Pith. sign in

REVIEW 4 major objections 6 minor 1 cited by

Extended Yard Sale model of wealth distribution on Erd\H{o}s-R\'enyi random networks

T0 review · 4 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read A networked Extended Yard Sale model leads to local wealth condensation, not a single oligarch, while preserving the continuous phase transition.

desk verdict The qualitative local-condensation result is credible and consistent with prior work, but the N^{-1/4} scaling law and 'only fully connected' corollary are fitted rather than derived, and the corollary overstates the paper's own equation. read the letter →

arxiv 2505.04032 v3 pith:QNXM5VGS submitted 2025-05-07 cond-mat.stat-mech cond-mat.dis-nn

Nicolás Vazquez Von Bibow , Juan I. Perotti This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.dis-nn
keywords wealthcondensationExtendedYardSalemodelErdős–Rényirandomnetworksassetexchangelocalphasetransitionquenchedmeanfieldeconophysics
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

The paper introduces a networked version of the Extended Yard Sale model, a random asset-exchange model in which wealth redistribution competes with wealth-attained advantage. On Erdős–Rényi random networks, it argues, the model still undergoes a continuous phase transition, but the condensation is local rather than global: no single agent takes a macroscopic share of total wealth. The load-bearing quantitative claim is a finite-size scaling law: at small nonzero temperature the expected relative wealth of the richest agent scales as $N^{-1/4}g_T(\bar{k}/N)$, so in the thermodynamic limit a nonzero share for the richest agent is possible only when the network is fully connected. Instead, wealth concentrates in a set of mutually non-adjacent rich agents whose individual wealth grows as $N^{3/4}$, while poor agents hold constant wealth. If this is right, interaction structure—not just the exchange rule—determines whether an economy produces one oligarch or a dispersed rich class.

What carries the argument

The central object is the Quenched Mean Field evolution equation for the expected relative wealth $x_i$ of each node: $$\dot{x}_i = \frac{T}{N}\left(-\beta_i x_i + \frac{1}{N}\sum_j \beta_j x_j\right) + \sum_j \kappa_{ij}(x_i-x_j)\min\{x_i,x_j\},$$ with taxation rates $\beta_i$ and interaction rates $\kappa_{ij}$ set by the choice of interaction and taxation mode. A two-population reduction splits agents into $R$ rich and $N-R$ poor agents; the marginal-stability condition of that reduced dynamics gives the critical temperature $T_c=N/(N-R^*)$. The quantitative punchline is carried by the scaling hypothesis $\langle x_1\rangle_a(\lambda^\nu\bar{k},\lambda^\mu N)\approx \lambda \langle x_1\rangle_a(\bar{k},N)$, whose data collapse gives $N^{-1/4}\langle x_1\rangle_a$ as a function of $\bar{k}/N$ at small nonzero temperature, together with the observation that rich agents must be non-adjacent, which bounds $R$ through the maximal independent set.

What would settle it

Run the same model at $N=1000$ or larger with many network samples and error bars, and test directly whether the richest agents' wealth grows as $N^{3/4}$ and whether $\langle x_1\rangle_a$ collapses under $N^{-1/4}g_T(\bar{k}/N)$; if the fitted exponents drift with $N$, the local-condensation scaling law is not asymptotic. Deriving the exponents from the Quenched Mean Field equations would also settle it.

Watch

Extended reading notes

Core claim

The central claim is that local wealth condensation replaces global wealth condensation when the Extended Yard Sale model is placed on Erdős–Rényi random networks, while the continuous condensation transition survives. For the four interaction/taxation variants studied, Monte Carlo simulation, Quenched Mean Field approximation, and a two-population Mean Field theory agree that rich agents cannot be neighbors: connected wealthy agents compete until one is ruined, so the rich form an independent set. Quantitatively, for small nonzero temperature $T$, the network-averaged relative wealth of the richest agent obeys $\langle x_1\rangle_a \approx N^{-1/4}g_T(\bar{k}/N)$, with $g_T(p)=-\frac{1}{3}+p^{1/3}(1-p)^{-1/4}$ as a phenomenological fit, so a finite fraction of total wealth in one agent's hands survives only as $\bar{k}\to N-1$, the fully connected limit. In the condensed phase the average wealth of each rich agent grows as $N^{3/4}$ and that of poor agents stays $O(1)$; the number of rich agents is bounded by the network's maximal independent set, which on Erdős–Rényi graphs gives roughly $2\ln N/\ln(1/(1-p))$.

