Percolation on branching simplicial and cell complexes and its relation to interdependent percolation

Ginestra Bianconi; Ivan Kryven; Robert M. Ziff

REVIEW 5 major objections 3 minor 49 references

Percolation on branching simplicial and cell complexes and its relation to interdependent percolation

T0 review · 5 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Branching cell complexes can have more than two percolation transitions.

desk verdict Solid exact mapping and intermediate-transition results, but the anomalous scaling claim rests on critical-point formulas that are internally inconsistent as printed. read the letter →

arxiv 1908.09392 v2 pith:F6DKKC4I submitted 2019-08-25 cond-mat.dis-nn cond-mat.stat-mechphysics.soc-ph

Ginestra Bianconi , Ivan Kryven , Robert M. Ziff This is my paper · ORCID

classification cond-mat.dis-nncond-mat.stat-mechphysics.soc-ph

keywords percolationbranchingsimplicialcomplexescellnon-amenablegraphsphasetransitionsmultiplexnetworksrenormalizationgroupfractalexponent

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

Link percolation on two-dimensional branching simplicial and cell complexes—networks built by repeatedly gluing triangles or polygons onto links—is governed by a single recursive equation whose fixed points can bifurcate more than once. The paper establishes that, in addition to the usual lower threshold at $p^*=0$ and upper threshold at $p_c$, a multimodal distribution of the number of polygons glued per link produces one or more intermediate phase transitions. At each intermediate transition both the percolation probability and the fractal exponent jump discontinuously, yet the largest connected component remains smaller than the full network volume on both sides. The paper also proves an exact mapping between this percolation problem and the mutually connected giant component in correlated multiplex networks, and a renormalization-group analysis shows that the upper transition can be discontinuous, of Berezinskii-Kosterlitz-Thouless type, or continuous with anomalous exponential scaling. A sympathetic reader would care because this unifies two previously separate sources of explosive percolation and predicts a much richer phase structure for network geometry.

What carries the argument

The central object is the recursive percolation map $T_{n+1}=1-(1-p)\sum_{k\ge 1}r_k(1-T_n^{m-1})^k$, encoding the probability that the two initial nodes stay connected after $n$ rounds of gluing and link removal. Its fixed-point equation $f(T)=0$ is studied with bifurcation theory; the type of bifurcation (transcritical, saddle-node, or pitchfork of order $s$) sets the upper-transition universality class, while multi-modal $r_k$ creates additional saddle-node bifurcations that give intermediate transitions. The fractal exponent is obtained from the largest eigenvalue $\lambda_n$ of the Jacobian of the generating-function recursion via $\psi_n=\ln\lambda_n/\ln[\langle k\rangle(m-1)]$, and the RG flow of the map near $(p_c,T_c)$ yields the order-parameter scaling. The exact mapping identifies $T$ with $1-S'$ (one minus the MCGC reach probability), $p$ with $1-\tilde p$, polygon side count $m$ with $\kappa+1$, and $r_k$ with the multiplex activity distribution $P(B)$, transferring known results from correlated multiplex networks.

What would settle it

Iterate the exact generating-function recursions (Eq. (38)) for a branching simplicial complex with $r_k=0.62\delta_{k,1}+0.07\delta_{k,2}+0.31\delta_{k,20}$ to very large $n$ (e.g., $n=10^4$) and compute $T$ and $\psi_n$ from Eq. (47). If no discontinuity appears near $p\approx 0.0395$ with $T<1$ and $\psi<1$ on both sides, the intermediate-transition claim fails. Separately, at a critical point of order $s=5$, plot $-\ln P_\infty$ versus $\ln(p-p_c)$ over several decades; the predicted slope is $\sigma=1/2$, so a cleanly different slope would falsify the anomalous scaling.

