Pith. sign in

REVIEW 4 major objections 4 minor 25 references

The principle of least action for random graphs

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read The paper aims to show that the evolution of a simple random graph can be described by a least-action principle, with the graph's vertex degrees treated as a scalar field.

desk verdict The Gaussian distribution of action values is plausible, but the identification of the mode with Hamilton's principle is a non-sequitur because independent resampling at each step kills any notion of nearby paths. read the letter →

arxiv 2507.01468 v1 pith:E3Y7XYDC submitted 2025-07-02 cond-mat.dis-nn gr-qchep-latquant-ph

Ioannis Kleftogiannis , Ilias Amanatidis This is my paper
classification cond-mat.dis-nngr-qchep-latquant-ph
keywords randomgraphsdegreefieldgraphLaplacianDirichletenergyleastactionprincipleGaussiandistributionemergentspacetimeregularandirregular
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 aims to show that a least-action principle can be derived for the evolution of simple random graphs. The idea is to treat the number of neighbors at each vertex — the degree — as a scalar field, define a Lagrangian through the graph Laplacian, and call the Dirichlet energy of that field the action S. The paper then builds an ensemble of evolution paths in which one edge is removed per time step and, at each step, a random graph configuration is drawn with equal probability. It reports that the action over these paths is Gaussian for sufficiently dense graphs, that the most probable action values are exactly those with zero spacing between action values ($\Delta S=0$), and that those paths pass through graph structures whose degree variance sits midway between regular and irregular extremes. If correct, this gives a concrete mechanism by which a physical action principle could emerge from purely combinatorial randomness.

What carries the argument

The key object is the degree field $\phi_i = d(i) - 2m/n$ at each vertex, together with the graph Laplacian $L = D - A$. The Lagrangian per vertex is $L_i = \sum_j \phi_i L_{ij} \phi_j$, and summing it over vertices gives $S = \frac{1}{2}\sum_{ij} A_{ij}(\phi_i - \phi_j)^2$, i.e. the Dirichlet energy of the degree field. The argument is carried by the random evolution mechanism: at each time step the graph configuration (run) is chosen uniformly among all graphs with the current $n$ and $m$, so a full evolution path is a sequence of independently drawn runs; the path action $S_p$ is the sum of the actions of the runs on that path. The statistics of $S_p$ over all paths, together with the degeneracy $D(S_p)$ of paths sharing the same action, determine where $\Delta S=0$ and hence which paths the paper identifies as classical.

What would settle it

Evolve a graph by literally removing one randomly chosen edge from the previous configuration at each time step, with the same $n$, $m$, $T$, and number of runs, and compute the path-action distribution; if its maximum does not sit at zero spacing $\Delta S=0$ or the distribution is not Gaussian, the central claim fails. A cheaper check is to test a small graph exactly: under contiguous edge removal, the most probable action path should still have degree variance at the midpoint between the regular and irregular extremes.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the action $S$ built from the degree field $\phi_i = d(i) - 2m/n$ is a graph-theoretic quantity (the Dirichlet energy of the degree field with respect to the graph Laplacian), and its distribution over random evolution paths is normal; the mode of that distribution occurs where $\Delta S=0$, which the paper identifies with Hamilton's least-action principle. A second, coupled finding is that the paths realizing the most probable action values have degree variance $\sigma_p^2(d(i))$ equal to the midpoint between the minimum variance (regular graphs) and the maximum variance (irregular graphs), so the 'classical' graph evolution is through balanced regular-irregular configurations. The paper also notes that adding a mass term of the form $\frac{1}{2}(m_v \phi_i)^2$ shifts the action distributions without changing their Gaussian form.

Load-bearing premise

The load-bearing premise is that each evolution step is an independent uniform draw over all configurations with the current $n$ and $m$, so the path action is a sum over unrelated graphs rather than a single graph actually losing one edge at a time.

Editorial extensions

If this is right

  • The Gaussian character of $P(S_p)$ means that for dense graphs the action behaves like a normal random variable; the width grows with edge count until the graph approaches uniformity.
  • The identification of the mode of $P(S_p)$ with $\Delta S=0$ turns the classical least-action principle into a statistical statement: the classically followed path is the one with the largest number of equally-actioned alternatives.
  • Paths with the most probable action have degree variance midway between regular and irregular graph values, implying the classical evolution is through intermediate, balanced structures rather than toward either regular lattices or maximally disordered graphs.
  • Adding a mass term for the degree field preserves the Gaussian form and merely shifts the distributions, so the qualitative conclusions are robust to that modification.
  • Interpreting equal-probability path sampling with phase factors $e^{iS/\hbar}$ gives a path-integral reading of random graph evolution, opening a discrete model for emergent quantum spacetime.