Load-bearing premise

The specific numbers in the central scaling law come from fitting simulation curves, not from a proof, and the largest simulated systems have only 256 agents with no reported error bars.

Editorial extensions

If this is right

  • None of the four interaction/taxation modes on Erdős–Rényi networks produces a single agent holding a macroscopic share of total wealth at nonzero temperature; local condensation is generic across modes.
  • The condensation transition remains continuous, with critical temperature $T_c\approx 1$ in the large-$N$ limit, so the critical-point phenomenology of the fully connected model carries over to sparse networks.
  • At any nonzero temperature the relative wealth of the richest agent vanishes as $N^{-1/4}$ times a function of $\bar{k}/N$, so a nonzero macroscopic share requires $\bar{k}=N-1$.
  • Wealth concentrates in an independent set of rich agents: the number of rich agents is bounded by the maximal independent set, roughly $2\ln N/\ln(1/(1-p))$ on dense Erdős–Rényi networks.
  • In the condensed phase individual rich agents hold wealth $\sim N^{3/4}$ while poor agents hold $O(1)$, producing a two-peaked wealth distribution.

Reading between the lines

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

  • If adding links among the rich destroys their advantage, then network rewiring that raises clustering among the wealthy should measurably lower the condensation threshold; this is a testable consequence the paper leaves implicit.
  • The fitted zero-temperature scaling has degenerate exponents, $\nu=\mu=\infty$ with only $\nu/\mu=1$ fixed, so a direct measurement of $\langle x_1\rangle_a$ versus $N$ at fixed $\bar{k}$ would determine whether the $T=0$ law is truly $\bar{k}/N$ or only an apparent collapse.
  • The hysteresis loops reported for some taxation modes imply the stationary condensed state is not unique; policy conclusions drawn from uniform-initial-condition runs may miss trajectories in which the richest agent's share stays high.
  • Extending the same local-competition mechanism to scale-free or small-world topologies, where hubs cannot be adjacent yet dominate connectivity, may yield a different condensation scale than $N^{3/4}$; that is a natural next test.
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 / 6 minor

Summary. The manuscript introduces a networked version of the Extended Yard Sale (EYS) model on Erdős–Rényi random graphs, combining Monte Carlo simulations, a Quenched Mean Field approximation, and a two-population Mean Field theory. The central claim is that, unlike the fully connected case, the networked model does not exhibit global wealth condensation but instead local wealth condensation, in which a set of mutually non-adjacent rich agents holds a submacroscopic share each, while the model still undergoes a continuous phase transition at a temperature Tc. The paper derives the QMF dynamical equations, reduces them to a two-population ODE, and proposes a finite-size scaling law for the expected wealth of the richest agent, x1 ~ N^{-1/4} g_T(kbar/N) at small nonzero temperature, concluding that nonzero macroscopic shares are possible only in fully connected networks. The qualitative claim is supported by simulations and by comparison with the prior result of Börgers and Greengard, but the quantitative scaling rests on exponents and a scaling function obtained by visual inspection and by fitting.

Significance. If the quantitative claims held rigorously, this would be a valuable extension of a well-known econophysics model to structured populations, showing that network topology prevents single-agent oligarchy while preserving a continuous transition. The paper's strengths include a clear formulation of the networked dynamics, a transparent QMF derivation, a two-population MF that recovers the fully connected limit, and a comparative study of four interaction/taxation modes. The qualitative finding of local condensation agrees with an existing rigorous theorem, which lends credibility to the overall picture. However, the quantitative content—especially the N^{-1/4} law and the N^{3/4} rich-agent wealth scaling—is not derived from first principles and is supported only by finite-size collapses with no error bars and with data up to N=256. Those quantitative statements, rather than the mere existence of local condensation, are the main new contributions claimed, so the current support is insufficient for the strength of the abstract and conclusions.

