Pith. sign in

REVIEW 4 major objections 5 minor 1 cited by

Dynamical Phase Transitions in Non-equilibrium Networks

T0 review · 4 major / 5 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 The paper's central claim is that a minimal random-network model with quadratic triadic edge interactions produces a finite-time divergence of the mean degree, $z(t) \sim (t_c - t)^{-1}$, at a dynamical phase transition.

desk verdict Clean minimal model of triadic closure with finite-time blow-up predictions, but the uncontrolled moment truncation leaves the central claim not fully rigorous. read the letter →

arxiv 2412.06704 v1 pith:NSSU6A3B submitted 2024-12-09 physics.soc-ph cond-mat.stat-mechnlin.AO

Jiazhen Liu , Nathaniel M. Aden , Debasish Sarker , Chaoming Song This is my paper · ORCID
classification physics.soc-phcond-mat.stat-mechnlin.AO PACS 89.75.Hc05.70.Ln64.60.-i
keywords dynamicalphasetransitionnon-equilibriumnetworkstriadicclosurefinite-timesingularityhyperbolicscalingscale-freeclusteringcoefficientrandomgraphmodel
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 finite-time dynamical phase transitions in classical complex systems have a simple minimal cause: nonlinear (triadic) interactions between network edges. It shows that adding a quadratic edge-formation term $\beta(A^2)_{ij}$ to a standard random graph produces, above a critical coupling $\beta > \gamma^2/(4\alpha)$, a mean degree $z(t)$ that blows up as $(t_c - t)^{-1}$, the same hyperbolic divergence empirically observed in social, financial, and deadline-driven systems. At the critical time the degree distribution becomes $P(k) \sim k^{-1}$ and the clustering coefficient $C(k) \sim 1/k$, so scale-free structure emerges from the transition itself. If correct, this provides a solvable theoretical foundation for abrupt, collective changes in networks and places classical DPTs on the same footing as their quantum counterparts.

What carries the argument

The load-bearing object is the quadratic interaction term $\beta(A^2)_{ij}$, interpreted as triadic closure, and the one-dimensional equation it produces for the mean degree: $dz/dt = \alpha - \gamma z + \beta(z^2 - \Delta)$. At the tree level $\Delta$ is dropped, leaving $dz/dt = \alpha - \gamma z + \beta z^2$; the sign of $\gamma^2 - 4\alpha\beta$ determines whether solutions approach the stable fixed point $\alpha/\gamma$ or run away to a pole at $t_c$. The generating-function equation for the degree distribution, $\partial_t G = (x-1)(\alpha G - (\gamma-\beta x z)\partial_x G)$, carries the $P(k) \sim k^{-1}$ result.

What would settle it

Run continuous-time simulations of the transition rates in Eqs. (7a)-(7b) at $\alpha=1$, $\gamma=1$, $\beta=0.5$ with $N=2000$, measuring $z(t)$ and $\Delta(t)$. The central claim fails if $z(t)$ leaves the Riccati solution before the theoretical $t_c \approx 5.9$, or if $\Delta$ becomes comparable to $z^2$ before $t_c$.

Watch

Extended reading notes

Core claim

The paper's central claim is that a random directed network with edge turnover becomes critically singular purely through the leading nonlinearity of edge formation. Writing the rate of new edges as $\alpha/N + \beta(A^2)_{ij}$ (triadic closure), the expected mean degree obeys $dz/dt = \alpha - \gamma z + \beta(z^2 - \Delta)$; neglecting the triangle density $\Delta$ at tree level gives a Riccati equation whose solution diverges as $z(t) \sim 1/(t_c - t)$ whenever $\beta > \gamma^2/(4\alpha)$, with $t_c = \omega^{-1}(\arccot(2\omega) + \pi/2)$ and $\omega = \sqrt{\alpha\beta - \gamma^2/4}$. The paper calls this a first-order dynamical phase transition: the order parameter $q = z/N$ jumps from zero to a nonzero value at $t_c$, and after $t_c$ the network condenses into a complete-graph core with $q(t) = (1 - p_0 e^{-\alpha(t-t_c)})^2$. Near criticality the degree distribution becomes $P(k) \sim k^{-1}$ and the clustering coefficient $C(k) \sim 1/k$, so scale-free structure and hierarchical clustering emerge from critical dynamics rather than preferential attachment. An adiabatic treatment of triangle density renormalizes the coupling to $\beta' = 3\beta\gamma/(3\gamma + \beta)$, and the corrected phase boundary $\beta' = \gamma^2/(4\alpha)$ matches simulations even at moderate $\beta$.

