Emergence of Scaling in Random Networks
Science, 1999 · paper · Computer science
Reports that vertex connectivities in many large networks, such as the web and actor collaborations, follow a scale-free power-law distribution. Shows a model with network growth and preferential attachment to well-connected vertices reproduces this distribution.
Checked: PDF from arxiv.org, fetched 5 Oct 2026. Page numbers are those of that PDF.
3 findings
Substance The ER degree distribution is given as "P(k) = e^−λλ^k/k!, where λ = N (N−1 choose k) p^k(1−p)^(N−1−k)". The λ printed depends on k (it is the expected number of degree-k vertices), so it cannot be the Poisson parameter of a single vertex's degree distribution, which would be the mean degree p(N−1).
- p. 4 Paragraph on ER random graph model
P(k) = e−λλk/k!
Only the authors can say what is right here.
Judge's note: Checked on the page 4 image (crop C001-a). My own working: a vertex's degree in G(N,p) is Binomial(N−1,p), so (N−1 choose k)p^k(1−p)^(N−1−k) is already P(k), and its Poisson limit has the constant parameter λ = p(N−1) ≈ pN. The printed λ contains k and has an extra factor N. It matches the ER/Bollobás result for the expected number of vertices of degree k (the count of such vertices is Poisson with that mean), not the probability that one vertex has k edges. As printed, the formula is not a valid Poisson degree law. Only the authors can say which statement they meant.
Second check: another model agreed
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Reference or label "we obtain γ(p) = 3 − p, which is supported by numerical simulations (16)", but note 16 reads "Note that for most networks the connectivity m of the newly added vertices is not constant. However, choosing m randomly will not change the exponent γ [Y. Tu, private communication]". It says nothing about directed links or simulations of γ(p).
- p. 7 Discussion of modified exponents
we obtain γ(p) = 3 −p, which is supported by numerical simulations (16)
- p. 9 Note 16
However, choosing m randomly will not change the exponent γ [Y. Tu, private
- p. 9 Note 16
choosing m randomly will not change the exponent γ [Y. Tu, private
Only the authors can say what is right here.
Judge's note: Read the page 7 sentence (crop C006-a) and note 16 on page 9 (crops C006-b, C006-c). Note 16 is only about choosing m at random and does not mention directed links, the formula 3 − p, or simulations of that variant. The text cites (16) only at this one place, so the pointer does not support the claim it is attached to. The note may have been meant for another sentence, such as the definition of m; only the authors can say.
Second check: another model agreed
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Reference or label Ref. 15 reads "M. Barthélémy and L. A. N. Amaral, Phys. Rev. Lett. 82, 15 (1999).". The '15' appears to be the issue number, not the article's page.
- p. 9 Reference 15
Phys. Rev. Lett. 82, 15 (1999).
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: the Barthélémy & Amaral paper 'Small-world networks: evidence for a crossover picture' appeared in Phys. Rev. Lett. 82, 3180 (1999), in issue 15. The page 9 image confirms the entry prints '82, 15 (1999)', with no page number other than 15. The files cannot confirm the true page.
Second check: another model agreed
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3 minor (typos and the like; most checked by the judge only)
Wording Model B: "In model B we start with N vertices and no edges. At each time step we randomly select a vertex and connect it with probability Π(ki) = ki/Σj kj". With every ki = 0, this rule is 0/0 at the first step, and no seeding rule is given.
- p. 6 Description of model B
In model B we start with N vertices and no edges.
Only the authors can say what is right here.
Judge's note: Checked on the page 6 image. When there are no edges, every ki = 0 and Σj kj = 0, so the printed attachment probability is undefined at the start, and the text gives no offset or seed. The extension to the main model is not established: page 5 says only "starting with a small number (m0) of vertices" and does not say they have no edges, so the statement is limited to model B. This is an ambiguity in the methods description that the authors would need to resolve.
Second check: another model agreed
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Typo "P(k) is independent of time (and, subsequently, independent of the system size m0 + t)". 'subsequently' is used where 'consequently' is meant.
- p. 5 Paragraph after model definition, Fig. 2A discussion
(and, subsequently, independent of the system size m0 + t)
Judge's note: Seen on the page 5 image. Independence of system size follows from independence of time because the size is m0 + t, so the intended sense is logical consequence, not a later time. This is a word-choice slip, and the meaning can be recovered from context.
Second check: none (judge only)
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Typo The opening quotation mark in a ”rich-gets-richer” phenomenon is typeset as a closing double quote.
- p. 6 Paragraph on growth rate of ki
a ”rich-gets-richer” phenomenon
Judge's note: The page 6 image (crop C005-a) shows the opening mark before 'rich' with the same closing-quote glyph as the mark after 'richer'. This is the usual LaTeX " vs `` slip. Cosmetic.
Second check: none (judge only)