A Simple Model of Global Cascades on Random Networks
Proceedings of the National Academy of Sciences, 2002 · paper · Sociology
Presents a threshold model of agents on sparse random networks to explain rare, very large cascades from small shocks. It identifies two regimes with different cascade-size distributions, and finds heterogeneous thresholds raise vulnerability while heterogeneous connectivity lowers it.
Checked: PDF from europepmc.org, fetched 5 Oct 2026. Page numbers are those of that PDF.
3 findings
Substance The Fig. 1 simulation is described in the text (page 3) as "cascades are observed over 1,000 realizations of the dynamics" but in the Fig. 1 caption (page 4) as "averaged over 100 random single-node perturbations".
- p. 3 Results and Discussion, first paragraph
cascades are observed over 1,000 realizations of the dynamics
- p. 4 Fig. 1 caption
(averaged over 100 random single-node perturbations)
- p. 4 Fig. 1 caption
averaged over 100 random single-node perturbations
- p. 3 Results and Discussion, para 1
observed over 1,000 realizations of the dynamics
Only the authors can say what is right here.
Judge's note: Both passages read on the page images. The text defines a realization as one randomly constructed network of 10,000 vertices with a single vertex switched on, which is the same unit as a single-node perturbation. I searched the article for a nested design (several perturbations per network) and found none; Fig. 2, a cross section of Fig. 1, also says 1,000 realizations. Only the author can say which count applies to Fig. 1.
Second check: another model agreed
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Substance Fig. 3 caption (page 5) gives "for n = 1,000 and z = 1.05 (open squares) and z = 6.14 (solid circles)", but the plotted cascade sizes (fractions of the system) start near 0.0001-0.0002, which is not possible for a 1,000-node network; other simulations use n = 10,000.
- p. 5 Fig. 3 caption
critical points, for n = 1,000 and z = 1.05 (open squares) and z = 6.14 (solid
- p. 5 Fig. 3, x axis (Cascade Size)
0.0001
- p. 4 Results, left column
10,000 nodes averaged over 1,000 random realizations
- p. 5 Results and Discussion, right column
Fig. 3 shows only a single cascade occurring in 1,000
Only the authors can say what is right here.
Judge's note: Looked at Fig. 3 on the page 5 image: the x axis (Cascade Size) runs from 0.0001 to 1, the global cascade sits at 1 (so size is a fraction of the system), and the first markers of both series lie between 0.0001 and 0.001 at a cumulative count of about 1000. With n = 1,000 the smallest cascade is 0.001. The text and Figs. 1 and 2 use n = 10,000, and the text says Fig. 3 shows "a single cascade occurring in 1,000 random trials", matching the y-axis maximum of 1000, so the caption's 1,000 may be the number of trials or a misprint for 10,000. No statement in the article says a smaller system was used for Fig. 3.
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Reference or label Page 2: "majority voting (27, 28), spreading activation (29), and self-organized criticality (8, 29)" cites ref. 29 (Bak, Tang & Wiesenfeld) for spreading activation; ref. 28 (Shrager, Hogg & Huberman, Science 236) appears to be the intended citation.
- p. 2 Model Specification, third paragraph
lation (25, 26), majority voting (27, 28), spreading activation (29),
- p. 2 Model Specification, third paragraph
and self-organized criticality (8, 29).
- p. 5 left column
organized criticality (29)
- p. 6 References 28 and 29
29. Bak, P., Tang, C. & Wiesenfeld, K. (1987) Phys. Rev. Lett. 59, 381–384.
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: that ref. 28 (Shrager, Hogg & Huberman 1987, Science 236, 1092) is a paper on spreading activation networks and ref. 29 (Bak, Tang & Wiesenfeld 1987) is the self-organized criticality paper; the reference list prints no titles. Checked on the page 2 and page 6 images that the citation numbers and entries are as quoted. Internal support: the paper itself cites 29 for self-organized criticality in the same sentence and again on page 5 ("models of self-organized criticality (29)"). The likely intended numbering is majority voting (27), spreading activation (28).