Reading between the lines

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

  • The paper's evolution mechanism draws an independent configuration at every time step; a natural testbed is to replace this with genuine edge-removal from the current configuration, which would make $S_p$ a random walk on graph space and could change or sharpen the least-action statement.
  • Because the action is the Dirichlet energy of the degree field, its ensemble distribution may be derivable from the spectrum of the graph Laplacian; if so, the Gaussian result would follow from spectral averaging rather than simulation.
  • The balanced-regular-irregular finding suggests a quantifiable target: the most probable classically followed configurations have degree variance at the midpoint of possible variances. One could test whether a dynamically edge-deleting graph drifts toward that midpoint as it evolves.
  • Applying the same degree-field action construction to other graph ensembles (scale-free, spatial, or small-world) would reveal whether Gaussian action statistics and least-action maxima are generic or specific to uniform random graphs.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper assigns to each vertex of a uniform random graph a scalar degree field φ_i = d(i) − 2m/n, defines a graph Lagrangian via the graph Laplacian, and takes the action S of a configuration to be the Dirichlet energy S = (1/2) Σ_ij A_ij(φ_i−φ_j)^2. It then introduces an evolution in which one edge is removed per fundamental time step, but at each step the graph configuration is chosen independently and uniformly among all configurations with the current number of edges. The path action S_p is the sum of the actions of the configurations along such a path. For n = 3000, Ω_ts = 20, and T = 3, the paper reports that the histogram of S_p is Gaussian, identifies the mode of P(S_p) with Hamilton's least-action principle via the condition ΔS_p = 0, and concludes that the most probable paths have a balanced regular-irregular degree structure.

Significance. If the central identification were valid, the paper would offer a concrete connection between random graph statistics and least-action/path-integral ideas, with potential relevance to emergent-spacetime models. The manuscript is transparent about its setup and provides explicit formulas for the action, the constraint Σ_i φ_i = 0, and the path sum; the Gaussian observations, if properly supported, would be a falsifiable statistical statement. However, the paper has no machine-checked proofs and, more importantly, the leap from the mode of P(S_p) to a variational least-action principle is not justified. The numerical evidence is thin and the balanced-structure conclusion rests on an underspecified comparison. In its present form the paper is better read as a suggestive numerical observation than as a demonstration of Hamilton's principle for random graphs.