Watch

Extended reading notes

Core claim

The central claim is that branching cell complexes are not limited to the two percolation transitions previously known for non-amenable graphs. When the branching distribution $r_k$ has more than one mode, the fixed-point equation for the percolation probability develops additional hybrid (saddle-node) bifurcations below the upper threshold; at these, $T$ and the fractal exponent $\psi$ both jump while remaining below one, so the giant component stays sub-extensive. At the upper threshold the transition can take several universality classes depending on the bifurcation: transcritical bifurcations give a discontinuous transition, saddle-node bifurcations give a BKT singularity $P_\infty\sim\exp[-\alpha/(\Delta p)^{1/2}]$, and pitchfork critical points of order $s$ give continuous transitions with $P_\infty\sim\exp[-A/(\Delta p)^\sigma]$, $\sigma=(s-3)/(s-1)$, generalizing BKT to arbitrary $\sigma$. These results follow from analyzing the recursive map $T_{n+1}=1-(1-p)\sum_k r_k(1-T_n^{m-1})^k$ and from an exact mapping, for a modified construction, to the mutually connected giant component equation of multiplex networks with maximally correlated degrees.

Load-bearing premise

The load-bearing assumption is that, in the recursion for the component size $M_n$, the term that does not depend on earlier component sizes is exponentially smaller than the term that does; if both grew at the same exponential rate, the formula for the fractal exponent and the predicted universality classes would need revision.

Editorial extensions

If this is right

A multimodal branching distribution $r_k$ produces one or more intermediate discontinuous transitions at which $T$ and $\psi$ jump but remain below 1; the paper gives explicit phase diagrams for trimodal distributions on triangles.
The universality class of the upper percolation transition is controlled by the topology of the branching complex: discontinuous, BKT, and anomalous continuous transitions can all occur as $r_k$ varies.
At a critical point of order $s$, the percolation probability approaches its critical value as $|\Delta T|\sim|\Delta p|^{1/(s-1)}$, so the effective critical exponent can be tuned by the branching distribution.
Because of the exact mapping, any transition found for these branching cell complexes has a counterpart in correlated multiplex networks with perfect inter-layer degree correlation, and vice versa.
The lower percolation threshold is always $p^*=0$ in these complexes, so every intermediate transition occurs while the largest component is still sub-extensive.

Reading between the lines

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

The mapping suggests that the intermediate transitions in branching complexes are the geometric analogue of cascading-failure events in interdependent networks; if so, intuitions about fragility from multiplex percolation could transfer to designing or avoiding multi-stage failure in hierarchical materials.
The anomalous scaling family $P_\infty\sim\exp[-A/(\Delta p)^\sigma]$ with $\sigma=(s-3)/(s-1)$ interpolates between BKT and exponential singularities; this family may appear in other hierarchical or non-amenable geometries where an order-$s$ pitchfork bifurcation controls percolation.
These complexes are two-dimensional; the same recursive-equation approach might extend to higher-dimensional branching simplicial complexes, where new intermediate transitions or modified anomalous scaling could appear.
The paper's phase diagrams come from fixed-point analysis; a finite-size scaling theory for the intermediate jumps (how $M_n$ grows with $n$ exactly at $p_c^*$) is not developed and would be a natural next test of whether the discontinuities survive finite-size corrections.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

Here is my honest read of 1908.09392. Two things to know. First, the paper has real content: an exact equation-level mapping between link percolation on branching cell complexes and interdependent percolation in maximally correlated multiplex networks, plus a clear demonstration that multimodal branching distributions r_k produce intermediate hybrid transitions where both the percolation probability and the fractal exponent jump while remaining below one. The recursive equations are derived cleanly, no parameters are fitted, and the phase diagrams are self-consistent. Second, the headline new universality class, P_inf ~ exp[-A/(Delta p)^sigma] at higher-order critical points, rests on critical-point conditions that are internally inconsistent as printed.