major comments (4)
  1. [III.B, Fig. 3 and Eq. 13] The scaling law in Eq. 13 is the quantitative backbone of the central claim, but the exponents are obtained 'by visual inspection' with no error bars, and only the ratio nu/mu is constrained by the data collapse variable N^{-nu/mu} kbar; the prefactor exponent 1/mu, which fixes the N^{-1/4} decay, is not independently established. With Monte Carlo data only up to N=256, the asymptotic decay is not robust. The authors should either derive this exponent, provide a statistically grounded collapse with error estimates, or explicitly moderate the claim from a determined law to a fitted finite-size observation.
  2. [III.B, paragraph after Eq. 13] The statement that a nonzero average <x1> as N -> infinity 'is possible only for fully connected networks' is stronger than what Eq. 13 implies. Even if Eq. 13 holds, taking p = 1 - c/N with constant c gives kbar/N -> 1 and <x1> = O(1) for any g_T(1) finite, so the equation itself allows macroscopic shares on dense non-fully-connected sequences. The wording should be revised to describe the limit kbar/N -> 1 rather than the exclusive case kbar = N-1.
  3. [III.B, Fig. 5c and Eq. 17] The comparison in Fig. 5c against 'predictions of Eq. 17' is circular: the value R* used to evaluate Eq. 17 is read from the very same MC/QMF simulations whose u* is being fit. The same issue applies to the critical temperature Tc = N/(N-R*). These comparisons demonstrate consistency with the two-population ansatz, but they are not independent tests of the MF theory. The text should label them as consistency checks and avoid the word 'prediction'.
  4. [Appendix B, Fig. 10] The claimed N^{3/4} scaling for the average wealth of the rich agents is presented as a guide-to-the-eye fit without error bars, a derivation, or a scaling collapse analogous to Fig. 3. Since this scaling is presented as a concrete manifestation of local condensation, it needs either an analytical argument or a more careful finite-size analysis with uncertainties; as written, it does not support the conclusion that this exponent is universal across modes and network sizes.
minor comments (6)
  1. [III.B, first paragraph] The sentence 'Appendix A sheeds light on the origin of this discrepancy' contains a typo: 'sheeds' should be 'sheds'.
  2. [Appendix A, Fig. 7 caption] The phrase 'the heathing phase' should be 'the heating phase'.
  3. [II.E, paragraph on taxation modes] The phrase 'Without loosing generality' should be 'Without loss of generality'.
  4. [II.B] The convention aii in {0,2,4,...} for non-directed networks is unusual and could confuse readers; consider a brief clarification that it encodes the factor 2 for self-loops.
  5. [III.B, Fig. 5d] The inset of Fig. 5d reports that MC tends to overestimate R* due to stochastic fluctuations; this is an important caveat and should be stated in the main text near the discussion of Fig. 5c as well, since it affects how seriously the consistency between MC and QMF should be taken.
  6. [II.D, Eq. 6] The derivation of Eq. 6 jumps from the full expectation to the factorized form without explicitly stating the factorization assumption P(wi,wj|a) ≈ P(wi|a)P(wj|a) at the point where the noise average is taken; making that step explicit would improve readability.

Circularity Check

3 steps flagged · score 5.0 of 10

The MF 'predictions' of Fig. 5c are evaluated with R* read from the same simulations, and the N^{-1/4} law is a visual fit; the quantitative claims are partly circular, while the qualitative local-condensation result retains independent support.

  1. fitted input called prediction [Sec. III.B, discussion of Figure 5c (around Eq. 17 and Eq. 19)]
    "For comparison, the predictions of Eq. 17, evaluated using the network-averaged number R∗ of rich agents obtained from simulations, are also plotted (colored lines matching the corresponding data). The inset displays the network average of the fraction of rich agents, r∗ = R∗/N, as a function of T, from which the values of R∗ are derived."

    Eq. 17 is u* = (1/R*)(1 - T(N-R*)^2/N^2). R* is not independently predicted; it is read from the same MC/QMF simulations whose u* curve is being 'predicted.' The agreement in Fig. 5c is therefore a self-consistency check, not an independent test: the comparison curve is constructed from the very data it is compared with. The associated critical temperature Tc = N/(N-R*) inherits the same measured R* input, so it is likewise not an independent prediction.

