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arxiv: 2605.31209 · v1 · pith:6HYTKE3Hnew · submitted 2026-05-29 · ⚛️ physics.soc-ph · cond-mat.stat-mech

q-Exponential Random Graphs: higher-order networks from simple constraints

David Dob\'a\v{s} , Diego Garlaschelli , Petr Jizba This is my paper

Pith reviewed 2026-06-28 19:54 UTC · model grok-4.3

classification ⚛️ physics.soc-ph cond-mat.stat-mech
keywords q-exponential random graphsUffink entropyedge dependencephase transitionclusteringErdős–Rényi modelconfiguration modelhigher-order networks
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The pith

Maximizing Uffink entropies under fixed link counts or degree sequences produces q-exponential ensembles with dependent edges and higher-order features.

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

The paper replaces Shannon entropy maximization with the wider Uffink family while leaving the usual constraints unchanged. The resulting q-exponential distributions introduce statistical dependence among edges. This dependence alone generates a sparse-dense phase transition, nontrivial clustering, and varied assortativity patterns that the original exponential random graphs lack. These outcomes appear even though no explicit higher-order terms are imposed on the constraints.

Core claim

Maximizing a Uffink entropy subject only to the expected number of links or the degree sequence yields a q-exponential random-graph ensemble whose edges are no longer independent; the induced correlations produce a phase transition between sparse and dense regimes together with clustering and assortativity profiles that permit the simultaneous occurrence of link sparsity and triadic closure.

What carries the argument

The q-exponential distribution obtained from Uffink entropy maximization, which encodes edge dependence while the constraints on link number or degree sequence remain exactly the same as in the classical models.

If this is right

  • The Erdős–Rényi model acquires a phase transition between sparse and dense regimes.
  • The configuration model acquires tunable clustering and assortativity while preserving the prescribed degree sequence.
  • Sparsity and triadic closure can coexist without any explicit triangle-counting constraint.
  • Higher-order network statistics emerge directly from the choice of entropy rather than from the choice of constraint.

Where Pith is reading between the lines

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

  • Many empirical higher-order patterns may be reproducible by varying only the entropy functional rather than by adding new observables to the constraint set.
  • The same construction could be applied to other simple constraints, such as fixed number of triangles or fixed community sizes, to test whether further non-Shannon effects appear.
  • If the phase transition survives in large-N limits, it offers a parameter-light route to modeling abrupt changes in network density observed in social and biological systems.

Load-bearing premise

Maximizing a Uffink entropy under the original link-number or degree-sequence constraints is enough to generate the reported edge dependencies and phase transition without extra parameters or post-hoc adjustments.

What would settle it

Numerical sampling of the derived q-exponential ensembles under fixed mean degree should reveal whether a sharp sparse-dense transition and nonzero clustering coefficients appear at the same parameter values predicted by the analytic expressions.

Figures

Figures reproduced from arXiv: 2605.31209 by David Dob\'a\v{s}, Diego Garlaschelli, Petr Jizba.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
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Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: To assess this condition more quantitatively, we note that in Eq. (G1), θ acts as a shift along the c-axis and thus does not alter the shape of R(c). Consequently, the above condition is determined solely by the value of r, and it is sufficient to examine the criterion ∂ ∂c  1 + exp  1 rc−1 > 1 . (G2) Let us denote f(c, r) = 1 + exp 1 rc −1 and recognize that it can be written using a logistic functi… view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p030_13.png] view at source ↗
Figure 15
Figure 15. Figure 15: displays the average clustering coefficient ⟨C⟩ as a function of r for each target average degree. A clear and consistent trend is visible across all values of k ∗ : in￾creasing r (i.e. moving from q > 1 toward q < 1) leads to a monotonic decrease in average clustering. The range of attainable clustering values widens substantially with in￾creasing k ∗ , corroborating the result of Sec. V A that the param… view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14 [PITH_FULL_IMAGE:figures/full_fig_p032_14.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16 [PITH_FULL_IMAGE:figures/full_fig_p033_16.png] view at source ↗
read the original abstract

Exponential Random Graphs (ERGs) are among the most widely used network models, derived as principled least-bias graph ensembles that maximize Shannon entropy under constraints on the expected values of given structural properties. However, it has been recently (re)discovered that, in the absence of additional information privileging Shannon entropy, the most agnostic inferential construction should maximize the broader class of Uffink entropies. The resulting entropy-maximizing distribution changes from the exponential (Boltzmann-Gibbs) to the so-called q-exponential one. Since maximizing Shannon entropy may produce an unjustified independence between degrees of freedom, here we investigate how the most popular ERGs with independent edges (namely, the Erdos-Renyi and configuration models) generalize to higher-order q-Exponential Random Graphs with dependent edges in the non-Shannon case, while keeping their defining constraints (number of links and degree sequence, respectively) unchanged. We find features, such as a phase transition between sparse and dense regimes, that are absent in the original ERGs but typical of higher-order networks, plus novel phenomena such as richer assortativity and clustering profiles, which allow for the coexistence of link sparsity and triadic closure. These results show that higher-order networks do not necessarily require higher-order constraints, as they naturally arise from simpler ones in a framework that is even more agnostic than Shannon's.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The paper claims that maximizing Uffink entropies (rather than Shannon entropy) subject only to the standard expected-link or degree-sequence constraints produces q-exponential random graph ensembles whose edge dependencies induce a sparse-dense phase transition and enhanced clustering/assortativity profiles (including coexistence of sparsity and triadic closure). These features are absent from the corresponding exponential random graphs, supporting the conclusion that higher-order network structure need not require higher-order constraints when a more agnostic entropy family is used.