The problem is concrete. For m=3, the tricritical condition obtained by setting f''(1)=0 is r1=4r2, but Eq. (25) prints r1=5r2. And Eq. (33), under any natural reading of the fraction bars, does not reproduce the ratios (8/11, 2/11, 1/11) that the paper itself uses for the order-4 critical point in Figs. 7 and 10. Since those examples are the evidence for the anomalous scaling, the paper's central claim is not yet backed by its own formulas. This looks like a fixable error in the closed-form parametrization rather than a flaw in the underlying RG expansion, which is set up correctly in terms of derivatives of f. But it has to be fixed, and the s=5,6 numerics rechecked, before the claim stands.

Two smaller concerns. The numerical verification is deterministic iteration of the recursion, not independent stochastic simulation; that is fine as a consistency check but weaker than a direct Monte Carlo test. And the assertion in Sec. VIII that the inhomogeneous term in Eq. (42) is subleading is stated without proof. I think that one is likely harmless for the leading exponential growth rate, but a sentence of justification would close the gap. The mapping to multiplex percolation uses Ref. [46] from the same group, but the relation is derived here rather than assumed, so the self-citation is not a problem.

Who benefits: anyone working on percolation in hierarchical or non-amenable networks, and the multiplex-percolation community will care about the formal bridge. The intermediate transitions and the mapping are solid contributions worth publishing on their own. I would send this to referees, with a clear request that the authors correct Eqs. (25) and (33) and re-examine Fig. 10 after doing so.

Best.

Referee Report

5 major / 3 minor

Summary. The paper studies link percolation on random branching simplicial and cell complexes, where each link is incident to k m-polygons drawn from a distribution r_k. The authors derive a one-dimensional recursion for the percolation probability T, show that its fixed-point equation is formally analogous to that of interdependent percolation in a correlated multiplex network, and use this mapping plus a renormalization-group treatment to characterize the possible phase transitions. The main claims are: (i) such complexes can display one or more intermediate hybrid transitions in addition to the lower and upper percolation thresholds, at which T and the fractal exponent ψ jump while remaining below one; and (ii) the upper transition can be discontinuous, BKT-like, or continuous with an anomalous singular dependence P∞ ∼ exp[−A/(Δp)^σ] with σ = (s−3)/(s−1) at a critical point of order s. The central recursive equations and the RG framework are clearly presented, and the numerical phase diagrams are self-consistent. However, the paper contains inconsistencies in the stated conditions for the higher-order critical points on which claim (ii) is built.

Significance. If the claims are correct, the paper identifies new universality classes in a solvable hierarchical network model and establishes a conceptually interesting bridge between percolation on branching cell complexes and interdependent percolation in multiplex networks. The derivation is self-contained: Eq. (10) follows from the model, no parameters are fitted to data, and the RG scaling calculation is explicit. The prediction of anomalous continuous transitions with exponent σ = (s−3)/(s−1) is a falsifiable statement and is a genuinely new result for this class of models, if it survives the corrections to the critical-point conditions. The numerical confirmation for s = 4 in Fig. 10 is encouraging, but the deviations reported for s = 5, 6 cannot be interpreted until the critical-manifold conditions are fixed and the numerical parameters are verified to lie on the true manifold.

major comments (5)