  2. fitted input called prediction [Sec. III.B, scaling analysis leading to Eq. 13 and the 'only fully connected' corollary]
    "By visual inspection, it is found that the scaling occurs for ν=μ=∞ such that ν/μ=1 when T=0 ... and for ν=μ=−4 when T=0.06. ... A better fit is provided by the phenomenological proposition gT≳0(p)≈−1/3+p1/3(1−p)−1/4 ... Observe that if ⟨x1⟩a=c for some constant 0<c<1, then Eq. 13 implies ¯k/N≈1−c−4/N. Consequently, for non-zero temperatures, a non-zero average ⟨x1⟩a in the limit N→∞ is possible only for fully connected networks."

    The N^{-1/4} law and the function g_T are obtained by visually collapsing the same MC/QMF data to which Eq. 13 is then applied; no independent derivation fixes the exponents. The corollary that only fully connected graphs give O(1) ⟨x1⟩ is thus a restatement of the fitted scaling form rather than a prediction. It also overstates that form: for p=1-O(1/N) (not literally fully connected), g_T diverges as N^{1/4}, so Eq. 13 itself permits an O(1) ⟨x1⟩. The scaling is a legitimate empirical collapse, but it is fit, not derived, and the strong universality claim inherits that status.

1 more flagged steps
  1. self definitional [Sec. III.B, paragraph introducing the order parameter φ (Eq. 19)]
    "Based on this, it is natural to introduce the order parameter ϕ:=1−2r, which captures the imbalance between poor and rich agents. By construction, ϕ→0 as T→∞, and ϕ≈1 in the low-temperature and high-connectivity limit T→0, ¯k→N−1. These predictions for ϕ are corroborated by both MC simulations and QMF approximations, as shown in Figs. 5a) and b)."

    The stated limits of φ are written into its definition and into the assumed high-T/low-T behavior of r, so calling the data's approach to those values a 'corroboration' of predictions is tautological. This step is not central to the main scaling claim, but it is a clear instance of a by-construction statement being presented as a confirmed prediction.

full rationale

The paper's main qualitative claim—that the networked EYS model shows local rather than global wealth condensation and a continuous transition—is not itself manufactured: it is directly observed in MC/QMF data and is aligned with the independent Börgers-Greengard result [41], so it would be wrong to call the paper wholly circular. The circularity is concentrated in the validation machinery. Figure 5c's 'predictions' of Eq. 17 use R* measured from the same simulations, making the agreement a consistency check rather than an out-of-sample test; the N^{-1/4} scaling of Eq. 13 is a visual finite-size collapse with fitted exponents and a phenomenological g_T, yet it is used as the basis for the 'only fully connected networks' corollary, which moreover is stronger than Eq. 13 implies for p=1-O(1/N). There is no load-bearing self-citation or imported uniqueness theorem; the Perotti self-citations are tangential. Overall: partial circularity in the quantitative claims, with the qualitative local-condensation result retaining independent content.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central results rest on three fitted inputs: the scaling exponents and the function g_T, fixed by visual collapse and phenomenological fitting (Fig. 3), and the rich-agent count R*, imported from the same simulations into Eqs. 17 and the critical temperature. Four structural assumptions are load-bearing: the peaked-marginal factorization behind Eq. 7, the Poissonian event limit, the non-adjacency plus k_i approx k-bar reduction behind Eq. 16, and the cited independence-number bound. No new physical entities are introduced; the rich/poor split and the order parameter phi are diagnostics. The direct Monte Carlo evidence for the qualitative claim does not depend on the fitted inputs, which is the main reason the circularity burden is only partial.

