Pith. sign in

REVIEW 3 major objections 5 minor 65 references

Sequential epidemics on random graphs are solved exactly for any finite number of strains, in both cross-immunity and coinfection regimes.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-01 23:06 UTC pith:UX2F2J6Y

load-bearing objection The collaborative N-strain binary-tree construction is the real contribution; the competitive extension is expected. But the printed threshold equations have several load-bearing algebraic errors, especially Eq 56, so the paper needs a careful revision before the critical-point claims can be trusted. the 3 major comments →

arxiv 2607.15513 v1 pith:UX2F2J6Y submitted 2026-07-16 physics.soc-ph

An exact N-strain epidemic model using bond percolation

Peter Mann , V. Anne Smith , John B.O. Mitchell , Simon Dobson This is my paper
classification physics.soc-ph MSC 05C8060K3592D30 PACS 89.75.Hc
keywords bond percolationepidemic modelingsequential strainscross-immunitycoinfectiongenerating functionsrandom graphsscale-free networks
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Sequential epidemics — seasonal strains that follow one another through a population — have previously been analysed for two strains. This paper extends that analysis to any finite number N of strains and claims to solve the model exactly on random graphs using generating functions. Two interaction regimes are treated: competitive (each new strain spreads only among nodes that escaped all earlier strains, perfect cross-immunity) and collaborative (each new strain spreads only among nodes infected by all earlier strains, perfect coinfection). The authors derive recursive self-consistent equations for the probability that a neighbour remains uninfected, from which they obtain the expected outbreak size of every strain, the percolation/epidemic threshold of every strain, and the degree distribution of the residual or giant-component structure left behind. Because the formulas are exact in the locally tree-like random-graph regime, they give a precise quantitative picture of seasonal outbreak burnout, threshold escalation, and coinfection-limited spread.

Core claim

On the paper's own terms, the central discovery is that repeated bond percolation — performed N times on the residual graph (competitive) or on the giant component (collaborative) — can be described exactly by cascades of generating functions. For competitive strains, the probability g_i that an edge fails to connect to the i-th giant component obeys a one-dimensional recursion whose fixed points yield outbreak size A_i = ∏_{j<i} G0(g_j) − G0(g_i) and threshold T_{i,c} = 1/G'_1(g_{i-1}); thresholds provably increase with strain index. For collaborative strains, the required per-neighbour probabilities multiply as a perfect binary tree of 'infection histories' (2^{i−1} histories for strain i)

What carries the argument

The load-bearing object is the excess-degree generating function G_1(x) = G'_0(x)/G'_0(1) together with the cavity-method assumption that neighbour infection states are iid. In the competitive process the machinery is the recursive edge-failure probability g_i = u_{i−1} ar g_i + (1−u_{i−1})(1−T_{i−1}) and the self-consistent hierarchy u_i = G_1(g_i)/∏_{j<i} u_j. In the collaborative process the machinery is a perfect binary tree of infection histories: each generation doubles the number of distinct neighbour states, and the common factor C_i = ar C_{i−1}(ar f_i) multiplies history probabilities H_{i_h} and priors Q_{i_h} to form P_{i_h}=C_i H_{i_h}. These objects convert the percolation prob

Load-bearing premise

The generating-function construction assumes that the infection states of a node's neighbours are independent and identically distributed along every edge (the cavity method), which is exact only on locally tree-like random graphs with negligible clustering and no degree correlations; on real clustered networks the formulas are approximations and the paper does not test the size of the error.