Load-bearing premise

The calculation assumes that the density of triangles stays negligible until the critical time and is then approximated by a closure that ignores higher-order shapes; if triangles build up substantially before that time, the predicted blow-up could be an artifact of the approximation.

Editorial extensions

If this is right

  • When $\beta > \gamma^2/(4\alpha)$, the network's mean degree approaches the universal hyperbolic law $z(t) \sim (t_c - t)^{-1}$, the same scaling reported in Eq. (1) for social and financial systems.
  • The phase boundary $\beta' = \gamma^2/(4\alpha)$ separates an equilibrium random-graph regime from a non-equilibrium regime; the dressed coupling $\beta'$ extends the tree-level prediction to larger $\beta$.
  • Near $t_c$ the degree distribution is $P(k) \sim k^{-1}$, so the transition itself generates scale-free structure without preferential attachment.
  • The transition is first order in $q = z/N$, with a latent heat set by the fraction $p_0$ of isolated nodes at $t_c$; for very strong coupling $p_0$ grows and the first-order jump may soften into a continuous transition.
  • Arbitrarily small nonlinearity $\beta$ can still drive a DPT if the random edge-creation rate $\alpha$ is large enough, so the classical random-graph baseline is structurally unstable to even weak triadic closure.

Reading between the lines

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

  • Because the same Riccati structure appears in any process with linear decay plus quadratic positive feedback, the finite-time singularity may be a generic property of a broad class of growth-removal dynamics, not only of this network model.
  • The $P(k) \sim k^{-1}$ and $C(k) \sim 1/k$ predictions provide a signature that empirical time series could be checked before an observed collapse: degree distributions should fatten and clustering should become inversely proportional to degree in the approach to $t_c$.
  • A next-order closure that keeps 4-node shapes would shift the dressed coupling further; measuring that shift in simulations would tell how much of the phase boundary is an artifact of the adiabatic truncation.
  • The mechanism offers a possible micro-foundation for hyperbolic discounting and deadline-driven submission surges, treating them as finite-time singularities of an underlying interaction-driven process rather than as purely behavioral regularities.
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 / 5 minor

Summary. The manuscript proposes a minimal stochastic network model in which edges are added at rate α/N plus a triadic-closure interaction β(A^2)_ij and removed at rate γ. The central claim is that for β > γ^2/(4α) the expected average degree z(t) diverges at a finite critical time t_c as z(t) ∼ (t_c − t)^{-1}, reproducing the empirical hyperbolic scaling in Eq. (1). The paper further derives a phase diagram, a first-order transition order parameter q(t), a tree-level degree distribution with P(k) ∼ k^{-1} near t_c, and clustering scaling C(k) ∼ 1/k, comparing these predictions with numerical simulations.