free parameters (3)
  • Scaling exponents nu, mu at T = 0.06 = nu = mu = -4
    Chosen by visual inspection of the data collapse in Fig. 3 (Sec. III.B); no derivation from the QMF or MF equations. At T = 0 the paper uses nu = mu = infinity with nu/mu = 1, a degenerate limit that fixes only a ratio.
  • Scaling function g_T(p) for T just above zero = g_T(p) = -1/3 + p^{1/3}(1-p)^{-1/4}
    Introduced as a 'phenomenological proposition' (Sec. III.B, Eq. 14) to fit the collapsed data; the (1-p)^{-1/4} branch is anchored to the fully connected limit p close to 1, and the p^{1/3} term is empirical.
  • Number of rich agents R* = Measured from simulations, Fig. 5c inset
    R* enters the equilibrium solution u* = (1/R*)(1 - T(N-R*)^2/N^2) and the critical temperature T_c = N/(N-R*), but it is imported from the same MC and QMF runs, making Eq. 17 a consistency relation rather than a closed-form prediction (Sec. III.B).
assumptions (4)
  • domain assumption Wealth marginals factorize and are sharply peaked: P(w_i,w_j|a) approx P(w_i|a)P(w_j|a), so min{w_i,w_j} is replaced by min{<w_i>,<w_j>} in the QMF equation.
    Invoked in Sec. II.D to pass from Eq. 6 to Eq. 7. The assumption is weakest near the critical point and in the condensed phase, where Fig. 6 shows broad, power-law-like wealth distributions.
  • domain assumption Poissonian limit: multiple taxation or interaction events within one time step contribute negligibly as Delta t goes to zero.
    Sec. II.D: 'The Poissonian limit is assumed. Therefore, multiple taxation and interaction events within a time step will contribute negligibly as Delta t goes to zero.'
  • domain assumption In the steady state, rich agents are mutually non-adjacent and all nodes have k_i approx k-bar, so the interaction sum in Eq. 15 collapses to (u-v)v.
    Sec. III.B: 'a key observation in the stationary regime is the lack of connections among rich agents'; the k_i approx k-bar approximation is used in Eq. 15. This makes an empirical observation the input of the two-population theory and of the bound r <= r_max.
  • standard math Independence-number bound R_max approx 2 ln N / ln(1/(1-p)) for Erdos-Renyi graphs.
    External theorem (Janson, Luczak, Rucinski, Random Graphs, Thm 7.1, cited as [46]) used in Sec. III.B, Eq. 18, to bound the number of mutually non-adjacent rich agents.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Extended Yard Sale model of wealth distribution on Erd\H{o}s-R\'enyi random networks." pith.science (2026). https://pith.science/paper/QNXM5VGS

@misc{pith2026250504032,
  author       = {Pith},
  title        = {Pith review of: Extended Yard Sale model of wealth distribution on Erd\Hos-R\'enyi random networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QNXM5VGS}},
  note         = {Machine review of arXiv:2505.04032}
}
read the original abstract

Excessive wealth concentration can undermine economic and social development. Random Asset Exchange (RAE) models provide valuable tools to investigate this phenomenon. Assuming that economic systems may operate optimally near the critical point of a continuous phase transition, the Extended Yard Sale (EYS) model introduced by Boghosian et al.~[Physica A 476, 15 (2017)] offers a compelling framework. This model captures the interplay between wealth redistribution and accumulation, exhibiting a continuous phase transition marked by a broad wealth distribution at criticality, separating a condensed phase -- where a microscopic fraction of agents holds a macroscopic share of total wealth -- from a distributed phase with a light-tailed wealth distribution. While the original EYS model assumes fully connected interactions, this work introduces and studies a networked variant where agents interact over Erd\H{o}s-R\'enyi random networks. The analysis combines Monte Carlo simulations with Quenched Mean Field and Mean Field approximations, exploring a variety of interaction and taxation schemes. A scaling analysis shows that, although the networked model also undergoes a continuous phase transition, it leads to local wealth condensation rather than the global condensation found in the fully connected case. These results deepen our understanding of wealth dynamics in structured populations and may help inform the development of more effective economic and social policies.

Figures

Figures reproduced from arXiv: 2505.04032 by the authors.

Figure 1
Figure 1. FIG. 1. Behavior of EYS model on fully connected networks. [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The expected value [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Scaling of the network average of the expected relative wealth of the richest agent [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Scaling of the network-averaged order parameter [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Network average of the normalized wealth distribution [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 2
Figure 2. Figure 2: In Fig. 7a) an hysteresis cycle is shown for MC sim￾ulations of the IMB/TMB case for one ER network of size N = 64 and average degree ¯k = 8. The cycle be￾gins at T = 0 starting from an uniform initial condition, xi = 1/N. As can be seen in the figure, a loop between …
Figure 7
Figure 7. Figure 7: FIG. 7. Hysteresis analysis of the EYS model on ER [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 10
Figure 10. Figure 10: FIG. 10. Scaling of the average wealth [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The fraction of rich agents [PITH_FULL_IMAGE:figures/full_fig_p014_8.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. Wealth Thermalization Hypothesis and Social Networks

    cond-mat.stat-mech 2025-06 conditional novelty 6.0 of 10

    Societal wealth inequality is claimed to be the Rayleigh-Jeans condensation of a conserved 'wealth energy' in a thermalized social system, reproducing Lorenz curves for countries and stock exchanges.