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12 minor (typos and the like; most checked by the judge only)
Typo Page 1: "the decision of whether or not to join is essential a binary one" should read "essentially a binary one".
- p. 1 Model Motivation, second paragraph
of whether or not to join is essential a binary one
- p. 1 Model Motivation, paragraph 2
is essential a binary one
Judge's note: Read on the page 1 image (Model Motivation, second paragraph). The word is printed in full as "essential"; the adverb is needed.
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Typo Page 2: "where we define a global cascade formally as cascade that occupies a finite fraction of an infinite network" lacks the article: "as a cascade that".
- p. 2 Exact Solution on an Arbitrary Random Graph, first paragraph
where we define a global cascade formally as cascade that
- p. 2 Exact Solution on an Arbitrary Random Graph, first paragraph
means more than a fixed fraction of large, but finite network
- p. 2 Exact Solution, paragraph 2
formally as cascade that
- p. 2 Exact Solution
fixed fraction of large, but finite network
Judge's note: Read on the page 2 image (Exact Solution section, first paragraph). The article is missing in print. The second missing article in the same paragraph ("of large, but finite network") is handled in C004.
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Typo Page 2: "this means more than a fixed fraction of large, but finite network" lacks the article: "of a large, but finite network".
- p. 2 Exact Solution on an Arbitrary Random Graph, first paragraph
means more than a fixed fraction of large, but finite network
- p. 2 Exact Solution, paragraph 2
fixed fraction of large, but finite network
Judge's note: Read on the page 2 image (Exact Solution section, end of first paragraph). A separate slip from the one in C003, a few lines later in the same paragraph.
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Typo The vulnerable fraction is defined as "Pv = G0(1)" (page 3) but appears without the subscript in Eq. 4, "= P + zv²/(z − G0″(1))", and on page 4, "Sv = 1 − H0(1) = P − G0(H1(1))".
- p. 3 left column, after Eq. 1b
fraction of the population Pv = G0(1)
- p. 3 Eq. 4
= P + zv2/(z − G0″(1))
- p. 4 left column, first paragraph under Fig. 1
Sv = 1 − H0(1) = P −
- p. 3 Eq. 4
= P + zv
Judge's note: Checked on the page 3 and page 4 images: the definition carries a subscript v, while Eq. 4 and the Sv expression print a bare P, which is never defined; zv keeps its subscript in the same equation. From Eq. 3b at x = 1 the term is G0(1), i.e. Pv. A dropped subscript, not an extraction loss.
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Typo Page 3: "Because the generation function approach requires the largest vulnerable cluster to be a pure branching structure" should read "generating function approach".
- p. 3 Results and Discussion, second paragraph
Because the generation function approach requires the largest
- p. 3 left column, first full paragraph
solved using a generating function approach
- p. 3 Results and Discussion, p. 5768 (read on PDF page 3), 'solve exactly' paragraph
the generation function approach
- p. 3 first paragraph
generating function approach
Judge's note: Read on the page 3 image. The same page earlier has "solved using a generating function approach" and page 5 has "the same generating function method"; "generation function" is a slip.
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Typo The extended-vulnerable-cluster curve of Fig. 2b is called "the corresponding (dotted) curve in Fig. 2b" in the text and "extended vulnerable cluster (short dashes)" in the caption (both page 4).
- p. 4 left column, second paragraph under Fig. 1
as the corresponding (dotted) curve in Fig. 2b demonstrates
- p. 4 Fig. 2 caption
able cluster (short dashes)
- p. 4 text on Fig. 2b
the corresponding (dotted) curve in Fig. 2b
- p. 4 Fig. 2 caption
extended vulner- able cluster (short dashes)
Judge's note: Both phrases read on the page 4 image. The same curve is given two line-style names; the panel also has a long-dashed curve, so the names matter a little, but a short-dashed line can loosely be called dotted and the curve is identifiable.
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Typo Fig. 1 caption (page 4) has "a homogenous threshold distribution" while the text on the same page and elsewhere has "homogeneous thresholds".