Significance. If the central claim is correct, the paper provides an attractive analytic mechanism for finite-time singularities in classical network dynamics, connecting triadic closure to hyperbolic scaling and to critical scale-free structure. Its strengths include a closed-form solution at tree level, no parameter fitting, an explicit phase boundary, and qualitative simulation support for the main transition. However, the analytic results rest on a moment truncation that is not controlled precisely in the regime where the singularity is predicted, and the numerical evidence lacks error bars and finite-size scaling. The conceptual payoff is therefore real, but the evidence currently falls short of establishing the universality claim.

major comments (4)
  1. [§Theoretical Framework, Eqs. (2)–(4) and Methods] The finite-time divergence is obtained by first dropping the triangle density Δ in Eq. (2) and then, for the quantitative predictions, using the adiabatic closure Δ = β/(3γ+β) z^2 from Eq. (3). The paper justifies the truncation by saying that higher-order terms involve coefficients proportional to β^{n−1}, but near t_c the relevant expansion parameter is not β alone; it is β z/γ, which diverges as z ∼ 1/(t_c−t). Indeed, from Eq. (3), if Δ ∼ z^2 then dΔ/dt ∼ β z^3, so the omitted four-node term □ must contribute at O(z^3) to balance the equation. Thus the hierarchy is uncontrolled in exactly the regime where the singularity is predicted. The authors should either provide a controlled asymptotic limit (e.g., a large-N limit with rates scaled so that the closed equation is exact) or an explicit bound showing that higher moments cannot alter the leading singularity. Without this, the hyperbolic scaling could be an artifact of the closure rather than a property of the full model.
  2. [§CRITICAL BEHAVIOR, Eq. (6) and Figure 4] The master equation for the degree distribution P(k,t) is written at tree level and inherits the same truncation as Eq. (2). The advertised P(k) ∼ k^{-1} therefore is not independent of the closure. Near t_c the triangle density is not small, and the factorization ⟨Σ_k A_{ik}A_{kj}⟩ = z^2 used in the Methods assumes uncorrelated in- and out-degrees, an assumption that can fail precisely as clustering builds up. The agreement between theory and simulation in Fig. 4 is between the simulation and the same truncated equation, so it does not by itself validate the closure. Please provide a direct test of Eq. (6) against simulations at larger N and at several times approaching t_c, with ensemble error bars, or show that the tree-level master equation is exact in some limit.
  3. [§FIRST-ORDER PHASE TRANSITION, Eq. (5)] The post-transition prediction q(t) = (1 − p0 e^{−α(t−tc)})^2 is based on a complete-core-plus-isolated-nodes ansatz that is asserted rather than derived from the microscopic rates. The argument that an isolated node, after forming one edge to the core, immediately connects to all core members because the number of common neighbors is O(N) is plausible but is not a quantitative derivation; moreover p0 is computed from the tree-level P(0,t_c), so the latent heat prediction also depends on the uncontrolled closure. The manuscript should explicitly state this ansatz as an assumption or derive it from the transition rates, and should discuss how sensitive the latent heat is to the closure.
  4. [Figs. 2–4 and numerical methods] The numerical evidence is reported without error bars, confidence intervals, or the number of independent realizations. Since the central prediction is a finite-time singularity and the simulations are for finite N (up to N = 2,000), finite-size effects could mimic or mask the divergence. The authors should report ensemble statistics and a finite-size scaling analysis—for example, how the apparent t_c, the height of q(t) at t_c, and the degree-distribution tail depend on N. This is necessary to support the claim that the observed behavior is the N → ∞ singularity rather than a finite-size crossover.
minor comments (5)
  1. [Title and headings] The heading 'THEORICAL FRAMEWORK' should read 'THEORETICAL FRAMEWORK'.
  2. [§FIRST-ORDER PHASE TRANSITION] The phrase 'first-oder DPT' in the caption of Fig. 2 should read 'first-order DPT'.
  3. [Eq. (9) and surrounding text] In Eq. (9), the notation ω = sqrt(|αβ − γ^2/4|) is used for three branches, but the sign choices and the relation to the critical time t_c are not fully explained. Please define the branch parameters explicitly and state the formula for t_c when the dressed coupling β′ is used.
  4. [Symbols throughout] The symbol γ is used both for the edge-removal rate and for the claimed degree-distribution exponent (P(k) ∼ k^{−γ} with γ = 1). This double use is confusing and should be resolved, for instance by using another symbol for the exponent.
  5. [§Methods, Generating Function Method] The derivation of Eq. (11) from Eq. (10) is not shown; the method of characteristics is mentioned only in passing. Please include a short derivation or a reference so that the generating-function solution can be checked.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the finite-time singularity and critical scaling are derived from the stated transition rates, not from fitted inputs or self-citation.