major comments (4)
  1. [Section 'We adopt an evolution mechanism', Eq. (15), Fig. 2] The evolution path is not a sequence of graphs related by edge deletion: at each step 'one of the possible configurations (runs) of the graph, determined by n and m, is chosen at random', so G_{t+1} is independent of G_t and is not obtained from G_t by removing one of its edges. Consequently S_p in Eq. (15) is a sum of independent random variables, one for each edge count, and there is no notion of a small variation of a path or a variational derivative δS/δ(path). The statement ΔS_p = 0 at the mode of P(S_p) describes coincidences of equal sums, i.e., degeneracy, not stationarity of an action functional. The identification of this degeneracy with Hamilton's principle is therefore a non-sequitur and is the central scientific claim of the paper.
  2. [Paragraph beginning 'Classically based on the principle of least action' and Eq. (16)] The assertion that 'the differences ΔS_p between the values of S_p become minimum at the maxima of the distributions P(S_p)' is asserted rather than derived. For the discrete distribution constructed here, P(S_p) = D(S_p)/N_p is by definition proportional to the degeneracy, so the mode is the value with maximal degeneracy; this is a combinatorial statement, not a stationarity condition. For a continuous distribution, a mode does not imply zero spacing of nearby action values. To connect the mode to least action one would need to define a space of paths, a notion of nearby paths, and show that the action is stationary there; none of these is present.
  3. [Fig. 3 and the paragraph describing the numerical setup] The numerical evidence consists of a single system size n = 3000, a fixed number of runs Ω_ts = 20, and T = 3 evolution steps, with Gaussian curves fitted to histograms and no error bars, goodness-of-fit tests, or scaling analysis. The claim that the distribution 'approaches the normal (Gaussian) form as the graph becomes denser' is not supported by this dataset: there is no variation of n, T, or Ω_ts, and no demonstration that the Gaussian shape is robust rather than a finite-ensemble artifact. Given that the least-action identification relies on the detailed shape of P(S_p), this thinness is load-bearing.
  4. [Inset of Fig. 3b and Eqs. (18)–(19)] The balanced regular-irregular conclusion is based on comparing μ = (σ²_min + σ²_max)/2 with the average variance over paths having 'the highest S_p degeneracy'. With only 20 runs per step and T = 3, the number of paths in the highest-degeneracy class is small, and the coincidence of the two curves is not accompanied by any statistical significance measure or dependence on n, T, or Ω_ts. Moreover, Eq. (18) defines a midpoint of the observed range, not a median, so the terminology 'medium value' and the abstract's 'median-like variance' are misleading; this is not merely a typo because the choice of central value affects the claimed balance.
minor comments (4)
  1. [Numerical setup, Eq. (13)] The text states that for T = 3 and Ω_ts = 20 the number of paths is N_p = 20^4 = 204; the correct value is 160,000. Please correct the typographical rendering of the exponent and verify all derived counts.
  2. [Eq. (18) and text around Fig. 3b] The quantity μ is called the 'medium value' and later the 'median', but the formula is the midpoint between the minimum and maximum variances. Please choose consistent terminology and clarify whether the intended object is the midrange, the median, or something else.
  3. [Abstract and text near Eq. (15)] The phrase 'spacing between the values of S becomes zero ΔS = 0' is undefined; spacing among discrete observed values is always zero for equal values, and this is not the same as a variational stationarity condition. Please define ΔS precisely.
  4. [Introduction, use of t_f] The 'fundamental quantum of time t_f' is introduced but never used in any equation or calculation; its physical meaning and any dependence of the results on t_f should be stated or the concept should be removed.

Circularity Check

2 steps flagged · score 8.0 of 10

Mode = least-action path is baked into the definitions: P=D/N makes ΔS=0 a synonym for degeneracy, and the 'classical path' assumption is then restated as the paper's main result.

  1. self definitional [Abstract; Section 'Classically based on the principle of least action' (Eq. 16)]
    "The maximum of the probability distribution of the action corresponds to graph configurations whose spacing between the values of S becomes zero ΔS=0, corresponding to the least-action (Hamilton) principle... The number of paths in each set can be thought as the degeneracy of the set D(Sp). Then the probability of Sp is simply P (Sp) = D(Sp) / Np."

    P(Sp) is defined as D(Sp)/Np, so maximizing P is equivalent by definition to maximizing the number of paths with identical Sp. The paper calls that equality 'ΔS=0'; hence the asserted correspondence between the mode and ΔS=0 is a restatement of the definition of P. The further identification with Hamilton's principle (stationarity under small path variations) is an additional physical assumption, not a consequence of the computation.

  2. self definitional [Section 'Classically based on the principle of least action'; Conclusion]
    "Classically we can assume that the system follows paths corresponding to the highest probability P (Sp), at the maxima of the probability distributions shown in Fig. 3a... To conclude we have shown how the least action principle manifests in random graphs, by interpreting the degree at each vertex as a scalar field."

    The main conclusion is exactly this assumption repeated. No variational argument connects the mode of P(Sp) to stationarity of the action, so labeling the most probable paths 'least-action' makes the central physical claim an input rather than a derived output.

full rationale

The central circular step is the identification of the mode of P(Sp) with ΔS=0 and the least-action principle. Because Eq. 16 defines P(Sp) as D(Sp)/Np, the maximum of P is, by construction, the action value with the largest number of exactly equal paths; calling that equality 'ΔS=0' is a tautology. The paper then inserts the classical principle as an assumption ('we can assume') and presents it as the result. The Gaussian statistics of Sp and the variance comparison are empirical and not themselves circular, but they do not rescue the physical conclusion. Self-citations [1,2] are background for emergent dimension and are not load-bearing for the main claim. The resampling evolution mechanism (independent random graph at each step) further undermines the variational reading of 'small deviations,' but that is a correctness concern rather than an additional circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 2 invented entities

The simulation constants (n=3000, Ω=20, T=3) are hand-picked for computational convenience; Gaussian fit parameters are fitted to histograms. The axioms show that the central result rests on the independent-draw evolution model and on the asserted equivalence between the mode of P(S_p) and least action.

