link to. It packages these into a stationary master equation whose solution yields the growth rate, the fitness distribution, the degree distribution, the joint degree distribution, the nearest-neighbor degree, and the assortativity coefficient. Applied to four Web of Science citation networks spanning biology, chemistry, mathematics, and physics, the predictions match empirical measurements without per-node fitting. If the paper is right, topological correlations in a directed acyclic graph are a fingerprint of growth mechanics rather than incidental structural noise.

What carries the argument

The machinery is the stationary master equation $\psi(s) = \frac{1}{\langle k\rangle} \int_0^\infty d\tau \int ds'\, K(s|s',\tau)\, \partial_\tau k(s',\tau)\, e^{-r\tau}\, \psi(s')$, fed by two inputs: the Green's function $G_s(k,\tau|0,0)$ for individual degree growth (dynamic correlation) and the causal kernel $K(s|s',\tau)$ for state transmission along edges (causal correlation). For the reinforced Poisson process used here, the Green's function is a negative binomial distribution, and the kernel is factorized as $K(\lambda|\lambda',t) = \frac{1}{\bar{\lambda}(\lambda',t)} \tilde{K}\left(\frac{\lambda}{\bar{\lambda}(\lambda',t)}\right)$, with the mean fitness $\bar{\lambda}$ an exponential function of $k(\lambda',t)/k_\infty(\lambda')$ and $\tilde{K}$ a mixed Weibull fitted from the data. Solving the master equation self-consistently determines the growth rate $r$ and stationary state distribution $\psi(s)$, from which all network observables follow by averaging Green's functions over the joint state-age distribution.

What would settle it

Fit G and K on one causal network, say physics citations, and use them with Eq. (4) to predict P(k,k') and assortativity on an out-of-sample causal network such as patent citations without refitting the kernel; if the predicted assortativity falls outside empirical error bars, the claim that degree correlations emerge from these two inputs fails. Alternatively, hold fitness values fixed but shuffle the parent-child pairing in the data while preserving the univariate marginals of K: if the predicted P(k,k') is unchanged, then the kernel is not carrying the causal correlation the paper attributes to it.

Watch

Extended reading notes

Core claim

The central claim is that in a growing directed acyclic graph, the joint state-age distribution $\Psi(s,\tau;s',\tau') = \frac{1}{\langle k\rangle} K(s|s',\tau'-\tau)\, \partial_{\tau'} k(s',\tau'-\tau)\, \Psi(s',\tau')$ fully encodes degree correlations. Because degrees are conditionally independent given state and age, the joint degree distribution factors as an average of Green's functions, $P(k,k') = \left\langle \frac{k'}{k(s',\tau')} G_{s'}(k',\tau'|0,0)\, G_s(k,\tau|0,0)\right\rangle$, so all topological correlation is mediated by marginal dependencies in the state-age distribution. The paper derives closed forms for the nearest-neighbor degree $k_{\mathrm{nn}}(k')$ and the assortativity coefficient $r_{\mathrm{corr}}$ from the same average, and validates them against citation networks from four disciplines.

Load-bearing premise

The load-bearing premise is that the causal kernel factorizes as in Eq. (8) with a single universal mixed-Weibull shape, so the kernel fitted on the four observed networks is the actual mechanism generating degree correlations rather than an absorbing fit that merely encodes those correlations.

Editorial extensions

If this is right

  • The number of free parameters needed to model a causal network drops from O(N) to O(1): latent fitness values are not assigned per vertex but generated by the causal kernel.
  • The observed saturation of k_nn(k') at large k' is a genuine prediction of the causal kernel combined with the finite ultimate degree k_infinity, not a finite-size artifact.
  • Assortativity in citation networks is reproduced quantitatively, so the framework explains the sign and magnitude of degree correlation without invoking dynamic rewiring.
  • Because stationarity requires time-translation invariance, the same framework predicts a super-exponential, winner-takes-all regime when the causal kernel has sufficiently broad tails, analogous to Bose-Einstein condensation in fitness models.
  • The framework is written for general causal systems, so the same equations apply to social media event cascades, biological evolution, and economic growth wherever a directed acyclic causal structure grows in time.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the fitted mixed-Weibull kernel is universal, the same kernel shape should reproduce degree correlations in an out-of-sample causal system such as patent citations or legislative citations; relative success or failure there would test the universality claim.
  • Because the derivation only needs degrees to be conditionally independent given state and age, the mechanism likely extends to any growing DAG with a low-dimensional latent state, not just citation networks.
  • A practical use the authors do not spell out is forecasting: measuring G and K on a young network may predict its eventual assortativity, turning the framework into a prediction tool rather than only a fitting tool.
  • A subtle circularity check would be to ask whether the mixed-Weibull fit is already absorbing the degree correlations the paper claims to derive; this can be probed by holding univariate marginals fixed while destroying the parent-child pairing in the data.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 7 minor