Reference graph

Works this paper leans on

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

  1. [41]

    Onnela, J

    J.-P. Onnela, J. Saram¨ aki, J. Hyv¨ onen, G. Szab´ o, D. Lazer, K. Kaski, J. Kert´ esz, and A.-L. Barab´ asi, Struc- ture and tie strengths in mobile communication networks, Proc. Natl. Acad. Sci.104, 7332 (2007)

    work page 2007

  2. [1]

    wheref(x 1) is proportional to the r.h.s

    +f′(x∗ 1)ϵ+... wheref(x 1) is proportional to the r.h.s. of Eq. 11. Since f(x∗

    work page

  3. [2]

    = 0 becausex ∗ 1 is an equilibrium solution, then the dynamics is stable iff ′(x∗ 1)<0, linearly marginal iff ′(x∗

    work page

  4. [3]

    The condition for the marginal case can be used to identify the critical point at which the condensation phase transition occurs

    = 0 and unstable iff ′(x∗ 1)>0. The condition for the marginal case can be used to identify the critical point at which the condensation phase transition occurs. It predicts a critical temperatureT c =N/(N−1)≈1 as N→∞, withx ∗ 1(T=T c) = 1/Nfor both equilibrium solutionsx ∗

    work page

  5. [4]

    However, the stability of these solutions differs: forT <T c,f ′(x∗ 1)>0, and forT >T c,f ′(x∗ 1)< 0 whenx ∗ 1 = 1/N, while the reverse holds for the other equilibrium point. As expected, these results indicate that aboveTc, all agents share approximately equally the same wealthW/N, while, in contrast, belowT c, a single oligarch emerges, owning a macrosc...

    work page 2023

  6. [5]

    Piketty,Capital in the Twenty-First Century(Belk- nap Press, Cambridge, MA, 2014)

    T. Piketty,Capital in the Twenty-First Century(Belk- nap Press, Cambridge, MA, 2014)

    work page 2014

  7. [6]

    Angle, The Surplus Theory of Social Stratification and the Size Distribution of Personal Wealth, Soc

    J. Angle, The Surplus Theory of Social Stratification and the Size Distribution of Personal Wealth, Soc. Forces65, 293 (1986)

    work page 1986

  8. [7]

    Ispolatov, P

    S. Ispolatov, P. L. Krapivsky, and S. Redner, Wealth dis- tributions in asset exchange models, Eur. Phys. J. B.2, 267 (1998)

    work page 1998

Show all 51 references
  1. [8]

    Dragulescu and V

    A. Dragulescu and V. Yakovenko, Statistical mechanics of money, Eur. Phys. J. B.17, 723 (2000)

    2000

  2. [9]

    V. M. Yakovenko and J. B. Rosser, Colloquium: Statisti- cal mechanics of money, wealth, and income, Rev. Mod. Phys.81, 1703 (2009)

    2009

  3. [10]

    Chakraborti, I

    A. Chakraborti, I. Muni Toke, M. Patriarca, and F. Abergel, Econophysics review: I. empirical facts, Quant. Finance11, 991 (2011)

    2011

  4. [11]

    Chakraborti, I

    A. Chakraborti, I. Muni Toke, M. Patriarca, and F. Abergel, Econophysics review: II. agent-based models, Quant. Finance11, 1013 (2011)

    2011

  5. [12]

    B. K. Chakrabarti, A. Chakraborti, S. R. Chakravarty, and A. Chatterjee,Econophysics of Income and Wealth Distributions(Cambridge University Press, 2013)

    2013

  6. [13]

    Greenberg and H

    M. Greenberg and H. O. Gao, Twenty-five years of ran- dom asset exchange modeling, Eur. Phys. J. B.97, 69 (2024)

    2024

  7. [14]

    Bouchaud and M

    J.-P. Bouchaud and M. M´ ezard, Wealth condensation in a simple model of economy, Physica A282, 536 (2000)

    2000

  8. [15]