[Sec. IV, Eq. (19)] The condition for a second-order (transcritical) critical point at T_c = 1 is misstated. Differentiating f(T) = T − 1 + (1−p)R(1−T^{m−1}) at p_c = 1 − 1/[(m−1)r1] gives f''(1) = [2(m−1)r2 − (m−2)r1]/r1, so the condition f''(1) < 0 is r1 > 2(m−1)/(m−2) r2, not r1 > (2m−1)/(m−2) r2 as printed. For m = 3 the printed inequality requires r1 > 5r2, which would exclude the s = 4 example (r1,r2,r3) = (8/11,2/11,1/11) used in Figs. 7 and 10, since that example satisfies r1 = 4r2.
[Sec. IV, Eq. (25)] The tricritical condition is printed as r1 = (2m−1)/(m−2) r2, but the correct condition f''(1) = 0 gives r1 = 2(m−1)/(m−2) r2, i.e., r1 = 4r2 for m = 3. The paper itself later uses the correct form in Sec. IX D (the line 'when r1 = 2r2(m−1)/(m−2), pc = ...'), so the two parts of the manuscript contradict each other. This inconsistency must be resolved because the tricritical manifold determines where the universality class of the upper transition changes.
[Sec. IV, Eq. (33)] The stated formula for r_k at a critical point of order s is not compatible with the example used in the paper. For m = 3 and k = 2, Eq. (33) gives r2 = Γ(1/2)Γ(1/2)/Γ(3) r1 = (π/2)r1, whereas vanishing of f''(1) requires r2 = r1/4. The coefficients (0.727273, 0.181818, 0.0909091) shown for the s = 4 case in Figs. 7 and 10 satisfy the algebraic relations r1 = 4r2 and r2 = 2r3, not Eq. (33). Consequently, the s = 5 and s = 6 parameter sets used in Fig. 10 may not lie on the true critical manifold, and the reported deviations from the theoretical σ could be an artifact of off-manifold parameters. The authors should either correct Eq. (33) to the actual solution of Eqs. (31)–(32) or provide the explicit parameter values used and verify them against the corrected conditions.
[Sec. IX D and Fig. 10] The numerical evidence for the anomalous exponent σ = (s−3)/(s−1) is incomplete. The s = 4 data agree with σ = 1/3, but the s = 5 and s = 6 exponents are reported to deviate from the prediction, and the paper attributes this to finite sizes or the continuous approximation. Given that the critical-point conditions in Sec. IV are misstated, the s = 5 and s = 6 runs may have been performed at parameters that are not critical points of order s. The authors should recompute these cases using the corrected conditions and then reassess whether the deviations persist; without this, the claim that the anomalous scaling generalizes beyond s = 4 is not supported.
[Sec. VIII A, Eq. (42)] The derivation of the fractal exponent assumes that the non-homogeneous term ∂F_n/∂x in Eq. (42) is subleading compared with the homogeneous term, so that M_n is proportional to ∏ λ_n for large n. This is stated without proof. The conclusion is plausible—a bounded forcing term cannot change the exponential growth rate—but the manuscript should justify the claim, for example by bounding ∂F_n/∂x uniformly in n and noting that the contribution from the first source term already grows at the rate ∏ λ_n. Since Eq. (47) and the subsequent RG expression for P∞ rest on this step, a short argument or a reference to a derivation is needed.

minor comments (3)

[Sec. III] The phase enumeration is numbered (1), (3), (3); the second item should read (2).
[Eq. (3)] The formula M_n = (\bar N_n)^{\psi_n} is typeset in a way that could be misread as (\bar N_n)\psi_n; please clarify with an explicit exponent.
[Fig. 10] The figure shows the theoretical prediction as a thin dotted line, but it is unclear whether this line corresponds to s = 4 only or is meant for all s. For a useful comparison, the predicted slopes for s = 5 and s = 6 should be included or clearly described in the caption.

Circularity Check

0 steps flagged · score 2.0 of 10

Minor non-load-bearing self-citation; the derivation itself is self-contained and not circular.

full rationale

The derivation is self-contained. The percolation probability T is defined by the recursive branching equation (6), whose fixed point is Eq. (7)/(10); no parameter is fitted to output data. The intermediate transitions in Sec VI are found by direct numerical solution of this fixed-point equation (Figs. 3-6), not inherited from Ref. [46]. The fractal exponent in Eq. (47) follows from the Jacobian of the generating-function recursion (49), and the RG scaling P_infty ~ exp[-A/(Delta p)^sigma] in Sec IX is obtained by expanding the same exact RG map (60); the higher-order conditions (31)-(34) are derivative conditions on f, so the exponents are mathematical consequences rather than inputs. The only self-citation is Ref. [46] (Kryven and Bianconi), used to motivate the mapping in Sec V: 'Since the considered multiplex networks have been shown to display multiple percolation phase transitions, it follows that the mapping described above suggests that also in the modified random branching network one may expect multiple critical points in the equation determining the linking probability.' The paper then explicitly states that the original model 'needs to be explored in detail and will be addressed in the following section,' and Sec VI verifies the multiple-transition claim by direct solution of Eq. (10). The explicit caveats, namely the unproved subleading assumption on partial F_n/partial x in Sec VIII and the reported s=5,6 deviations in Fig. 10, are accuracy limitations rather than circular steps. The score reflects only the minor, non-load-bearing self-citation; the central derivation is independent.