free parameters (4)
  • Gaussian mean and standard deviation of P(S_p) = not reported
    Gaussian curves in Fig. 3a are fitted to action histograms; the Gaussian form is then presented as a finding, but the parameters are fit, not predicted.
  • Number of runs per time step Ω_ts = 20
    Chosen for computational efficiency; the distribution and degeneracy structure depend on it.
  • Number of evolution steps T = 3
    Chosen for computational efficiency; with T=3 the Gaussian claim rests on only four summed action values per path.
  • Number of vertices n = 3000
    Chosen for the simulation; no scaling analysis over n is provided.
assumptions (5)
  • domain assumption All graph configurations with fixed n and m are equally probable (uniform random graph ensemble).
    Used in the definition of runs and path probabilities; standard for G(n,m), cited from refs [14-20].
  • standard math The degree deviations φ_i = d(i) - 2m/n sum to zero at each configuration.
    Follows from Σ d(i) = 2m, Eqs. (2)-(3); used to define the field but cancels in the action Eq. (10).
  • ad hoc to paper The physical action of a graph is the Dirichlet energy S = (1/2) Σ A_ij(φ_i - φ_j)^2.
    Chosen by analogy with lattice QFT; no variational principle or physical dynamics motivates this specific choice (Eq. 10).
  • ad hoc to paper At each time step the graph is chosen independently from all configurations with the current edge count.
    This is the evolution mechanism described around Fig. 2; it makes path actions sums of independent random variables rather than actions of a connected edge-removal sequence.
  • ad hoc to paper The maximum of P(S_p) corresponds to the least-action (Hamilton) principle.
    Asserted in the text ('for a statistical physical system the differences ΔS_p become minimum at the maxima of the distributions'); no derivation connects the mode of a random ensemble to stationarity of an action.
invented entities (2)
  • Degree field φ_i
    purpose: To cast graph degrees in the language of lattice quantum field theory.
    It is a mathematical rewriting of the degree (Eq. 1), not a new physical entity; it has no predictions beyond the graph degrees themselves.
  • Fundamental quantum of time t_f
    purpose: Unit for the edge-removal evolution steps; mentioned but never used numerically.
    Introduces a time scale without any falsifiable handle or observable prediction.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The principle of least action for random graphs." pith.science (2026). https://pith.science/paper/E3Y7XYDC

@misc{pith2026250701468,
  author       = {Pith},
  title        = {Pith review of: The principle of least action for random graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E3Y7XYDC}},
  note         = {Machine review of arXiv:2507.01468}
}
abstract

We study the statistical properties of the physical action $S$ for random graphs, by treating the number of neighbors at each vertex of the graph (degree), as a scalar field. For each configuration (run) of the graph we calculate the Lagrangian of the degree field by using a lattice quantum field theory(LQFT) approach. Then the corresponding action is calculated by integrating the Lagrangian over all the vertices of the graph. We implement an evolution mechanism for the graph by removing one edge per a fundamental quantum of time, resulting in different evolution paths based on the run that is chosen at each evolution step. We calculate the action along each of these evolution paths, which allows us to calculate the probability distribution of $S$. We find that the distribution approaches the normal(Gaussian) form as the graph becomes denser, by adding more edges between its vertices. The maximum of the probability distribution of the action corresponds to graph configurations whose spacing between the values of $S$ becomes zero $\Delta S=0$, corresponding to the least-action (Hamilton) principle, which gives the path that the physical system follows classically. In addition, we calculate the fluctuations(variance) of the degree field showing that the graph configurations corresponding to the maximum probability of $S$, which follow the Hamilton's principle, have a balanced structure between regular and irregular graphs.

Figures

Figures reproduced from arXiv: 2507.01468 by the authors.