Summary. This manuscript proposes a mean-field growth framework for causal (directed acyclic) networks in which each vertex is endowed with a latent state (fitness) and an individual-level growth process described by a Green's function. The authors derive a stationary master equation, Eq. (4), that self-consistently determines the state distribution and growth rate from the causal kernel and the Green's function. From that solution they compute the degree distribution, joint degree distribution, k-nearest-neighbor function, and assortativity, via Eqs. (5), (6), (9), and (10). The theory is applied to four Web of Science citation networks (biology, chemistry, mathematics, physics), and the authors report quantitative agreement for network growth rates, fitness distributions, degree distributions, joint degree distributions, knn(k'), and assortativity. The central mathematical claim is that degree correlations are not directly imposed but emerge from marginal dependencies in the joint state-age distribution. The paper argues that its framework reduces the number of free parameters from O(N) in fitness models to O(1), because causal dependencies among fitness values are encoded in a universal causal kernel.

Significance. If the claims hold, the paper would provide a valuable analytic unification: a single self-consistent equation linking individual growth (dynamic correlations) and edge-formation rules (causal correlations) to macroscopic degree correlations in DAGs. The derivation of Eqs. (5)-(6) is nontrivial and internally consistent, and the conditional-independence observation is a clean conceptual contribution. The empirical validation on four large citation networks is a strength, and the paper is generally clear about the distinction between dynamic and causal correlation. However, the significance of the empirical results is currently tempered by the fact that the central input to the theory, the causal kernel K in Eq. (8), is fitted on the same four networks that are later used for validation, and the fitted parameter values are not reported in the main text. The 'O(1) parameters' claim is therefore not yet fully supported. Still, the theoretical framework is sufficiently well posed that an out-of-sample test or a fully specified kernel could substantially raise its value.