    B. M. Boghosian, A. Devitt-Lee, M. Johnson, J. Li, J. A. Marcq, and H. Wang, Oligarchy as a phase tran- sition: The effect of wealth-attained advantage in a Fokker–Planck description of asset exchange, Physica A 476, 15 (2017), 1511.00770

    2017 arXiv

  9. [16]

    B. M. Boghosian, M. Johnson, and J. A. Marcq, An H Theorem for Boltzmann’s Equation for the Yard- Sale Model of Asset Exchange, J. Stat. Phys.161, 1339 (2015)

    2015

  10. [17]

    Francisco Cardoso, S

    B.-H. Francisco Cardoso, S. Gon¸ calves, and J. R. Igle- sias, Why equal opportunities lead to maximum inequal- ity? The wealth condensation paradox generally solved, Chaos Solit. Fractals168, 113181 (2023)

    2023

  11. [18]

    Chakraborti, Distributions of money in model markets of economy, Int

    A. Chakraborti, Distributions of money in model markets of economy, Int. J. Mod. Phys. C13, 1315 (2002)

    2002

  12. [19]

    Hayes, Computing science: Follow the money, Am

    B. Hayes, Computing science: Follow the money, Am. Sci.90, 400 (2002)

    2002

  13. [20]

    Boghosian, Fokker–Planck description of wealth dy- namics and the origin of Pareto’s law, Int

    B. Boghosian, Fokker–Planck description of wealth dy- namics and the origin of Pareto’s law, Int. J. Mod. Phys. C25, 1441008 (2014)

    2014

  14. [21]

    B. M. Boghosian, Kinetics of wealth and the Pareto law, Phys. Rev. E89, 042804 (2014)

    2014

  15. [22]

    Ne˜ ner, B.-H

    J. Ne˜ ner, B.-H. F. Cardoso, M. F. Laguna, S. Gon¸ calves, and J. R. Iglesias, Study of taxes, regulations and inequality using machine learning algorithms, Philos. Trans. R. Soc. A380, 20210165 (2022)

    2022

  16. [23]

    J. Li, B. M. Boghosian, and C. Li, The Affine Wealth Model: An agent-based model of asset exchange that al- lows for negative-wealth agents and its empirical valida- tion, Physica A516, 423 (2019)

    2019

  17. [24]

    Chatterjee, B

    A. Chatterjee, B. K. Chakrabarti, and S. Manna, Pareto law in a kinetic model of market with random saving propensity, Physica A335, 155 (2004)

    2004

  18. [25]

    Iglesias, S

    J. Iglesias, S. Gon¸ calves, G. Abramson, and J. Vega, Cor- relation between risk aversion and wealth distribution, Physica A342, 186 (2004), proceedings of the VIII Latin American Workshop on Nonlinear Phenomena

    2004

  19. [26]

    Ne˜ ner and M

    J. Ne˜ ner and M. F. Laguna, Optimal risk in wealth ex- change models: Agent dynamics from a microscopic per- spective, Physica A566, 125625 (2021). 14 FIG. 7. Hysteresis analysis of the EYS model on ER networks with average connectivity ¯k/N= 0.25, for the IMA/TMB variant. Al...

    2021

  20. [27]

    C. F. Moukarzel, S. Gon¸ calves, J. R. Iglesias, M. Rodr´ ıguez-Achach, and R. Huerta-Quintanilla, Wealth condensation in a multiplicative random asset ex- change model, Eur. Phys. J. Spec. Top.143, 75 (2007)

    2007

  21. [28]

    Ne˜ ner and M

    J. Ne˜ ner and M. F. Laguna, Wealth exchange models and machine learning: Finding optimal risk strategies in multiagent economic systems, Phys. Rev. E104, 014305 (2021). 15

    2021

  22. [29]

    Bak,How Nature Works: The Science of Self- Organized Criticality(Springer-Verlag, New York, 1996)

    P. Bak,How Nature Works: The Science of Self- Organized Criticality(Springer-Verlag, New York, 1996)

    1996

  23. [30]

    J. I. Perotti, O. V. Billoni, F. A. Tamarit, D. R. Chialvo, and S. A. Cannas, Emergent self-organized complex net- work topology out of stability constraints, Phys. Rev. Lett.103, 108701 (2009)

    2009

  24. [31]

    D. R. Chialvo, Emergent complex neural dynamics, Nat. Phys.6, 744 (2010)

    2010

  25. [32]