Figure 1
Figure 1. Random graph structures for n = 3000 number of vertices and various numbers of edges m distributed among the vertices. In all cases a giant graph component appears containing most of the vertices and edges, along with many small disconnected components with tree-like structures. The emergent spatial dimension D of the giant component increases as the graph becomes denser with increas￾ing m. For the sparser graph wit… view at source ↗
Figure 2
Figure 2. The evolution mechanism of the graph represented as a [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. a)The probability distributions of the action [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

25 extracted references · 18 canonical work pages

  1. [1]

    Kleftogiannis and I

    I. Kleftogiannis and I. Amanatidis, Phys. Rev. E 105, 024141 (2022)

    work page 2022

  2. [2]

    Kleftogiannis and I

    I. Kleftogiannis and I. Amanatidis, arXiv:2210.00963 (2022)

    arXiv 2022

  3. [3]

    Quantum Grav

    Class. Quantum Grav. 28, 153002 (2011)

    work page 2011

  4. [4]

    Rovelli, Living Rev

    C. Rovelli, Living Rev. Relativ. 11, 5 (2008)

    work page 2008

  5. [5]

    Bombelli, J

    L. Bombelli, J. Lee, D. Meyer and R. D. Sorkin, Phys. Rev. Lett. 59, 521-524 (1987)

    work page 1987

  6. [6]

    Dowker, Annals of the New York Academy of Sciences1326, 18-25, (2014)

    F. Dowker, Annals of the New York Academy of Sciences1326, 18-25, (2014)

    2014

  7. [7]

    Dowker and S

    F. Dowker and S. Zalel, C R Physique 1, 8:246-253 (2017)

    work page 2017

  8. [8]

    Surya, Living Rev

    S. Surya, Living Rev. Rel. 22, 5 (2019), arXiv:1903.11544 [gr- qc]

    arXiv 2019

Show all 25 references
  1. [9]

    Wolfram, Complex Systems 29, (2) pp

    S. Wolfram, Complex Systems 29, (2) pp. 107-536 (2020)

    2020

  2. [10]

    Gorard, Complex Systems, 29, (2) pp

    J. Gorard, Complex Systems, 29, (2) pp. 599-654 (2020)

    2020

  3. [11]

    Konopka, F

    T. Konopka, F. Markopoulou and S. Severini, Phys. Rev. D 77, 104029 (2008)

    2008

  4. [12]

    Trugenberger, J

    C.A. Trugenberger, J. High Energ. Phys. 45 (2017)

    2017

  5. [13]

    Kelly, C

    C. Kelly, C. Trugenberger, and F. Biancalana, Class. Quantum Grav. 38, 075008 (2021)

    2021

  6. [14]

    Erd ˝os, T

    P. Erd ˝os, T. Gallai, ”Gr ´afokel˝o´ırt foksz ´am´u pontokkal”, Matematikai Lapok, 11: 264-274 (1960)

    1960

  7. [15]

    Aigner, E

    M. Aigner, E. Triesch, Discrete Math. 136, 3-20 (1994)

    1994

  8. [16]

    I. J. Farkas, I. Der ´enyi, A.-L. Barab ´asi, and T. Vicsek, Phys. Rev. E 64, 026704 (2001)

    2001

  9. [17]

    M. E. J. Newman Networks: An Introduction Oxford Univer- sity Press, Oxford (2010)

    2010

  10. [18]

    Frieze, M

    A. Frieze, M. Karonski, Introduction to random graphs. Cam- bridge University Press (2015)

    2015

  11. [19]

    Berg and M

    J. Berg and M. Lassig., Phys. Rev. Lett. 89, 228701 (2002)

    2002

  12. [20]

    Mizutaka and T

    S. Mizutaka and T. Hasegawa, Journal of Physics: Complexity 1, 035007 (2020)

    2020

  13. [21]

    Dirac, Physikalische Zeitschrift der Sowjetunion, 5 Band,3, Heft 1 (1933)

    P.A.M. Dirac, Physikalische Zeitschrift der Sowjetunion, 5 Band,3, Heft 1 (1933)

    1933

  14. [22]

    Feynman, R. P., Rev. Mod. Phys. 20, 367 (1948)

    1948

  15. [23]

    R. P. Feynman, The principle of least action in quantum me- chanics. In Feynman’s Thesis—A New Approach to Quantum Theory,1–69 (World Scientific, 2005)

    2005

  16. [24]

    Infeld L, Equations of motion in general relativity theory and the action principle, Rev Mod Phys 29:398–411 (1957)

    1957

  17. [25]

    Schwinger, Quantum Kinematics and Dynamics (1st edition: Benjamin, 1970; 2nd edition: Addison-Wesley, 1991; 3rd edi- tion: Perseus Books Group, 2000)

    J. Schwinger, Quantum Kinematics and Dynamics (1st edition: Benjamin, 1970; 2nd edition: Addison-Wesley, 1991; 3rd edi- tion: Perseus Books Group, 2000)

    1970

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.