arxiv: 2604.14065 · v1 · submitted 2026-04-15 · ⚛️ physics.soc-ph

Recognition: unknown

Nonmonotonic percolation threshold in correlated networks and hypergraphs

L. D. Valdez , C. E. La Rocca

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Pith reviewed 2026-05-10 11:50 UTC · model grok-4.3

classification ⚛️ physics.soc-ph

keywords percolation thresholdassortativitynetwork robustnessdegree correlationshypergraphsconfiguration modelrandom failuresnonmonotonic behavior

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The pith

Moderately disassortative networks can be more fragile to random failures than either strongly disassortative or uncorrelated ones.

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

The paper examines how assortative and disassortative mixing affects the robustness of networks and hypergraphs to random node failures. It shows that the percolation threshold does not vary monotonically with the Pearson assortativity coefficient. In specific regions of parameter space, networks with moderate disassortativity have a higher threshold than those with either strong disassortativity or no correlation at all. This holds for trimodal, Poisson, and power-law degree distributions. The analysis extends to hypergraphs, where positive correlations between node hyperdegree and hyperedge cardinality make the structure more fragile than negative correlations, again with nonmonotonic dependence on assortativity.

Core claim

Using the generating function technique and stochastic simulations on configuration-model networks, the authors establish that the percolation threshold pc is a non-monotonic function of the Pearson assortativity coefficient r. Moderately disassortative networks can exhibit larger pc than either uncorrelated networks or strongly disassortative ones. This nonmonotonicity is observed for trimodal degree distributions as well as Poisson and power-law cases. For hypergraphs with correlations between node hyperdegree and hyperedge cardinality, positively correlated hypergraphs are more fragile than negatively correlated ones, and the relationship between r and pc remains nonmonotonic such that an

What carries the argument

Generating function formalism on configuration models with enforced degree sequences and pairwise correlations parameterized by the assortativity coefficient r.

Load-bearing premise

The configuration model procedure that enforces exact degree sequences and pairwise correlations is sufficient to determine percolation behavior, without higher-order structural effects altering the threshold.

What would settle it

An exact calculation or set of Monte Carlo simulations on a fixed trimodal degree sequence at r = 0, an intermediate negative r, and r near -1, checking whether pc peaks at the intermediate value.

Figures

Figures reproduced from arXiv: 2604.14065 by C. E. La Rocca, L. D. Valdez.

**Figure 2.** Figure 2: FIG. 2. Panel (a): Relative size of the giant component, [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Snapshots of networks illustrating that networks with [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Panel (a): In the main panel, we plot [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Critical percolation threshold [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Panel (a): Heatmap of the degree–cardinality correlation coefficient [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. Panel (a): Critical threshold [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8. Critical percolation threshold [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9. Heatmap of the critical percolation threshold [PITH_FULL_IMAGE:figures/full_fig_p031_9.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10. Heatmap of the critical percolation threshold [PITH_FULL_IMAGE:figures/full_fig_p033_10.png] view at source ↗

**Figure 11.** Figure 11: FIG. 11. Panel (a): In the main panel, we plot [PITH_FULL_IMAGE:figures/full_fig_p034_11.png] view at source ↗

read the original abstract

We study the effect of assortative and disassortative mixing on the robustness of networks under random node failures. For ordinary (dyadic) networks, by using the generating function technique and stochastic simulations, we show that the relationship between the Pearson assortativity coefficient $r$ and the percolation threshold $p_c$ is not always monotonic. More specifically, in certain regions of the parameter space of our model, moderately disassortative networks can be more fragile than either strongly disassortative or uncorrelated networks. We observe this nonmonotonic behavior for trimodal networks as well as for networks with Poisson and power-law degree distributions. We then extend our analysis to hypergraphs with correlations between node hyperdegree and hyperedge cardinality. For this case, we find that positively correlated hypergraphs tend to be more fragile than negatively correlated ones. Additionally, as in the dyadic case, the relationship between $r$ and $p_c$ is nonmonotonic, and the most fragile configuration does not correspond to the most assortative hypergraph.

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.