major comments (4)
  1. [Empirical Validation, Eq. (8)] The empirical validation is partially in-sample: the causal kernel factorization K(λ|λ',t) = (1/λ̄(λ',t)) K̃(λ/λ̄(λ',t)) is measured and the mixed-Weibull form of K̃ is fitted on the same four WOS networks that are then used to report agreement in Figs. 1-3 and Table I. If the kernel absorbs the degree correlations the theory claims to derive, the reported agreement is not a strong test of the emergence mechanism. The manuscript should provide an out-of-sample validation (e.g., calibrate the kernel on three disciplines and predict the fourth) or otherwise demonstrate that the kernel is universal and not merely an efficient parametric fit to each target network.
  2. [Eq. (8) and preceding text] The mean fitness λ̄(λ',t) is stated to be 'an exponential function of the ratio k(λ',t)/k∞(λ'), as shown in our previous work [42]', but the exact functional form is not given here. Because Eq. (8) is one of the two central inputs to the stationary master equation, the paper is not self-contained and the predictions cannot be reproduced without consulting a separate arXiv preprint. Please state the explicit form of λ̄(λ',t) or reproduce it in the Supplemental Material, and give the fitted values of k0, m, and the mixed-Weibull parameters used for each network or for the assumed universal kernel.
  3. [Discussion, parameter-count claim] The claim that the framework reduces parameters from O(N) to O(1) is not yet substantiated. The manuscript fixes σ=1 and μ=2.3 globally, but k0, m, and the mixed-Weibull parameters are not tabulated, and it is unclear whether they are re-fitted for each of the four disciplines. If these parameters are refitted per network, the effective number of free parameters is O(1) per network, not O(1) across all causal networks. The Discussion should either state one universal parameter vector valid for all four networks or clarify precisely which parameters are shared and which are discipline-specific.
  4. [Eq. (4) and Section Network Characteristics] Equation (6) shows that k and k' are conditionally independent given state and age, with correlations mediated by Ψ(s,τ;s',τ'). This is a correct conditional-independence statement, but the phrase 'degree correlation is entirely encoded in the state-age correlation within Ψ' may overstate the content: the joint state-age distribution Ψ itself is determined by the same causal kernel K that is fitted to data. The manuscript should clarify that the emergence claim is about the conditional factorization, not about the origin of K. This would help the reader distinguish the analytic decomposition (which is new and useful) from the empirical claim that the kernel is fundamental rather than phenomenological.
minor comments (7)
  1. [Abstract and Introduction] There is a typo in the Introduction: 'a general correlated growth framework for casual networks' should read 'causal networks'.
  2. [Theoretical Framework, paragraph after Eq. (4)] 'Substituting Eq. (2) into Eq. (1) and taking t → ∞ yields leads the stationary master equation' should be 'yields' or 'leads to'.
  3. [Network Characteristics, Eq. (6)] The notation for the joint state-age distribution Ψ(s,τ;s',τ') is introduced in the text but the formula immediately after it is not assigned an equation number; giving it a number would make the cross-references in Eqs. (5) and (6) easier to follow.
  4. [Empirical Validation, discussion of ψ(λ)] The sentence 'the empirical ψ(λ) is obtained by fitting the RPP model to individual papers' raises a question: is that fit performed per paper (which would reintroduce O(N) fitting for the empirical baseline), and how is the empirical ψ(λ) then compared with the theory? A brief clarification of the fitting procedure would remove ambiguity.
  5. [Eq. (9)] In the sentence preceding Eq. (9), 'up to a normalization factor' is vague; the exact normalization of the conditional distribution P(λ,τ;λ',τ'|k') should be specified in the Supplemental Material to allow reproduction of the knn curves.
  6. [Table I caption] Table I would benefit from a column or footnote stating the number of vertices and the citation counts for each discipline, so the reader can judge the statistical weight of the reported errors.
  7. [Figure 1] In Figure 1, panel (d) shows the rescaled kernel for physics only; the caption notes similar plots are in the Supplementary Material. Since the kernel universality is a load-bearing assumption, it would be helpful to show the rescaled kernels for all four disciplines in the main text, even in a small inset.

Circularity Check

1 steps flagged · score 6.0 of 10

The causal kernel K in Eq. (8) is fitted to the same four WOS networks used for validation, so the predicted degree correlations are partially in-sample rather than an independent out-of-sample prediction.

  1. fitted input called prediction [Empirical Validation, Eq. (8)]
    "Here, the mean fitness λ(λ′, t) is an exponential function of the ratio k(λ′, t)/k∞(λ′), as shown in our previous work [42]. Moreover, the universal distribution K is best fitted by the mixed Weibull distribution function (see SM Section 2.3)."

    The causal kernel K in Eq. (8) is not derived from first principles; its scaling form and the mixed Weibull shape are measured and fitted from the same four WOS citation networks whose degree correlations are then reported as theoretical predictions. Eq. (4) is solved using this empirical K to determine ψ(s) and r, and Eqs. (5)–(6), (9)–(10) compute P(k), P(k,k′), knn(k′), and assortativity from ψ and G. Thus the agreement in Figs. 1–3 and Table I is partially an in-sample consistency check: the fitted kernel can encode the very degree correlations the paper claims to derive. The mean-fitness scaling is also imported from the authors' own prior work [42], so the universal-kernel assumption is not independently established here.