What would settle it

On a random-graph ensemble with tunable clustering (e.g., a configuration model augmented with triangles), run the N=3 competitive and collaborative percolation processes by Monte Carlo. Measure A_1, A_2, A_3 and the thresholds and compare them with Eqs 11, 16, 37 and 49. If the deviations exceed finite-size error, the cavity/iid assumption is the cause. More directly, sample the joint infection states of pairs of neighbours of a focal node after the process: if the joint distribution differs from the product of the marginal u-values, the independence assumption is violated.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • For N sequential cross-immune strains, each later strain needs a larger transmission probability than the previous one to trigger a macroscopic outbreak; the size of that outbreak is nevertheless smaller, so seasonal transmissibility must rise while attack sizes fall.
  • For collaborative (coinfection) strains, the i-th outbreak is strictly bounded by the (i−1)-th and can never exceed it, so coinfection-limited spread eventually burns out.
  • On scale-free networks, a cross-immune first strain destroys the power-law degree distribution of the residual graph, so the second strain's threshold can be nonzero even though the first strain has threshold zero; cooperative strains, by contrast, retain a self-similar high-degree core and can theoretically spread at arbitrarily low transmissibility in the infinite-size limit.
  • The total fraction infected across all strains is not monotonic in the first strain's transmissibility; there is a local minimum near the coexistence threshold where the first strain fractures the residual graph enough to prevent later strains.
  • The degree distribution and cumulative degree distribution of each generation's residual graph and giant component follow from the same generating functions, giving a complete topological description of the layered percolation structure.

Where Pith is reading between the lines

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

  • If the competitive threshold-ordering result carries to real seasonal pathogens, it predicts that any circulating strain that does not evolve upward transmissibility will fail to exceed threshold after the first season; this yields a concrete null model for interpreting observed transmissibility increases in influenza-like data.
  • The collaborative model's binary-tree structure suggests a computational short-cut: the 2^{i−1} coupled equations can be generated recursively from the tree, so an N-strain solver requires only tree traversal rather than hand-derived equations for each N.
  • A natural testable extension would be partial cross-immunity, interpolating between the competitive and collaborative extremes for N>2; the paper notes that such a model would not burn out, and one could check whether the competitive model's non-monotonic total-infection curve persists.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper claims to extend two-strain bond-percolation epidemic models to an arbitrary finite number N of sequential strains. Two branching processes are treated: a competitive process, in which each strain percolates on the residual graph left by previous strains (perfect cross-immunity), and a collaborative process, in which each strain percolates on the giant component formed by all previous strains (perfect coinfection). Generating functions are used to write self-consistent equations for the probabilities that edges fail to transmit each strain, from which outbreak sizes and critical transmissibilities are derived. Results are given for Erdős–Rényi and power-law (scale-free) networks and compared with Monte Carlo simulations for N=4–5. The authors also study degree and cumulative degree distributions of the residual graphs and GCC substructures.

Significance. The paper has a clear and useful goal: generalizing the well-known N=2 results of Karrer–Newman and Newman–Ferrario to arbitrary N, with explicit recursive formulas and binary-infection-history bookkeeping. The Monte Carlo validation of the main ER outbreak-size recursions (Fig. 2 for N=5) and the collaborative outbreak sizes (Fig. 6) is a genuine strength, as is the explicit reduction to known N=2 limits. If the algebraic problems in the printed general recursions and critical-point formulas are corrected, this would be a valuable reference for multi-season epidemic modelling on locally tree-like networks. However, several printed equations that are load-bearing for the critical-point and scale-free claims are internally inconsistent, so the manuscript in its present form cannot be used as a reliable recipe by readers.