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

The central derivation rests on the self-similar recursion, the smallest-solution rule, the subleading-source assumption in the generating function analysis, and the RG truncation. No new physical entities or fitted parameters are introduced. The model parameters r_k and m are inputs.

assumptions (4)

domain assumption The random branching cell complex is statistically self-similar: each link independently branches into k polygons drawn from r_k, so the connection probability T_n obeys the recursion Eq (6) in the infinite-size limit.
This defines the model and is the basis for the fixed-point equation Eq (10).
domain assumption The physical percolation probability is the smallest solution of Eq (10) in [0,1].
The paper states this in Sec IV; selecting the smallest solution is standard for such recursions but is an assumption about the physical branch.
domain assumption The non-homogeneous term ∂F_n/∂x in Eq (42) is subleading for n≫1, so M_n is governed by the product of largest eigenvalues.
Asserted without proof in Sec VIII.A; used to derive the fractal exponent ψ.
domain assumption Taylor expansions of Eq (60) truncated at leading order are valid close to the critical points, including the continuous approximation dx/dn = -C x(x^{s-1} + B Δp).
Standard RG practice, but it introduces uncontrolled corrections, which the paper acknowledges for s=5,6 in Fig 10.

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Cite this review

Pith. "Pith review of Percolation on branching simplicial and cell complexes and its relation to interdependent percolation." pith.science (2026). https://pith.science/paper/F6DKKC4I

@misc{pith2026190809392,
  author       = {Pith},
  title        = {Pith review of: Percolation on branching simplicial and cell complexes and its relation to interdependent percolation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F6DKKC4I}},
  note         = {Machine review of arXiv:1908.09392}
}

read the original abstract

Network geometry has strong effects on network dynamics. In particular, the underlying hyperbolic geometry of discrete manifolds has recently been shown to affect their critical percolation properties. Here we investigate the properties of link percolation in non-amenable two-dimensional branching simplicial and cell complexes, i.e., simplicial and cell complexes in which the boundary scales like the volume. We establish the relation between the equations determining the percolation probability in random branching cell complexes and the equation for interdependent percolation in multiplex networks with inter-layer degree correlation equal to one. By using this relation we show that branching cell complexes can display more than two percolation phase transitions: the upper percolation transition, the lower percolation transition, and one or more intermediate phase transitions. At these additional transitions the percolation probability and the fractal exponent both feature a discontinuity. Furthermore, by using the renormalization group theory we show that the upper percolation transition can belong to various universality classes including the Berezinskii-Kosterlitz-Thouless (BKT) transition, the discontinuous percolation transition, and continuous transitions with anomalous singular behavior that generalize the BKT transition.

Figures

Figures reproduced from arXiv: 1908.09392 by the authors.

**Figure 2.** FIG. 2. The result of the first [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** FIG. 3. The percolation probability [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: FIG. 4. The barycentric plot characterizing the phase diagram of link percolation for the branching simplicial complex with [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5. The barycentric plot characterizing the phase diagram of link percolation for the branching simplicial complex with [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: FIG. 6. The barycentric plot characterizing the phase diagram of link percolation for the branching simplicial complex with [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: FIG. 7. The percolation probability [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 8.** Figure 8: FIG. 8. The percolation order parameter [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

**Figure 9.** Figure 9: FIG. 9. The scaling of the exponent [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]

**Figure 10.** Figure 10: FIG. 10. The critical scaling of the order parameter [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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