- p. 4 Fig. 1 caption
for a uniform random graph with a homogenous threshold distribution
- p. 4 left column, first paragraph under Fig. 1
In the special case of a uniform random graph with homogeneous
Judge's note: Read on the page 4 image. "Homogenous" is an accepted variant spelling, so this is only an inconsistency within the paper.
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Typo Fig. 4 caption (page 5): "where the threshold distributions (φ) are normally distributed with mean φ and SD σ = 0.05 and σ = 0.1" drops the f of f(φ) and gives the mean as φ where the text has "threshold distributions f(φ) that are normally distributed with mean φ*".
- p. 5 Fig. 4 caption (a)
but where the threshold distributions (φ) are nor-
- p. 5 Fig. 4 caption (a)
mally distributed with mean φ and SD σ = 0.05
- p. 5 right column, last paragraph
threshold distributions f(φ) that are normally distributed with mean
- p. 5 Fig. 4 caption (a)
where the threshold distributions (φ) are nor-
Judge's note: Read on the page 5 image: the caption prints "(φ)" with no function symbol and "mean φ" with no asterisk, while the body text on the same page prints f(φ) and φ*. The text layer agrees (the f is present in the body and absent in the caption), so it is not an extraction loss. The meaning is recoverable.
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Typo The exponential cut-off of the power-law degree distribution is κ in the text (page 6: "pk = Ck^−τ e^−k/κ (k > 0), where C, τ, and κ are constants") and κ0 in the Fig. 4 caption (page 5: "exponential cut-off κ0 ... κ0 has been adjusted").
- p. 6 left column, first paragraph
where C, τ, and κ are constants that can be
- p. 5 Fig. 4 caption (b)
exponential cut-off κ0, where τ has been
- p. 6 Results and Discussion, left column
where C, τ, and κ are constants
- p. 5 Fig. 4 caption
mean φ and SD σ = 0.05
Judge's note: Both read on the page images; the caption prints a subscript 0 twice and the text none. The caption says κ0 is adjusted to vary z, so it is the same free parameter. A notation inconsistency only; the mean φ versus φ* point raised in one note is covered by C011.
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Typo Reference 25 (page 6): "Adler, J. (1991) Physics A 171, 453–470." gives the journal as "Physics A"; reference 26 has "Physica A 277, 239–247".
- p. 6 Reference 25
25. Adler, J. (1991) Physics A 171, 453–470.
- p. 6 Reference 26
Physica
- p. 6 Reference list, entry 26
A 277, 239–247.
- p. 6 Reference 25
Adler, J. (1991) Physics A 171, 453–470.
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: that the journal of ref. 25 is Physica A (no title is printed and the cited article is not in the files). The page 6 image shows "Physics A" in ref. 25 and "Physica A" in the next entry, so the list is internally inconsistent in a way that supports the reading. The other reference-list slips mentioned in two notes (Schheinkman, 02611.8) are judged in C015 and C014.
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Typo Reference 32 (page 6): "Newman, M. E. J., Strogatz, S. H. & Watts, D. J. (2001) Phys. Rev. E 64, 02611.8." has a full stop inside the article number.
- p. 6 Reference 32
Phys. Rev. E 64, 02611.8.
- p. 6 Reference 32
Phys. Rev. E 64, 02611.8
- p. 6 Reference 32
64, 02611.8
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: that the article number of this Phys. Rev. E paper is 026118 (six digits, no stop). The page 6 image shows "02611.8." with a stop inside the number and the entry's closing stop after it, so it is not an extraction artefact.
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Typo Reference 16 (page 6): "Glaeser, E. L., Sacerdote, B. & Schheinkman, J. A. (1996) Quart. J. Econ. 111, 507–548." spells the third author with a doubled h.
- p. 6 Reference 16
Glaeser, E. L., Sacerdote, B. & Schheinkman, J. A. (1996)
- p. 6 Reference 16
Schheinkman, J. A.
- p. 6 Reference 16
Schheinkman
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: that the third author of this paper is J. A. Scheinkman. The page 6 image shows "Schheinkman" as quoted, so the spelling is in the published PDF and not an extraction artefact.
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