major comments (3)
  1. [Section III, Eq. (9)] The printed recursion does not reproduce the nested expressions in Eqs. (12)–(13). For i=3, Eq. (9) gives g_3 = u_2[u_3+(1-u_3)(1-T_3)] + (1-u_2)(1-T_2), whereas Eq. (13) contains an additional factor u_1 multiplying the whole bracket. The later statement that G'_1(g_i) becomes G'_1(g_{i-1}) at u_i=1, which underlies Eq. (16), is true only for the nested form. The general recursion must be corrected, and the derivation of Eq. (16) re-verified.
  2. [Section III, Eq. (18)] The ER coexistence condition G'_1(g_i)=1 is not equivalent to the printed equation. Since G_1(x)=e^{<k>(x-1)} for Erdős–Rényi graphs, G'_1(g_i)=1 gives g_i=1 - ln<k>/<k>. Equation (18), with its ratio u_i / \prod_{j<i} u_j and right-hand side g_{i-1}, does not follow from that condition; for i=1 it would imply u_1=e^{<k>}. Consequently Eq. (19) and the coexistence-threshold curves in Fig. 5(right) need to be re-derived and re-plotted.
  3. [Section V, Eq. (56)] The scale-free critical-point formula is algebraically wrong. Using dLi_s(z)/dz = Li_{s-1}(z)/z, Eq. (16) applied to Eq. (55) gives T_{i,c} = g_{i-1}^2 Li_{α-1}(e^{-1/κ}) / [Li_{α-2}(g_{i-1}e^{-1/κ}) - Li_{α-1}(g_{i-1}e^{-1/κ})], not Eq. (56). The printed denominator g_{i-1}Li_{α-2}(g_{i-1}e^{-1/κ}) - Li_{α-1}(g_{i-1}e^{-1/κ}) can be negative for g_{i-1}<1 (e.g. α=2, large κ, g≈1/2), leading to negative thresholds. Therefore Eq. (58) and the associated 'rapid fracture' claims for scale-free networks are not justified as stated.
minor comments (5)
  1. [Abstract / Section II.A] The abstract and introduction use 'exact' without qualification. The cavity method at Eq. (7) assumes iid neighbour states along each edge, which is exact only in the locally tree-like, configuration-model limit. Please state that qualification explicitly in the abstract or introduction.
  2. [Fig. 1 caption] Typo: 'competative' should be 'competitive'.
  3. [Eqs. (24), (30c)] The general definition \bar C_i = f_1(f_2(... f_i()))) is notationally incomplete; the explicit examples (Eqs. 25–27) are much clearer. Please make the general notation precise.
  4. [Section V, around Eq. (56)] The polylogarithm derivative identity dLi_s(z)/dz = Li_{s-1}(z)/z should be stated explicitly before the scale-free derivation; this would have prevented the algebra error in Eq. (56).
  5. [Fig. 5(right)] The caption says the coexistence threshold is obtained by 'numerically solving Eq. 18'. If Eq. 18 is corrected, please state the corrected equation used to generate the figure, or the figure will need regeneration.

Circularity Check

0 steps flagged

No significant circularity: the N-strain derivation is self-contained, starts from the standard generating-function formalism, and is benchmarked against independent Monte Carlo simulations; self-citations are contextual.

full rationale

The paper's central derivation is not circular. It builds on the standard Newman-Strogatz-Watts generating-function formalism (Eqs 1-7) and extends it to N sequential strains via recursive self-consistency equations (Eqs 9-11 for the competitive process, Eqs 21-37 and A1-A25 for the collaborative process). The N=2 limits are attributed to external papers by Karrer and Newman [6], Newman [15], and Newman and Ferrario [16], not to the present authors. No fitted parameters are used to force agreement; the analytical outbreak sizes are verified against independent Monte Carlo simulations of bond percolation (Figs 2, 6, 7, 8). Self-citations to the authors' prior work [8,10,11,12] appear only as background or future-work pointers and are not load-bearing for the derivations. The cavity-method iid assumption stated around Eq. 7 is an explicit modeling assumption (valid in the locally tree-like configuration-model limit), not a circular step. Any algebraic inconsistencies in the scale-free critical-point formulas (e.g., Eq. 56 or Eq. 18) would be correctness risks, not instances of the derivation reducing to its own inputs. The derivation chain is therefore self-contained; no prediction is equivalent to an input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No free parameters are fitted; all T_i are externally controlled inputs. The model relies on the standard implicit assumptions of configuration-model generating functions and adds the domain assumption that strains act sequentially with no temporal dynamics. No invented particles, forces, or entities are introduced.