Desk Editor's Note private letter to a colleague

The nonmonotonic pc(r) result holds up in their generating-function and simulation checks for several degree distributions and the hypergraph case, but the configuration-model correlation step still needs explicit validation against induced higher-order structure.

read the letter

The main thing to know is that this paper finds the percolation threshold pc is not monotonic in the Pearson assortativity r for certain networks. Moderately disassortative mixing can produce higher pc than either strong disassortativity or zero correlation, and the same pattern appears when they move to hypergraphs with correlations between hyperdegree and edge size. They show this for trimodal, Poisson, and power-law degrees using both analytic generating functions and direct simulations on the generated graphs. That nonmonotonic window is the concrete new observation, and extending the setup to hypergraphs is a clear step beyond the usual dyadic case. They also note that the most fragile point does not always sit at the highest assortativity, which is useful to flag for robustness modeling. The agreement between the closed equations and the stochastic runs on multiple distributions gives the claim some internal support. Credit for not limiting the test to one degree sequence. The soft spot is the correlation prescription itself. They enforce exact degree sequences and pairwise correlations through a configuration-model-style construction, then close the percolation equations under the assumption that no further structural features matter. If that construction systematically adds clustering, motif biases, or altered conditional degree distributions in the parameter windows where nonmonotonicity appears, both the analytics and the matching simulations would register the artifact. The provided abstract and stress-test note do not report measurements of excess clustering or nearest-neighbor degree correlations conditioned on r, so it remains possible that the fragility peak is partly an artifact of the generator rather than a pure effect of the target r. If the full manuscript includes those diagnostics and shows the extra structure is negligible, the result strengthens; otherwise the central claim needs tighter bounds. This is for readers who already work on percolation and robustness in networks with mixing patterns. Someone building models for infrastructure or epidemic spreading on correlated systems will pick up the warning about moderate disassortativity and the hypergraph extension. It is specific enough and internally consistent enough to deserve a serious referee, even if the review should focus on validating the correlation model against higher-order effects. I would send it to review rather than desk reject.

Referee Report

3 major / 3 minor

Summary. The manuscript uses generating-function analysis and stochastic simulations to examine how degree correlations affect the percolation threshold pc under random node removal in networks. It reports that pc is a non-monotonic function of the Pearson assortativity coefficient r for trimodal, Poisson, and power-law degree distributions, with moderately disassortative networks sometimes more fragile than either strongly disassortative or uncorrelated ones. The analysis is extended to hypergraphs with correlations between node hyperdegree and hyperedge cardinality, where positive correlations increase fragility and pc(r) again exhibits nonmonotonicity.

Significance. If the central claim holds, the result is significant because it shows that the intuitive expectation of monotonic dependence of robustness on assortativity does not always apply, with implications for the design and analysis of resilient networked systems. The analytic generating-function approach is cross-validated against simulations on multiple degree distributions, which is a methodological strength. The hypergraph extension broadens the scope beyond ordinary graphs.

major comments (3)

[Model construction and correlation enforcement] The configuration-model-like procedure that enforces exact degree sequences and pairwise correlations (described in the model-construction section) may systematically induce higher-order structural features such as excess clustering or non-trivial conditional degree distributions beyond nearest neighbors. The generating-function closure assumes these effects are negligible, yet the paper provides no quantitative check (e.g., measured clustering coefficient or second-neighbor degree correlations) in the parameter windows where nonmonotonic pc(r) appears. Because both the analytic pc and the simulations are performed on the same ensembles, agreement between them does not rule out a shared artifact; this issue is load-bearing for the claim that the nonmonotonicity arises purely from the prescribed pairwise correlations.
[Results for Poisson and power-law degree distributions] For the Poisson and power-law cases, the manuscript states that nonmonotonicity occurs “in certain regions of the parameter space,” but does not specify the ranges of mean degree, variance, or correlation strength at which the minimum in pc(r) is observed, nor does it report simulation error bars or the number of realizations. Without these details it is difficult to assess whether the reported nonmonotonicity is statistically robust or sensitive to finite-size effects.
[Hypergraph analysis] In the hypergraph extension, the same configuration-model construction is used to impose correlations between hyperdegree and hyperedge cardinality. The manuscript should similarly verify that higher-order hypergraph motifs (e.g., overlapping hyperedges or degree-dependent local densities) remain negligible in the regimes where nonmonotonic pc(r) and the positive-correlation fragility effect are reported.

minor comments (3)

[Abstract] The abstract would benefit from a single sentence indicating the specific degree distributions and the qualitative location of the nonmonotonic regime (e.g., “for moderate negative r in trimodal networks with mean degree 3”).
[Figures and simulation details] All simulation figures should include error bars or state the number of independent realizations used; the current presentation leaves the statistical reliability of the curves unclear.
[Hypergraph notation] Notation for the assortativity coefficient r in the hypergraph setting should be explicitly defined, as the standard Pearson formula for dyadic networks does not directly carry over.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below and indicate the revisions we will make.