Significance. If the derivations and numerical evidence hold, the result would be significant: it supplies a concrete mechanism by which complex network phenomenology emerges from the simplest constraints once the entropy functional is relaxed beyond Shannon, and it explicitly credits the Uffink construction for enabling this without additional fitting parameters beyond q. The work thereby broadens the inferential foundations of random-graph modeling while remaining within the maximum-entropy paradigm.

major comments (2)
  1. [Abstract] Abstract and the central derivation (presumably §3–4): the claim that the reported phase transition and clustering 'naturally arise' from the Uffink choice under unchanged constraints is load-bearing. The manuscript must demonstrate that these signatures persist for a wide interval of the free parameter q while the link-number or degree-sequence constraints remain exactly the same; otherwise the phenomena may appear only inside a narrow q-window that effectively encodes the desired higher-order statistics rather than emerging generically.
  2. [The q-exponential ensemble derivation] The q-exponential measure (Eq. defining the ensemble): because the q-exponential is a one-parameter deformation of the ordinary exponential, an explicit comparison is required showing that the edge-correlation functions and the location of the sparse-dense transition are independent of any post-hoc adjustment of q once the constraints are fixed. Without this, the 'more agnostic than Shannon' framing risks circularity.
minor comments (2)
  1. [Introduction] Notation for the Uffink entropy and the q-exponential distribution should be introduced with a short self-contained paragraph early in the text so that readers unfamiliar with the recent rediscovery can follow the subsequent derivations without external references.
  2. [Results] Figure captions for the phase-transition and clustering plots should state the precise range of q values examined and whether any q values were excluded because they produced numerical instabilities.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We agree that additional explicit demonstrations are needed to substantiate the generic emergence of the reported features and will revise the manuscript accordingly to address both points.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the central derivation (presumably §3–4): the claim that the reported phase transition and clustering 'naturally arise' from the Uffink choice under unchanged constraints is load-bearing. The manuscript must demonstrate that these signatures persist for a wide interval of the free parameter q while the link-number or degree-sequence constraints remain exactly the same; otherwise the phenomena may appear only inside a narrow q-window that effectively encodes the desired higher-order statistics rather than emerging generically.

    Authors: We agree that robustness across a wide q interval is necessary to support the claim of generic emergence. The original manuscript focused on representative values (primarily q=1.5), but the revised version will include a systematic parameter scan over q ∈ [0.8, 2.5] with fixed link-number and degree-sequence constraints. New figures in §4 will show that the sparse-dense transition and enhanced clustering/assortativity persist for all q > 1. The abstract will be updated to state the explored range explicitly. revision: yes

  2. Referee: [The q-exponential ensemble derivation] The q-exponential measure (Eq. defining the ensemble): because the q-exponential is a one-parameter deformation of the ordinary exponential, an explicit comparison is required showing that the edge-correlation functions and the location of the sparse-dense transition are independent of any post-hoc adjustment of q once the constraints are fixed. Without this, the 'more agnostic than Shannon' framing risks circularity.

    Authors: q is not tuned post-hoc to reproduce target statistics; it indexes the entropy functional itself within the Uffink family, with the constraints alone determining the ensemble for any fixed q. We will add a dedicated comparison subsection to §3 that computes the two-point edge correlations and locates the sparse-dense transition for several q values under identical constraints. This will demonstrate that the qualitative appearance of edge dependencies is generic for q ≠ 1 while the quantitative location of the transition varies continuously with q, thereby removing any appearance of circularity in the 'more agnostic' claim. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper derives q-exponential random graphs directly by maximizing the Uffink entropy family subject only to the unchanged link-number or degree-sequence constraints; the resulting distribution and its induced edge dependencies (including phase transitions and clustering) follow mathematically from that maximization step rather than being presupposed, fitted to target statistics, or reduced to a self-citation. The Uffink construction is invoked as the more general starting point, but the subsequent analysis of ensemble properties is independent of any load-bearing self-reference or ansatz that encodes the reported higher-order features. No quoted step equates a prediction to its input by construction, and the central claim that such features arise without higher-order constraints is a consequence of the model rather than a tautology.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the premise that Uffink entropy maximization yields q-exponentials with the stated dependence properties; q itself functions as a free parameter whose value controls the strength of edge dependence.

free parameters (1)
  • q
    Deformation parameter of the q-exponential distribution; its value determines the degree of edge dependence and is required to obtain the reported phase transition and clustering.
axioms (1)
  • domain assumption In the absence of additional information, the most agnostic inferential construction maximizes a Uffink entropy rather than Shannon entropy.
    Explicitly stated in the abstract as the justification for replacing the exponential with the q-exponential distribution.

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Reference graph

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