axioms (5)
  • domain assumption Configuration-model / locally tree-like random graph approximation.
    All recursions are written in terms of G0 and G1 only, ignoring clustering and degree correlations. This is invoked in Section II and is the standard domain of the generating-function method.
  • domain assumption Cavity-method iid neighbour-state assumption.
    Eq 7 assumes neighbour-state probabilities are independent and identically distributed along each edge; every self-consistent u-equation (Eqs 10, 21) depends on this.
  • domain assumption SIR–bond-percolation equivalence with fixed infectious period.
    The interpretation of T_i and A_i as transmissibility and outbreak size relies on the mapping from bond percolation to SIR with a single-valued infectious period, as cited from [7,13,14].
  • domain assumption Temporal separation of the N strains.
    Each strain is assumed to complete its percolation before the next begins; the paper itself states this in the conclusion: 'a strong condition on the success of the model is the temporal separation of each generation.'
  • standard math Thermodynamic-limit exactness of the generating-function solutions.
    The 'exact' formulas are exact for infinite random graphs; finite-size Monte Carlo comparisons are approximate. This is standard for the method but still a domain assumption.

pith-pipeline@v1.3.0-alltime-deepseek · 26121 in / 21715 out tokens · 202825 ms · 2026-08-01T23:06:38.479571+00:00 · methodology

0 comments
read the original abstract

In this paper we examine the emergent structures of random networks that have undergone bond percolation an arbitrary, but finite, number of times. We define two types of sequential branching processes: a competitive branching process - in which each iteration performs bond percolation on the residual graph (RG) resulting from previous generations; and, a collaborative branching process - where percolation is performed on the giant connected component (GCC) instead. We investigate the behaviour of these models, including the expected size of the GCC for a given generation, the critical percolation probability and other topological properties of the resulting graph structures using the analytically exact method of generating functions. We explore this model for Erdos-Renyi and scale free random graphs. This model can be interpreted as a seasonal N-strain model of disease spreading.

Figures

Figures reproduced from arXiv: 2607.15513 by John B.O. Mitchell, Peter Mann, Simon Dobson, V. Anne Smith.

Figure 1
Figure 1. Figure 1: FIG. 1. A substrate network (left) is percolated to yield a GCC (red) and an RG. In the competative branching process, the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The outbreak fractions for five generations of the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (top) The degree distribution of the RG created once [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (Left) Theoretical results for the outbreak size of the [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The critical points of the first five strains of the com [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Four generations of the cooperative branching process with [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (top) The cumulative degree distribution of nodes in [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Four generations of the cooperative branching process with [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The graphical solution [18] of the generating function for strain 1 at [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The graphical solution of the generating functions for strain 2 at three different [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. A visualisation of the perfect binary tree of coinfected neighbours which a focal node could be surrounded by after [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. A visualisation of the perfect binary tree of coinfected neighbours which a focal node could be surrounded by and the [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. The graphical solution of the generating functions for strain 4 around the critical point of strain 1 (left) with [PITH_FULL_IMAGE:figures/full_fig_p022_13.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

65 extracted references · 48 canonical work pages

  1. [1]

    Epidemic Spreading in Scale-Free Networks , author =. Phys. Rev. Lett. , volume =. 2001 , month =. doi:10.1103/PhysRevLett.86.3200 , url =

    doi:10.1103/physrevlett.86.3200 2001

  2. [2]

    PLoS ONE , author=

    The shifting demographic landscape of pandemic influenza , volume=. PLoS ONE , author=. doi:10.1371/journal.pone.0009360 , number=

    work page doi:10.1371/journal.pone.0009360

  3. [3]