full rationale

The derivation chain is self-contained. The model is defined by the explicit transition rates in Eq. (7); the exact moment equation (8) is reduced to Eq. (2) using the stated factorization and the definition of triangle density. At tree level, Eq. (2) is a Riccati equation whose closed-form solution, Eq. (9), exhibits the finite-time divergence 1/(tc-t) for beta > gamma^2/(4alpha). The triangle correction is computed by an adiabatic closure, yielding a dressed coupling beta', not fitted to the target. The degree-distribution master equation (6), its generating-function solution (11), and the resulting P(k) ~ 1/k and C(k) ~ 1/k scalings are independent consequences of the same dynamics, not additional inputs. The empirical scaling law in Eq. (1) is cited as motivation and comparison, but it is not used as an input to the derivation, so the theoretical result stands on its own. The main caveat is the uncontrolled moment truncation near tc, where the effective expansion parameter beta z/gamma diverges; this is a correctness and rigor risk, not a circularity, and should be evaluated in a separate analysis.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central result rests on a hierarchy of mean-field closures. The paper acknowledges the hierarchy (Eq. 3) and truncates it at successively higher orders, but never establishes a controlled small parameter for the truncation near the critical time. The parameters α, β, γ are model inputs, not fitted to data.

free parameters (4)
  • α = model input (not fitted)
    Random edge formation rate per pair; appears in Eq. (7a). Chosen by modeler, not fit to data.
  • β = model input (not fitted)
    Coupling strength of triadic closure in Eq. (7a); controls the phase transition.
  • γ = model input (not fitted)
    Edge removal rate in Eq. (7b).
  • initial condition z(0) = 0
    The paper assumes an empty initial graph (G0(x)=1). A conventional choice, not derived from data.
assumptions (6)
  • domain assumption Mean-field factorization: (1/N)⟨Σ_{ijk} A_ik A_kj⟩ = z^2, assuming in- and out-degrees of a node are uncorrelated.
    Used to close Eq. (8) into Eq. (2). Near criticality, clustering may induce degree correlations that violate this assumption.
  • domain assumption Sparse network limit: z/N → 0, so the (1 - A_ij) factor and O(1/N) terms are dropped.
    Valid in the sparse regime t < t_c, but the transition point itself is where the network becomes dense, so the equation is used outside its strict validity range near t_c.
  • ad hoc to paper Tree-level truncation: triangle density Δ is neglected because Δ ~ O(β^2).
    This closure is uncontrolled; Δ may become significant as z diverges.
  • ad hoc to paper Adiabatic approximation for triangles: dΔ/dt = 0, giving Δ = β/(3γ+β) z^2.
    Assumes triangle density tracks the mean degree instantaneously; no timescale separation is established.
  • ad hoc to paper Post-transition ansatz: the network is a complete-graph core plus p(t) isolated nodes, with each isolated node joining the core at rate α.
    Used to derive Eq. (5) for q(t); the assumed structure is plausible but not derived from the microscopic rules.
  • ad hoc to paper Interaction expansion truncated at quadratic order: U_int ≈ β A^2, with the linear term neglected.
    The linear term is set to zero 'to isolate nonlinear effects', a modeling choice not justified by a small parameter.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Dynamical Phase Transitions in Non-equilibrium Networks." pith.science (2026). https://pith.science/paper/NSSU6A3B