    Mora and W

    T. Mora and W. Bialek, Are biological systems poised at criticality?, J. Stat. Phys.144, 268 (2011)

    2011

  26. [33]

    Tang, Y.-Y

    Q.-Y. Tang, Y.-Y. Zhang, J. Wang, W. Wang, and D. R. Chialvo, Critical fluctuations in the native state of pro- teins, Phys. Rev. Lett.118, 088102 (2017)

    2017

  27. [34]

    Cavagna, I

    A. Cavagna, I. Giardina, and T. S. Grigera, The physics of flocking: Correlation as a compass from experiments to theory, Phy. Rep.728, 1 (2018)

    2018

  28. [35]

    M. A. Mu˜ noz, Colloquium: Criticality and dynamical scaling in living systems, Rev. Mod. Phys.90, 031001 (2018)

    2018

  29. [36]

    D. R. Chialvo, S. A. Cannas, T. S. Grigera, D. A. Martin, and D. Plenz, Controlling a complex system near its crit- ical point via temporal correlations, Sci. Rep.10, 12145 (2020)

    2020

  30. [37]

    Zamponi, E

    N. Zamponi, E. Zamponi, S. A. Cannas, and D. R. Chialvo, Universal dynamics of mitochondrial networks: a finite-size scaling analysis, Sci. Rep.12, 17074 (2022)

    2022

  31. [38]

    D. J. Watts and S. H. Strogatz, Collective dynamics of ‘small-world’ networks, Nature393, 440 (1998)

    1998

  32. [39]

    Barab´ asi and R

    A.-L. Barab´ asi and R. Albert, Emergence of scaling in random networks, Science286, 509 (1999)

    1999

  33. [40]

    Newman,Networks(Oxford University Press, Oxford, 2018)

    M. Newman,Networks(Oxford University Press, Oxford, 2018)

    2018

  34. [42]

    V´ azquez-Montejo, R

    J. V´ azquez-Montejo, R. Huerta-Quintanilla, and M. Rodr´ ıguez-Achach, Wealth condensation in a Barabasi–Albert network, Physica A389, 1464 (2010)

    2010

  35. [43]

    Bustos-Guajardo and C

    R. Bustos-Guajardo and C. F. Moukarzel, Yard-Sale ex- change on networks: wealth sharing and wealth appro- priation, J. Stat. Mech.2012, P12009 (2012)

    2012

  36. [44]

    H. G. Lee and D.-S. Lee, Scaling in local to global con- densation of wealth on sparse networks, Phys. Rev. E 108, 064303 (2023)

    2023

  37. [45]

    B¨ orgers and C

    C. B¨ orgers and C. Greengard, Local wealth condensa- tion for yard-sale models with wealth-dependent biases (2024), arXiv:2406.10978 [q-fin.MF]

    2024 arXiv

  38. [46]

    HereM= P ijaij is preferred since it works for both, directed and non-directed networks

    Note that, usually, 2M= P ijaij is used for nondirected networks. HereM= P ijaij is preferred since it works for both, directed and non-directed networks

  39. [47]

    Pastor-Satorras, C

    R. Pastor-Satorras, C. Castellano, P. Van Mieghem, and A. Vespignani, Epidemic processes in complex networks, Rev. Mod. Phys.87, 925 (2015)

    2015

  40. [48]

    Klein, N

    W. Klein, N. Lubbers, K. K. L. Liu, T. Khouw, and H. Gould, Mean-field theory of an asset exchange model with economic growth and wealth distribution, Phys. Rev. E104, 014151 (2021)

    2021

  41. [49]

    Dall’Asta, P

    L. Dall’Asta, P. Pin, and A. Ramezanpour, Statistical mechanics of maximal independent sets, Phys. Rev. E 80, 061136 (2009)

    2009

  42. [50]

    Janson, T

    S. Janson, T. Luczak, and A. Rucinski,Random Graphs, Wiley Series in Discrete Mathematics and Optimization (John Wiley & Sons, Nashville, TN, 2000)

    2000

  43. [51]

    J. I. Perotti, Analysis of the inference of ratings and rank- ings in complex networks using discrete exterior calcu- lus on higher-order networks, Phys. Rev. E111, 034306 (2025)

    2025

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.