    Journal of Statistical Mechanics: Theory and Experiment , author=

    Multivariate generating functions for information spread on multi-type random graphs , volume=. Journal of Statistical Mechanics: Theory and Experiment , author=. 2022 , pages=. doi:10.1088/1742-5468/ac57b8 , number=

    work page doi:10.1088/1742-5468/ac57b8 2022

  4. [4]

    Journal of Physics: Complexity , abstract =

    Shogo Mizutaka and Takehisa Hasegawa , title =. Journal of Physics: Complexity , abstract =. doi:10.1088/2632-072x/abb4c5 , url =

    work page doi:10.1088/2632-072x/abb4c5

  5. [5]

    Symbiotic and antagonistic disease dynamics on networks using bond percolation , author =. Phys. Rev. E , volume =. 2021 , month =. doi:10.1103/PhysRevE.104.024303 , url =

    work page doi:10.1103/physreve.104.024303 2021

  6. [6]

    Disassortativity of percolating clusters in random networks , author =. Phys. Rev. E , volume =. 2018 , month =. doi:10.1103/PhysRevE.98.062314 , url =

    work page doi:10.1103/physreve.98.062314 2018

  7. [7]

    Structure of percolating clusters in random clustered networks , author =. Phys. Rev. E , volume =. 2020 , month =. doi:10.1103/PhysRevE.101.062310 , url =

    work page doi:10.1103/physreve.101.062310 2020

  8. [8]

    Rybski and H

    D. Rybski and H. D. Rozenfeld and J. P. Kropp , title =. doi:10.1209/0295-5075/90/28002 , url =

    work page doi:10.1209/0295-5075/90/28002

  9. [9]

    The impact of past epidemics on future disease dynamics , journal =

    Shweta Bansal and Lauren Ancel Meyers , keywords =. The impact of past epidemics on future disease dynamics , journal =. 2012 , issn =. doi:https://doi.org/10.1016/j.jtbi.2012.06.012 , url =

    work page doi:10.1016/j.jtbi.2012.06.012 2012

  10. [10]

    Wilf, Herbert S. , year=. Generatingfunctionology , publisher=

  11. [11]

    Networks , publisher=

    Newman, Mark E.J , year=. Networks , publisher=

  12. [12]

    The unreasonable effectiveness of tree-based theory for networks with clustering , author =. Phys. Rev. E , volume =. 2011 , month =. doi:10.1103/PhysRevE.83.036112 , url =

    work page doi:10.1103/physreve.83.036112 2011

  13. [13]

    General formulation of long-range degree correlations in complex networks , author =. Phys. Rev. E , volume =. 2018 , month =. doi:10.1103/PhysRevE.97.062308 , url =

    work page doi:10.1103/physreve.97.062308 2018

  14. [14]

    Finding community structure in networks using the eigenvectors of matrices , author =. Phys. Rev. E , volume =. 2006 , month =. doi:10.1103/PhysRevE.74.036104 , url =

    work page doi:10.1103/physreve.74.036104 2006

  15. [15]

    Dynamical and Correlation Properties of the Internet , author =. Phys. Rev. Lett. , volume =. 2001 , month =. doi:10.1103/PhysRevLett.87.258701 , url =

    work page doi:10.1103/physrevlett.87.258701 2001

  16. [16]

    arXiv e-prints , keywords =

    Network clique cover approximation to analyze complex contagions through group interactions. arXiv e-prints , keywords =

  17. [17]

    Nature , author=

    Self-similarity of complex networks , volume=. Nature , author=. 2005 , pages=. doi:10.1038/nature03248 , number=

    work page doi:10.1038/nature03248 2005

  18. [18]

    Observability transitions in clustered networks , journal =

    Takehisa Hasegawa and Yuta Iwase , abstract =. Observability transitions in clustered networks , journal =. 2021 , issn =. doi:https://doi.org/10.1016/j.physa.2021.125970 , url =

    arXiv 2021

  19. [19]