@misc{pith2026241206704,
  author       = {Pith},
  title        = {Pith review of: Dynamical Phase Transitions in Non-equilibrium Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NSSU6A3B}},
  note         = {Machine review of arXiv:2412.06704}
}
read the original abstract

Dynamical phase transitions (DPTs) characterize critical changes in system behavior occurring at finite times, providing a lens to study nonequilibrium phenomena beyond conventional equilibrium physics. While extensively studied in quantum systems, DPTs have remained largely unexplored in classical settings. Recent experiments on complex systems, from social networks to financial markets, have revealed abrupt dynamical changes analogous to quantum DPTs, motivating the search for a theoretical understanding. Here, we present a minimal model for nonequilibrium networks, demonstrating that nonlinear interactions among network edges naturally give rise to DPTs. Specifically, we show that network degree diverges at a finite critical time, following a universal hyperbolic scaling, consistent with empirical observations. Our analytical results predict that key network properties, including degree distributions and clustering coefficients, exhibit critical scaling as criticality approaches. These findings establish a theoretical foundation for understanding emergent nonequilibrium criticality across diverse complex systems.

Figures

Figures reproduced from arXiv: 2412.06704 by the authors.

Figure 1
Figure 1. FIG. 1. Illustration of network evolution over time for [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Theoretical predictions versus numerical simulations for ( [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

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

45 extracted references · 38 canonical work pages · cited by 1 Pith paper

  1. [1]

    Dynamical quantum phase transitions: a review.Reports on Progress in Physics, 81(5):054001, 2018

    Markus Heyl. Dynamical quantum phase transitions: a review.Reports on Progress in Physics, 81(5):054001, 2018

    2018

  2. [2]

    Thermalization after an interaction quench in the hubbard model.Physical review letters, 103(5):056403, 2009

    Martin Eckstein, Marcus Kollar, and Philipp Werner. Thermalization after an interaction quench in the hubbard model.Physical review letters, 103(5):056403, 2009

    work page 2009

  3. [3]

    Thermodynamics of quantum jump trajectories.Phys- ical review letters, 104(16):160601, 2010

    Juan P Garrahan and Igor Lesanovsky. Thermodynamics of quantum jump trajectories.Phys- ical review letters, 104(16):160601, 2010

    work page 2010

  4. [4]

    Dynamical phase transitions and instabilities in open atomic many-body systems.Physical review letters, 105(1):015702, 2010

    Sebastian Diehl, Andrea Tomadin, Andrea Micheli, Rosario Fazio, and Peter Zoller. Dynamical phase transitions and instabilities in open atomic many-body systems.Physical review letters, 105(1):015702, 2010

    work page 2010

  5. [5]

    Dynamical transitions and quantum quenches in mean-field models

    Bruno Sciolla and Giulio Biroli. Dynamical transitions and quantum quenches in mean-field models. Journal of Statistical Mechanics: Theory and Experiment, 2011(11):P11003, 2011

    work page 2011

  6. [6]

    Quantum quenches, dynamical transitions, and off- equilibrium quantum criticality.Physical Review B—Condensed Matter and Materials Physics, 88(20):201110, 2013

    Bruno Sciolla and Giulio Biroli. Quantum quenches, dynamical transitions, and off- equilibrium quantum criticality.Physical Review B—Condensed Matter and Materials Physics, 88(20):201110, 2013. 13

    work page 2013

  7. [7]

    Linear ramps of the mass in the o (n) model: Dynamical transition and quantum noise of excitations

    Anna Maraga, Pietro Smacchia, and Alessandro Silva. Linear ramps of the mass in the o (n) model: Dynamical transition and quantum noise of excitations. Physical Review B, 94(24):245122, 2016

    work page 2016

  8. [8]

    Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator.Nature, 551(7682):601–604, 2017

    Jiehang Zhang, Guido Pagano, Paul W Hess, Antonis Kyprianidis, Patrick Becker, Harvey Kaplan, Alexey V Gorshkov, Z-X Gong, and Christopher Monroe. Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator.Nature, 551(7682):601–604, 2017