full rationale

The analytical formalism (Eqs. (4)–(10)) is internally coherent: given a Green's function G and a causal kernel K, the stationary state distribution and all degree correlations follow by explicit marginalization. That part is not circular. The circularity concern is confined to the empirical validation. The paper estimates K from the same four Web of Science citation networks (biology, chemistry, mathematics, physics) through the collapse in Fig. 1d and a mixed Weibull fit, then solves Eq. (4) with that fitted K and reports the resulting ψ(λ), P(k), P(k,k′), knn(k′), and assortativity as predictions. Because the kernel is a direct empirical input rather than a parameter-free theoretical quantity, the validation is in-sample: the kernel can absorb causal correlations that reappear as the predicted degree correlations. The Discussion's claim of reducing parameters from O(N) to O(1) would be strong if the kernel were universal and fixed once, but the text does not show that the mixed Weibull parameters or the scaling in Eq. (8) are transferred to a held-out network. This is a fitted-input-called-prediction pattern, giving partial circularity (score 6), not full circularity, because the mathematical derivation does not itself reduce to the fitted parameters. No other load-bearing circular step was identified.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The central claim rests on a small set of domain assumptions, plus several fitted quantities. The most load-bearing are the time-translation invariance needed for the stationary solution, the adoption of the RPP growth model, and the empirical causal kernel scaling (8), whose mean is imported from the authors' prior work. The master equation's output is therefore only as independent as these inputs.

free parameters (5)
  • σ (log-normal aging width) = 1 (fixed globally)
    Set to 1 rather than fitted here; affects the aging function φ(t) in the RPP model and hence the individual growth k(λ,t).
  • μ (log-normal aging median) = 2.3 (fixed globally)
    Set to 2.3 rather than fitted here; together with σ it defines the long-tail dynamic correlation in the model.
  • k0 (initial attractiveness scale in Eq. (7)) = not given
    Needed to define k(λ,t) and k∞(λ); its value for the citation networks is not stated in the main text.
  • m (fixed outdegree) = not given
    The model assumes a fixed outdegree mi=m so that ⟨k⟩=m in Eq. (1); citation data actually has varying reference counts.
  • Mixed Weibull parameters for K̃ = fitted to data, values not in main text
    The universal kernel shape in Eq. (8) is fitted to the collapsed empirical kernel (SM Section 2.3); the parameters are not reported in the main text.
assumptions (7)
  • domain assumption Time-translation invariance (TTI): K(s,t|s',t') = K(s|s',t−t') and n(t)/n(t') ∼ f(t−t')
    Invoked in the Theoretical Framework to obtain the stationary solution and the exponential growth n(t) ∼ e^{rt}.
  • domain assumption Stationary solution to Eq. (4) exists with ∫ ds ψ(s)=1
    The paper assumes the dynamics reach a stationary state for the four empirical networks; it notes that wider kernels could break this assumption.
  • domain assumption Mean-field causal kernel K(s,t|s',t') describes the conditional probability of a new event's state given the state and age of the event it links to
    Standard mean-field approximation for network growth; used to write Eq. (1).
  • domain assumption Individual growth follows a reinforced Poisson process with rate Λ(k,τ)=λφ(τ)(k+k0)
    Adopted from ref [41]; defines the Green's function and k(λ,t) in Eq. (7).
  • domain assumption Aging function φ(t) is log-normal with σ=1 and μ=2.3
    Set globally; the choice is not derived in this paper.
  • ad hoc to paper Universal scaling of the causal kernel, Eq. (8), with λ̄(λ',t) exponential in k(λ',t)/k∞(λ')
    The scaling collapses the empirical kernel; the mean fitness relation is taken from the authors' prior work [42] and K̃ is fitted to a mixed Weibull, so the kernel shape is an empirical input.
  • ad hoc to paper Each new event connects to a fixed number m of existing events
    Used to set ⟨k⟩=m in Eq. (1); citation data actually has varying reference counts.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Correlated Growth of Causal Networks." pith.science (2026). https://pith.science/paper/T2TBB5V3

@misc{pith2026241216647,
  author       = {Pith},
  title        = {Pith review of: Correlated Growth of Causal Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T2TBB5V3}},
  note         = {Machine review of arXiv:2412.16647}
}
read the original abstract

The study of causal structure in complex systems has gained increasing attention, with many recent studies exploring causal networks that capture cause-effect relationships across diverse fields. Despite increasing empirical evidence linking causal structures to network topological correlations, the mechanisms underlying the emergence of these correlations in causal networks remain poorly understood. In this work, we propose a general growth framework for causal networks, incorporating two key types of correlations: causal and dynamic. We analytically demonstrate that degree correlations emerge as a consequence of marginal dependencies on these correlations. Our theoretical predictions align quantitatively with empirical data from four large-scale innovation networks. Our theory not only sheds light on the origins of topological correlations but also provides a general framework for understanding correlated growth across causal systems.