    Random graphs with arbitrary clustering and their applications , author =. Phys. Rev. E , volume =. 2021 , month =. doi:10.1103/PhysRevE.103.012309 , url =

    work page doi:10.1103/physreve.103.012309 2021

  20. [20]

    Exact formula for bond percolation on cliques , author =. Phys. Rev. E , volume =. 2021 , month =. doi:10.1103/PhysRevE.104.024304 , url =

    work page doi:10.1103/physreve.104.024304 2021

  21. [21]

    Percolation in random graphs with higher-order clustering , author =. Phys. Rev. E , volume =. 2021 , month =. doi:10.1103/PhysRevE.103.012313 , url =

    work page doi:10.1103/physreve.103.012313 2021

  22. [22]

    Two-pathogen model with competition on clustered networks , author =. Phys. Rev. E , volume =. 2021 , month =. doi:10.1103/PhysRevE.103.062308 , url =

    work page doi:10.1103/physreve.103.062308 2021

  23. [23]

    Jensen, J. L. W. V. , year=. Sur les fonctions convexes et les inégalités entre les valeurs moyennes , volume=. doi:10.1007/bf02418571 , journal=

    doi:10.1007/bf02418571

  24. [24]

    Cooperative coinfection dynamics on clustered networks , author =. Phys. Rev. E , volume =. 2021 , month =. doi:10.1103/PhysRevE.103.042307 , url =

    work page doi:10.1103/physreve.103.042307 2021

  25. [25]

    Contemporary Physics , author=

    Power laws, Pareto distributions and Zipfs law , volume=. Contemporary Physics , author=. 2005 , pages=. doi:10.1080/00107510500052444 , number=

    doi:10.1080/00107510500052444 2005

  26. [26]

    Competing epidemics on complex networks , author =. Phys. Rev. E , volume =. 2011 , month =. doi:10.1103/PhysRevE.84.036106 , url =

    work page doi:10.1103/physreve.84.036106 2011

  27. [27]

    PLoS ONE , author=

    Interacting Epidemics and Coinfection on Contact Networks , volume=. PLoS ONE , author=. doi:10.1371/journal.pone.0071321 , number=

    work page doi:10.1371/journal.pone.0071321

  28. [28]

    Physical Review Letters , author=

    Threshold Effects for Two Pathogens Spreading on a Network , volume=. Physical Review Letters , author=. 2005 , month=. doi:10.1103/physrevlett.95.108701 , number=

    work page doi:10.1103/physrevlett.95.108701 2005

  29. [29]

    Random Graphs with Clustering , author =. Phys. Rev. Lett. , volume =. 2009 , month =. doi:10.1103/PhysRevLett.103.058701 , url =

    work page doi:10.1103/physrevlett.103.058701 2009

  30. [30]

    Arxiv preprint cond-mat/0007235 , keywords =

    Newman, MEJ and Strogatz, SH and Watts, DJ , biburl =. Arxiv preprint cond-mat/0007235 , keywords =

    Pith/arXiv arXiv

  31. [31]

    Nature Physics , author=

    Avalanche outbreaks emerging in cooperative contagions , volume=. Nature Physics , author=. 2015 , pages=. doi:10.1038/nphys3457 , number=

    work page doi:10.1038/nphys3457 2015

  32. [32]

    Interacting epidemics on overlay networks , author =. Phys. Rev. E , volume =. 2010 , month =. doi:10.1103/PhysRevE.81.036118 , url =

    work page doi:10.1103/physreve.81.036118 2010

  33. [33]

    Percolation and epidemics in random clustered networks , author =. Phys. Rev. E , volume =. 2009 , month =. doi:10.1103/PhysRevE.80.020901 , url =

    work page doi:10.1103/physreve.80.020901 2009

  34. [34]

    , author=

    Spread of epidemic disease on networks. , author=. Physical review. E, Statistical, nonlinear, and soft matter physics , year=

  35. [35]