    2017

Show all 45 references
  1. [9]

    Dynamical quantum phase transitions in the transverse-field ising model.Physical review letters, 110(13):135704, 2013

    Markus Heyl, Anatoli Polkovnikov, and Stefan Kehrein. Dynamical quantum phase transitions in the transverse-field ising model.Physical review letters, 110(13):135704, 2013

    2013

  2. [10]

    Non-equilibrium early-warning signals for critical transitions in ecological systems.Proceedings of the National Academy of Sciences, 120(5):e2218663120, 2023

    Li Xu, Denis Patterson, Simon Asher Levin, and Jin Wang. Non-equilibrium early-warning signals for critical transitions in ecological systems.Proceedings of the National Academy of Sciences, 120(5):e2218663120, 2023

    2023

  3. [11]

    Synchro- nization within synchronization: transients and intermittency in ecological networks.National science review, 8(10):nwaa269, 2021

    Huawei Fan, Ling-Wei Kong, Xingang Wang, Alan Hastings, and Ying-Cheng Lai. Synchro- nization within synchronization: transients and intermittency in ecological networks.National science review, 8(10):nwaa269, 2021

    2021

  4. [12]

    Stock network stability in times of crisis.Physica A: Statistical Me- chanics and its Applications, 393:376–381, 2014

    Raphael H Heiberger. Stock network stability in times of crisis.Physica A: Statistical Me- chanics and its Applications, 393:376–381, 2014

    2014

  5. [13]

    Quantifying information flow during emergencies.Scientific reports, 4(1):3997, 2014

    Liang Gao, Chaoming Song, Ziyou Gao, Albert-László Barabási, James P Bagrow, and Dashun Wang. Quantifying information flow during emergencies.Scientific reports, 4(1):3997, 2014

    2014

  6. [14]

    The unruly power grid.IEEE Spectrum, 41(8):22–27, 2004

    Peter Fairley. The unruly power grid.IEEE Spectrum, 41(8):22–27, 2004

    2004

  7. [15]

    New online ecology of adversarial aggregates: Isis and beyond.Science, 352(6292):1459–1463, 2016

    Neil F Johnson, Minzhang Zheng, Yulia Vorobyeva, Andrew Gabriel, Hong Qi, Nicolás Velásquez, Pedro Manrique, Daniela Johnson, Eduardo Restrepo, Chaoming Song, et al. New online ecology of adversarial aggregates: Isis and beyond.Science, 352(6292):1459–1463, 2016

    2016

  8. [16]

    Hidden resilience and adaptive dynamics of the global online hate ecology.Nature, 573(7773):261–265, 2019

    Nicola F Johnson, Rhys Leahy, N Johnson Restrepo, Nicholas Velásquez, Minzhang Zheng, Pe- dro Manrique, Prajwal Devkota, and Stefan Wuchty. Hidden resilience and adaptive dynamics of the global online hate ecology.Nature, 573(7773):261–265, 2019

    2019

  9. [17]

    Generalized gelation theory describes onset of online extremist support.Physical Review Letters, 121(4):048301, 2018

    Pedro D Manrique, Minzhang Zheng, Zhenfeng Cao, Elvira Maria Restrepo, and Neil F John- son. Generalized gelation theory describes onset of online extremist support.Physical Review Letters, 121(4):048301, 2018

    2018

  10. [18]

    The 14 online competition between pro-and anti-vaccination views.Nature, 582(7811):230–233, 2020

    Neil F Johnson, Nicolas Velásquez, Nicholas Johnson Restrepo, Rhys Leahy, Nicholas Gabriel, Sara El Oud, Minzhang Zheng, Pedro Manrique, Stefan Wuchty, and Yonatan Lupu. The 14 online competition between pro-and anti-vaccination views.Nature, 582(7811):230–233, 2020