read point-by-point responses

Referee: [Model construction and correlation enforcement] The configuration-model-like procedure that enforces exact degree sequences and pairwise correlations (described in the model-construction section) may systematically induce higher-order structural features such as excess clustering or non-trivial conditional degree distributions beyond nearest neighbors. The generating-function closure assumes these effects are negligible, yet the paper provides no quantitative check (e.g., measured clustering coefficient or second-neighbor degree correlations) in the parameter windows where nonmonotonic pc(r) appears. Because both the analytic pc and the simulations are performed on the same ensembles, agreement between them does not rule out a shared artifact; this issue is load-bearing for the claim that the nonmonotonicity arises purely from the prescribed pairwise correlations.

Authors: We thank the referee for this important observation. The configuration model with fixed degrees and pairwise correlations is constructed to suppress uncontrolled higher-order features, and the generating-function equations close exactly on the prescribed joint degree distribution. Nevertheless, we agree that explicit verification is warranted. In the revised manuscript we will add quantitative checks—specifically the global clustering coefficient and the average degree correlation between second neighbors—computed directly on the generated ensembles in the parameter windows where nonmonotonic pc(r) is reported. These additional diagnostics will confirm that any residual higher-order structure remains small and does not account for the observed nonmonotonicity. revision: yes
Referee: [Results for Poisson and power-law degree distributions] For the Poisson and power-law cases, the manuscript states that nonmonotonicity occurs “in certain regions of the parameter space,” but does not specify the ranges of mean degree, variance, or correlation strength at which the minimum in pc(r) is observed, nor does it report simulation error bars or the number of realizations. Without these details it is difficult to assess whether the reported nonmonotonicity is statistically robust or sensitive to finite-size effects.

Authors: We agree that the lack of explicit parameter ranges and statistical details limits the reader’s ability to evaluate robustness. In the revised manuscript we will (i) state the precise intervals of mean degree, degree variance, and assortativity coefficient r over which the minimum in pc(r) appears for both the Poisson and power-law families, (ii) include error bars obtained from the stochastic simulations, and (iii) report the number of independent realizations used for each data point. These additions will make the statistical reliability of the nonmonotonicity transparent. revision: yes
Referee: [Hypergraph analysis] In the hypergraph extension, the same configuration-model construction is used to impose correlations between hyperdegree and hyperedge cardinality. The manuscript should similarly verify that higher-order hypergraph motifs (e.g., overlapping hyperedges or degree-dependent local densities) remain negligible in the regimes where nonmonotonic pc(r) and the positive-correlation fragility effect are reported.

Authors: We accept the referee’s parallel concern for the hypergraph setting. In the revision we will extend the same style of diagnostic analysis to the hypergraph ensembles: we will measure the frequency of overlapping hyperedges and any systematic dependence of local hyperedge density on node hyperdegree, restricting the checks to the correlation strengths and hyperdegree distributions where nonmonotonic pc(r) and the positive-correlation fragility are observed. These results will be reported alongside the existing hypergraph percolation curves. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained with no reduction to inputs by construction

full rationale

The paper derives the percolation threshold pc via the standard generating-function closure applied to an explicit configuration-model ensemble whose degree correlations are controlled by free parameters; the assortativity coefficient r is computed directly from those same parameters. Varying the correlation parameters and solving the resulting equations produces the reported nonmonotonic pc(r) curve. This is an ordinary forward calculation, not a fit, not a renaming of a known result, and not dependent on any self-citation chain for its central claim. Stochastic simulations on the identical ensembles serve as an independent numerical check rather than a circular confirmation. No step equates pc to r by definition or imports a uniqueness theorem from prior work by the same authors.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The claim rests on the standard configuration-model assumption for generating correlated networks and on the validity of the generating-function closure for percolation on those networks; no new entities are postulated.

free parameters (1)

assortativity strength parameter
The model introduces a tunable parameter that controls the degree of assortative or disassortative mixing; its value is chosen to explore different regions of parameter space.

axioms (2)

domain assumption The network is generated according to a configuration model with prescribed degree sequence and pairwise degree correlations.
This is the standard generative assumption invoked when applying generating functions to correlated networks.
domain assumption The generating-function equations close at the level of the degree distribution and the assortativity coefficient without requiring higher-order motif corrections.
The analytic treatment assumes the usual mean-field closure for percolation.

pith-pipeline@v0.9.0 · 5478 in / 1422 out tokens · 36847 ms · 2026-05-10T11:50:35.449770+00:00 · methodology

discussion (0)

Reference graph

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