    Epidemic size and probability in populations with heterogeneous infectivity and susceptibility , author =. Phys. Rev. E , volume =. 2007 , month =. doi:10.1103/PhysRevE.76.010101 , url =

    work page doi:10.1103/physreve.76.010101 2007

  36. [36]

    Second look at the spread of epidemics on networks , author =. Phys. Rev. E , volume =. 2007 , month =. doi:10.1103/PhysRevE.76.036113 , url =

    work page doi:10.1103/physreve.76.036113 2007

  37. [37]

    Percolation and Epidemic Thresholds in Clustered Networks , author =. Phys. Rev. Lett. , volume =. 2006 , month =. doi:10.1103/PhysRevLett.97.088701 , url =

    work page doi:10.1103/physrevlett.97.088701 2006

  38. [38]

    Properties of highly clustered networks , author =. Phys. Rev. E , volume =. 2003 , month =. doi:10.1103/PhysRevE.68.026121 , url =

    doi:10.1103/physreve.68.026121 2003

  39. [39]

    Double Percolation Phase Transition in Clustered Complex Networks , author =. Phys. Rev. X , volume =. 2014 , month =. doi:10.1103/PhysRevX.4.041020 , url =

    work page doi:10.1103/physrevx.4.041020 2014

  40. [40]

    and Zdeborová, Lenka , year=

    Karrer, Brian and Newman, M.E.J. and Zdeborová, Lenka , year=. Percolation on sparse networks , journal=

  41. [41]

    Spread of infectious disease through clustered populations , url=

    Miller, Joel C , year=. Spread of infectious disease through clustered populations , url=. Journal of the Royal Society, Interface , publisher=

  42. [42]

    Physical Review E , author=

    Random graphs with arbitrary degree distributions and their applications , volume=. Physical Review E , author=. doi:10.1103/physreve.64.026118 , number=

    doi:10.1103/physreve.64.026118

  43. [43]

    Random Structures & Algorithms , author=

    A critical point for random graphs with a given degree sequence , volume=. Random Structures & Algorithms , author=. 1995 , pages=. doi:10.1002/rsa.3240060204 , number=

    doi:10.1002/rsa.3240060204 1995

  44. [44]

    Physical Review E , author=

    Bond percolation on a class of clustered random networks , volume=. Physical Review E , author=. 2009 , month=. doi:10.1103/physreve.80.036107 , number=

    work page doi:10.1103/physreve.80.036107 2009

  45. [45]

    Science , author=

    Higher-order organization of complex networks , volume=. Science , author=. 2016 , month=. doi:10.1126/science.aad9029 , number=

    doi:10.1126/science.aad9029 2016

  46. [46]

    Physica A: Statistical Mechanics and its Applications , author=

    Higher order clustering coefficients in. Physica A: Statistical Mechanics and its Applications , author=. 2002 , pages=. doi:10.1016/s0378-4371(02)01336-5 , number=

    work page doi:10.1016/s0378-4371(02)01336-5 2002

  47. [47]

    Physical Review E , author=

    Propagation dynamics on networks featuring complex topologies , volume=. Physical Review E , author=. doi:10.1103/physreve.82.036115 , number=

    work page doi:10.1103/physreve.82.036115

  48. [48]

    Higher-order clustering in networks , author =. Phys. Rev. E , volume =. 2018 , month =. doi:10.1103/PhysRevE.97.052306 , url =

    work page doi:10.1103/physreve.97.052306 2018

  49. [49]

    Correlations in connected random graphs , author =. Phys. Rev. E , volume =. 2008 , month =. doi:10.1103/PhysRevE.77.036124 , url =

    work page doi:10.1103/physreve.77.036124 2008

  50. [50]

    Component sizes in networks with arbitrary degree distributions , author =. Phys. Rev. E , volume =. 2007 , month =. doi:10.1103/PhysRevE.76.045101 , url =

    work page doi:10.1103/physreve.76.045101 2007

  51. [51]