    2020

  11. [19]

    Uncertainty and hyperbolic discounting.American Eco- nomic Review, 95(4):1290–1299, 2005

    Partha Dasgupta and Eric Maskin. Uncertainty and hyperbolic discounting.American Eco- nomic Review, 95(4):1290–1299, 2005

    2005

  12. [20]

    Golden eggs and hyperbolic discounting.The Quarterly Journal of Economics, 112(2):443–478, 1997

    David Laibson. Golden eggs and hyperbolic discounting.The Quarterly Journal of Economics, 112(2):443–478, 1997

    1997

  13. [21]

    Global warming and hyperbolic discounting.Journal of public economics, 89(2- 3):261–282, 2005

    Larry Karp. Global warming and hyperbolic discounting.Journal of public economics, 89(2- 3):261–282, 2005

    2005

  14. [22]

    Quasi-hyperbolic discounting and retirement.Journal of Public Economics, 87(9-10):1839–1872, 2003

    Peter Diamond and Botond Köszegi. Quasi-hyperbolic discounting and retirement.Journal of Public Economics, 87(9-10):1839–1872, 2003

    2003

  15. [23]

    A universal law of procrastination.Physics Today, 69(2):11–12, 2016

    Tomasz Durakiewicz. A universal law of procrastination.Physics Today, 69(2):11–12, 2016

    2016

  16. [24]

    Metamagnetic anomalies near dynamic phase transitions

    Patricia Riego, Paolo Vavassori, and A Berger. Metamagnetic anomalies near dynamic phase transitions. Physical Review Letters, 118(11):117202, 2017

    2017

  17. [25]

    Persistent non-gaussian correlations in out-of-equilibrium rydberg atom arrays.PRX Quantum, 4(4):040339, 2023

    Aydin Deger, Aiden Daniel, Zlatko Papić, and Jiannis K Pachos. Persistent non-gaussian correlations in out-of-equilibrium rydberg atom arrays.PRX Quantum, 4(4):040339, 2023

    2023

  18. [26]

    First-order dynamical phase transitions

    Elena Canovi, Philipp Werner, and Martin Eckstein. First-order dynamical phase transitions. Physical Review Letters, 113(26):265702, 2014

    2014

  19. [27]

    Dy- namical phase transition in the open dicke model.Proceedings of the National Academy of Sciences, 112(11):3290–3295, 2015

    Jens Klinder, Hans Keßler, Matthias Wolke, Ludwig Mathey, and Andreas Hemmerich. Dy- namical phase transition in the open dicke model.Proceedings of the National Academy of Sciences, 112(11):3290–3295, 2015

    2015

  20. [28]

    Exploring dynamical phase transitions with cold atoms in an optical cavity.Nature, 580(7805):602–607, 2020

    Juan A Muniz, Diego Barberena, Robert J Lewis-Swan, Dylan J Young, Julia RK Cline, Ana Maria Rey, and James K Thompson. Exploring dynamical phase transitions with cold atoms in an optical cavity.Nature, 580(7805):602–607, 2020

    2020

  21. [29]

    Thermodynamic bounds on symmetry breaking in linear and catalytic biochemical systems

    Shiling Liang, Paolo De Los Rios, and Daniel Maria Busiello. Thermodynamic bounds on symmetry breaking in linear and catalytic biochemical systems. Physical Review Letters, 132(22):228402, 2024

    2024

  22. [30]

    Classes of critical avalanche dynamics in complex networks

    Filippo Radicchi, Claudio Castellano, Alessandro Flammini, Miguel A Muñoz, and Daniele Notarmuzi. Classes of critical avalanche dynamics in complex networks. Physical Review Research, 2(3):033171, 2020

    2020

  23. [31]