Figures

Figures reproduced from arXiv: 2412.16647 by the authors.

Figure 1
Figure 1. FIG. 1. Empirical results (scatter points) vs. theoretical pre [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Normalized joint degree distribution for (a–b) biology, [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. K-nearest-neighbor [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Beyond Equilibrium: Non-Equilibrium Foundations Should Underpin Generative Processes in Complex Dynamical Systems

    cs.CE 2025-05 conditional novelty 3.0 of 10

    A position paper arguing that non-equilibrium-physics-inspired generative models (like diffusion models) are, and should be, the foundation for modeling time-varying complex systems, supported by one 2D simulation.

Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages · cited by 1 Pith paper

  1. [42]

    Correlated Impact Dynamics in Science

    Jiazhen Liu, Tamang Kunal, Dashun Wang, and Chaom- ing Song. Correlated impact dynamics in science. arXiv preprint arXiv:2303.03646, 2023

    work page Pith review arXiv 2023

  2. [1]

    Emergence of scaling in complex substitutive systems

    Ching Jin, Chaoming Song, Johannes Bjelland, Geoffrey Canright, and Dashun Wang. Emergence of scaling in complex substitutive systems. Nature human behaviour , 3(8):837–846, 2019

    work page 2019

  3. [2]

    Random acyclic networks

    Brian Karrer and Mark EJ Newman. Random acyclic networks. Physical review letters , 102(12):128701, 2009

    work page 2009

  4. [3]

    Graphs: theory and algorithms

    Krishnaiyan Thulasiraman and Madisetti NS Swamy. Graphs: theory and algorithms. John Wiley & Sons, 2011

    work page 2011

  5. [4]

    From bayesian networks to causal networks

    Judea Pearl. From bayesian networks to causal networks. In Mathematical models for handling partial knowledge in artificial intelligence , pages 157–182. Springer, 1995

    work page 1995

  6. [5]

    Information thermo- dynamics on causal networks

    Sosuke Ito and Takahiro Sagawa. Information thermo- dynamics on causal networks. Physical review letters , 111(18):180603, 2013

    work page 2013

  7. [6]

    Experimental violation of local causality in a quantum network.Nature communications, 8(1):14775, 2017

    Gonzalo Carvacho, Francesco Andreoli, Luca Santodonato, Marco Bentivegna, Rafael Chaves, and Fabio Sciarrino. Experimental violation of local causality in a quantum network.Nature communications, 8(1):14775, 2017

    work page 2017

  8. [7]

    Cosmological networks

    Mari´ an Bogu˜ n´ a, Maksim Kitsak, and Dmitri Kri- oukov. Cosmological networks. New Journal of Physics , 16(9):093031, 2014

    work page 2014

Show all 42 references
  1. [8]

    Arrow of causality and quantum gravity

    John F Donoghue and Gabriel Menezes. Arrow of causality and quantum gravity. Physical Review Letters, 123(17):171601, 2019

    2019

  2. [9]

    Weak gravity conjecture from unitarity and causality

    Yuta Hamada, Toshifumi Noumi, and Gary Shiu. Weak gravity conjecture from unitarity and causality. Physical Review Letters, 123(5):051601, 2019

    2019

  3. [10]

    Causality in gravitational theories with second order equations of motion

    Harvey S Reall. Causality in gravitational theories with second order equations of motion. Physical Review D , 103(8):084027, 2021

    2021

  4. [11]

    Hot streaks in artistic, cultural, and scientific careers

    Lu Liu, Yang Wang, Roberta Sinatra, C Lee Giles, Chaoming Song, and Dashun Wang. Hot streaks in artistic, cultural, and scientific careers. Nature, 559(7714):396–399, 2018

    2018

  5. [12]

    Quantifying long-term scientific impact

    Dashun Wang, Chaoming Song, and Albert-L´ aszl´ o Barab´ asi. Quantifying long-term scientific impact. Sci- ence, 342(6154):127–132, 2013

    2013

  6. [13]

    Stochastic dy- namical model of a growing citation network based on a self-exciting point process