    Motif-based communities in complex networks , journal =

    A Arenas and A Fern. Motif-based communities in complex networks , journal =. doi:10.1088/1751-8113/41/22/224001 , url =

    work page doi:10.1088/1751-8113/41/22/224001

  52. [52]

    Physical Review E , author=

    General and exact approach to percolation on random graphs , volume=. Physical Review E , author=. 2015 , month=. doi:10.1103/physreve.92.062807 , number=

    work page doi:10.1103/physreve.92.062807 2015

  53. [53]

    Tuning clustering in random networks with arbitrary degree distributions , author =. Phys. Rev. E , volume =. 2005 , month =. doi:10.1103/PhysRevE.72.036133 , url =

    work page doi:10.1103/physreve.72.036133 2005

  54. [54]

    Physical Review E , author=

    Random graphs containing arbitrary distributions of subgraphs , volume=. Physical Review E , author=. doi:10.1103/physreve.82.066118 , number=

    work page doi:10.1103/physreve.82.066118

  55. [55]

    Cascades on a class of clustered random networks , author =. Phys. Rev. E , volume =. 2011 , month =. doi:10.1103/PhysRevE.83.056107 , url =

    work page doi:10.1103/physreve.83.056107 2011

  56. [56]

    How clustering affects the bond percolation threshold in complex networks , author =. Phys. Rev. E , volume =. 2010 , month =. doi:10.1103/PhysRevE.81.066114 , url =

    work page doi:10.1103/physreve.81.066114 2010

  57. [57]

    Beyond the locally treelike approximation for percolation on real networks , author =. Phys. Rev. E , volume =. 2016 , month =. doi:10.1103/PhysRevE.93.030302 , url =

    work page doi:10.1103/physreve.93.030302 2016

  58. [58]

    Revealing the microstructure of the giant component in random graph ensembles , author =. Phys. Rev. E , volume =. 2018 , month =. doi:10.1103/PhysRevE.97.042318 , url =

    work page doi:10.1103/physreve.97.042318 2018

  59. [59]

    , title = "

    Ritchie, Martin and Berthouze, Luc and Kiss, Istvan Z. , title = ". Journal of Complex Networks , volume =. 2016 , month =. doi:10.1093/comnet/cnw011 , url =

    work page doi:10.1093/comnet/cnw011 2016

  60. [60]

    Generating random networks that consist of a single connected component with a given degree distribution , author =. Phys. Rev. E , volume =. 2019 , month =. doi:10.1103/PhysRevE.99.042308 , url =

    work page doi:10.1103/physreve.99.042308 2019

  61. [61]

    Clustering determines the dynamics of complex contagions in multiplex networks , author =. Phys. Rev. E , volume =. 2017 , month =. doi:10.1103/PhysRevE.95.012312 , url =

    work page doi:10.1103/physreve.95.012312 2017

  62. [62]

    Proceedings of the National Academy of Sciences , author=

    Message passing on networks with loops , volume=. Proceedings of the National Academy of Sciences , author=. 2019 , month=. doi:10.1073/pnas.1914893116 , number=

    work page doi:10.1073/pnas.1914893116 2019

  63. [63]

    Physical Review E , author=

    Spectra of networks containing short loops , volume=. Physical Review E , author=. doi:10.1103/physreve.100.012314 , number=

    work page doi:10.1103/physreve.100.012314

  64. [64]

    Proceedings of the National Academy of Sciences , author=

    Spectral redemption in clustering sparse networks , volume=. Proceedings of the National Academy of Sciences , author=. 2013 , pages=. doi:10.1073/pnas.1312486110 , number=

    doi:10.1073/pnas.1312486110 2013

  65. [65]

    Physical Review E , author=

    Localization and centrality in networks , volume=. Physical Review E , author=. 2014 , month=. doi:10.1103/physreve.90.052808 , number=

    doi:10.1103/physreve.90.052808 2014