    Timelinesscriticalityincomplexsystems

    JoséMoran, MatthijsRomeijnders, PierreLeDoussal, FrankPPijpers, UtzWeitzel, Debabrata Panja, andJean-PhilippeBouchaud. Timelinesscriticalityincomplexsystems. Nature Physics, pages 1–7, 2024. 15

    2024

  24. [32]

    Cambridge University Press, 2010

    Pavel L Krapivsky, Sidney Redner, and Eli Ben-Naim.A kinetic view of statistical physics. Cambridge University Press, 2010

    2010

  25. [33]

    How nature works: the science of self-organized criticality

    Per Bak. How nature works: the science of self-organized criticality. Springer Science & Business Media, 2013

    2013

  26. [34]

    Self-organized criticality in the’game of life.Nature, 342(6251):780–782, 1989

    Per Bak, Kan Chen, and Michael Creutz. Self-organized criticality in the’game of life.Nature, 342(6251):780–782, 1989

    1989

  27. [35]

    Self-organized criticality

    Per Bak, Chao Tang, and Kurt Wiesenfeld. Self-organized criticality. Physical review A, 38(1):364, 1988

    1988

  28. [36]

    Engineering self-organized criticality in living cells

    Blai Vidiella, Antoni Guillamon, Josep Sardanyés, Victor Maull, Jordi Pla, Nuria Conde, and Ricard Solé. Engineering self-organized criticality in living cells. Nature communications, 12(1):4415, 2021

    2021

  29. [37]

    Signatures of self-organized criticality in an ultracold atomic gas.Nature, 577(7791):481–486, 2020

    S Helmrich, A Arias, G Lochead, TM Wintermantel, M Buchhold, S Diehl, and S Whitlock. Signatures of self-organized criticality in an ultracold atomic gas.Nature, 577(7791):481–486, 2020

    2020

  30. [38]

    Self-organized criticality in x-ray flares of gamma-ray-burst afterglows

    FY Wang and ZG Dai. Self-organized criticality in x-ray flares of gamma-ray-burst afterglows. Nature Physics, 9(8):465–467, 2013

    2013

  31. [39]

    Self-organized network evolution coupled to extremal dynamics.Nature Physics, 3(11):813–817, 2007

    Diego Garlaschelli, Andrea Capocci, and Guido Caldarelli. Self-organized network evolution coupled to extremal dynamics.Nature Physics, 3(11):813–817, 2007

    2007

  32. [40]

    Emergence of scaling in random networks.science, 286(5439):509–512, 1999

    Albert-László Barabási and Réka Albert. Emergence of scaling in random networks.science, 286(5439):509–512, 1999

    1999

  33. [41]

    Statistical mechanics of complex networks.Reviews of modern physics, 74(1):47, 2002

    Réka Albert and Albert-László Barabási. Statistical mechanics of complex networks.Reviews of modern physics, 74(1):47, 2002

    2002

  34. [42]

    Collective dynamics of ‘small-world’networks.nature, 393(6684):440–442, 1998

    Duncan J Watts and Steven H Strogatz. Collective dynamics of ‘small-world’networks.nature, 393(6684):440–442, 1998

    1998

  35. [43]

    Geometric description of clustering in directed networks.Nature Physics, 20(1):150–156, 2024

    Antoine Allard, M Ángeles Serrano, and Marián Boguñá. Geometric description of clustering in directed networks.Nature Physics, 20(1):150–156, 2024

    2024

  36. [44]

    Hierarchical organization in complex networks

    Erzsébet Ravasz and Albert-László Barabási. Hierarchical organization in complex networks. Physical review E, 67(2):026112, 2003

    2003

  37. [45]

    Hierarchical organization of modularity in metabolic networks.science, 297(5586):1551–1555, 2002

    Erzsébet Ravasz, Anna Lisa Somera, Dale A Mongru, Zoltán N Oltvai, and A-L Barabási. Hierarchical organization of modularity in metabolic networks.science, 297(5586):1551–1555, 2002. 16

    2002

Pith tools

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