    Michael Golosovsky and Sorin Solomon. Stochastic dy- namical model of a growing citation network based on a self-exciting point process. Physical Review Letters , 109(9):098701, 2012

    2012

  7. [14]

    Causality-driven slow-down and speed-up of diffusion in non-markovian temporal networks

    Ingo Scholtes, Nicolas Wider, Ren´ e Pfitzner, Anto- nios Garas, Claudio J Tessone, and Frank Schweitzer. Causality-driven slow-down and speed-up of diffusion in non-markovian temporal networks. Nature communica- tions, 5(1):5024, 2014

    2014

  8. [15]

    Infor- mation propagation in multilayer systems with higher- order interactions across timescales

    Giorgio Nicoletti and Daniel Maria Busiello. Infor- mation propagation in multilayer systems with higher- order interactions across timescales. Physical Review X , 14(2):021007, 2024

    2024

  9. [16]

    Predicting the effect of micro-stimulation on macaque prefrontal activity based on spontaneous circuit dynam- ics

    Amin Nejatbakhsh, Francesco Fumarola, Saleh Esteki, Taro Toyoizumi, Roozbeh Kiani, and Luca Mazzucato. Predicting the effect of micro-stimulation on macaque prefrontal activity based on spontaneous circuit dynam- ics. Physical Review Research, 5(4):043211, 2023

    2023

  10. [17]

    Predicting memory from the network structure of naturalistic events.Nature Com- munications, 13(1):4235, 2022

    Hongmi Lee and Janice Chen. Predicting memory from the network structure of naturalistic events.Nature Com- munications, 13(1):4235, 2022

    2022

  11. [18]

    Mendelian randomization analyses reveal causal relation- ships between brain functional networks and risk of psy- chiatric disorders

    Changgai Mu, Xinglun Dang, and Xiong-Jian Luo. Mendelian randomization analyses reveal causal relation- ships between brain functional networks and risk of psy- chiatric disorders. Nature human behaviour , pages 1–12, 2024

    2024

  12. [19]

    Temporal genetic association and temporal genetic causality methods for dissecting com- plex networks

    Luan Lin, Quan Chen, Jeanne P Hirsch, Seungyeul Yoo, Kayee Yeung, Roger E Bumgarner, Zhidong Tu, Eric E Schadt, and Jun Zhu. Temporal genetic association and temporal genetic causality methods for dissecting com- plex networks. Nature Communications, 9(1):3980, 2018

    2018

  13. [20]

    How popular is your paper? an empirical 6 study of the citation distribution

    Sidney Redner. How popular is your paper? an empirical 6 study of the citation distribution. The European Physi- cal Journal B-Condensed Matter and Complex Systems , 4(2):131–134, 1998

    1998

  14. [21]

    Degree distributions of growing networks

    Pavel L Krapivsky, Geoff J Rodgers, and Sidney Red- ner. Degree distributions of growing networks. Physical Review Letters, 86(23):5401, 2001

    2001

  15. [22]

    Statistical me- chanics of complex networks

    R´ eka Albert and Albert-L´ aszl´ o Barab´ asi. Statistical me- chanics of complex networks. Reviews of modern physics, 74(1):47, 2002

    2002

  16. [23]

    The nature and nurture of network evolution

    Bin Zhou, Petter Holme, Zaiwu Gong, Choujun Zhan, Yao Huang, Xin Lu, and Xiangyi Meng. The nature and nurture of network evolution. Nature Communications, 14(1):7031, 2023

    2023

  17. [24]

    Emergence of scaling in random networks

    Albert-L´ aszl´ o Barab´ asi and R´ eka Albert. Emergence of scaling in random networks. science, 286(5439):509–512, 1999

    1999

  18. [25]

    Bose- einstein condensation in complex networks

    Ginestra Bianconi and Albert-L´ aszl´ o Barab´ asi. Bose- einstein condensation in complex networks. Physical re- view letters , 86(24):5632, 2001

    2001

  19. [26]

    Degree-preserving network growth

    Shubha R Kharel, Tam´ as R Mezei, Sukhwan Chung, P´ eter L Erd˝ os, and Zoltan Toroczkai. Degree-preserving network growth. Nature Physics, 18(1):100–106, 2022

    2022

  20. [27]

    A topological mech- anism for robust and efficient global oscillations in bi- ological networks

    Chongbin Zheng and Evelyn Tang. A topological mech- anism for robust and efficient global oscillations in bi- ological networks. Nature Communications, 15(1):6453, 2024

    2024

  21. [28]

    Specificity and stability in topology of protein networks

    Sergei Maslov and Kim Sneppen. Specificity and stability in topology of protein networks. Science, 296(5569):910– 913, 2002

    2002

  22. [29]

    Origin of degree correlations in the internet and other networks

    Juyong Park and Mark EJ Newman. Origin of degree correlations in the internet and other networks. Physical Review E, 68(2):026112, 2003

    2003

  23. [30]

    Influence of assortativity and degree-preserving rewiring on the spectra of networks

    Piet Van Mieghem, Huijuan Wang, Xin Ge, Siyu Tang, and Fernando A Kuipers. Influence of assortativity and degree-preserving rewiring on the spectra of networks. The European Physical Journal B , 76(4):643–652, 2010

    2010

  24. [31]

    Economic complexity theory and ap- plications

    C´ esar A Hidalgo. Economic complexity theory and ap- plications. Nature Reviews Physics , 3(2):92–113, 2021

    2021

  25. [32]

    Papers and patents are becoming less disruptive over time

    Michael Park, Erin Leahey, and Russell J Funk. Papers and patents are becoming less disruptive over time. Na- ture, 613(7942):138–144, 2023

    2023

  26. [33]

    In- heritance patterns in citation networks reveal scientific memes

    Tobias Kuhn, Matjaˇ z Perc, and Dirk Helbing. In- heritance patterns in citation networks reveal scientific memes. Physical Review X , 4(4):041036, 2014

    2014

  27. [34]

    Forecasting the dynamics of a complex microbial community using integrated meta-omics

    Francesco Delogu, Benoit J Kunath, Pedro M Queir´ os, Rashi Halder, Laura A Lebrun, Phillip B Pope, Patrick May, Stefanie Widder, Emilie EL Muller, and Paul Wilmes. Forecasting the dynamics of a complex microbial community using integrated meta-omics. Nature Ecology & Evolutio...

    2024

  28. [35]

    Assortative mixing in networks

    Mark EJ Newman. Assortative mixing in networks. Phys- ical review letters , 89(20):208701, 2002

    2002

  29. [36]

    Why social net- works are different from other types of networks.Physical review E, 68(3):036122, 2003

    Mark EJ Newman and Juyong Park. Why social net- works are different from other types of networks.Physical review E, 68(3):036122, 2003

    2003

  30. [37]

    Dynamical and correlation properties of the internet

    Romualdo Pastor-Satorras, Alexei V´ azquez, and Alessan- dro Vespignani. Dynamical and correlation properties of the internet. Physical review letters, 87(25):258701, 2001

    2001

  31. [38]

    Scale-free networks from vary- ing vertex intrinsic fitness

    Guido Caldarelli, Andrea Capocci, Paolo De Los Rios, and Miguel A Munoz. Scale-free networks from vary- ing vertex intrinsic fitness. Physical review letters , 89(25):258702, 2002

    2002

  32. [39]

    Vertex intrinsic fitness: How to produce arbitrary scale- free networks

    Vito DP Servedio, Guido Caldarelli, and Paolo Butt` a. Vertex intrinsic fitness: How to produce arbitrary scale- free networks. Physical Review E—Statistical, Nonlinear, and Soft Matter Physics , 70(5):056126, 2004

    2004

  33. [40]

    Univer- sal behavior of load distribution in scale-free networks

    K-I Goh, Byungnam Kahng, and Doochul Kim. Univer- sal behavior of load distribution in scale-free networks. Physical review letters , 87(27):278701, 2001

    2001

  34. [41]

    Modeling and predicting pop- ularity dynamics via reinforced poisson processes

    Huawei Shen, Dashun Wang, Chaoming Song, and Albert-L´ aszl´ o Barab´ asi. Modeling and predicting pop- ularity dynamics via reinforced poisson processes. In Proceedings of the AAAI Conference on Artificial Intel- ligence, volume 28, 2014

    2014

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.