Complex Contagions and the Weakness of Long Ties
American Journal of Sociology, 2007 · paper · Sociology
Tests whether the "strength of weak ties" holds for complex contagions needing reinforcement from multiple sources. Using formal models, it finds that as adoption thresholds rise, long ties can impede diffusion, which depends on the width of bridges across a network.
Checked: PDF from repository.upenn.edu, fetched 5 Oct 2026. Page numbers are those of that PDF.
7 findings
Substance p. 16: "on a ring lattice with z = 10, Wmax = 15 and R = .75 ... (a = 2 and Wc = 3)". The paper's own eq. 5, R = (Wmax − Wc)/Wmax, and eq. 6, R = 1 − 4a(a+1)/(z(z+2)), both give R = .8 for these values.
- p. 16 Text after eq. 6
with z p 10, Wmax p 15 and R p .75
- p. 16 Ring lattice discussion, paragraph after eq. 6
on a ring lattice with z p 10, Wmax p 15 and R p .75
- p. 16 text after eq. 6
z p 10, Wmax p 15 and R p .75
- p. 16 text after eq. 6
with z = 10, Wmax = 15 and R = .75
Judge's note: Checked against the page 16 image and the definitions on pp. 12–15. Eq. 2 gives Wmax = z(z+2)/8, which is 15 for z = 10. Eq. 4 gives Wc = a(a+1)/2, which is 3 for a = 2. Eq. 5 then gives R = (15−3)/15 = 0.8, and eq. 6 gives 1 − 24/120 = 0.8. The printed .75 matches neither equation.
Second check: another model agreed
Report a problem with this finding
Reference or label Reference "McPherson, J. Miller, and Lynn Smith-Lovin. 1986. 'Women and Weak Ties: ...' American Journal of Sociology 87:883–904" (cited in text as McPherson and Smith-Lovin 1986) pairs year 1986 with AJS volume 87, inconsistent with the list's other AJS entries (e.g., McAdam 1986 in vol. 92).
- p. 32 References
McPherson, J. Miller, and Lynn Smith-Lovin. 1986.
- p. 3 Text
(McPherson and Smith-Lovin 1986)
- p. 32 References
McPherson, J. Miller, and Lynn Smith-Lovin. 1986. “Women and Weak Ties
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: that 'Women and Weak Ties' appeared in AJS vol. 87 in 1982. Within the file, AJS 1986 = vol. 92 (McAdam 1986) and AJS 1983 = vol. 89 implied by sequence (1981 Ruef 2004=109, 1994=99), so vol. 87 corresponds to ~1982; year and volume conflict. Only authors can say which is right; in-text citation also uses 1986.
Second check: another model agreed
Report a problem with this finding
Reference or label p. 7: "to threshold models (Granovetter 1973)". In the reference list, Granovetter 1973 is 'The Strength of Weak Ties'; the threshold-model paper is Granovetter 1978, 'Threshold Models of Collective Behavior'.
- p. 7 Mechanism 4, Emotional contagion
to threshold models (Gra- novetter 1973)
- p. 7
to threshold models (Granovetter 1973) to cybernetics
Judge's note: The page 7 image shows '(Granovetter 1973)' in the threshold-models clause. The page 31 reference list has separate 1973 (Strength of Weak Ties) and 1978 (Threshold Models of Collective Behavior) entries. Elsewhere the text cites Granovetter 1978 for thresholds (text-layer lines 168, 255, 418). The year in this citation is wrong.
Second check: another model agreed
Report a problem with this finding
Reference or label Reference list, p. 31: Gould 1993, "Collective Action and Network Structure." American Journal of Sociology 58:182–96. Volume 58 cannot be a 1993 AJS volume.
- p. 31 References, Gould 1993
“Collective Action and Network Structure.” American Journal of Sociology 58:182–96.
- p. 31 References
1993. “Collective Action and Network Structure.” American Journal of Sociology 58:182–96.
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: that the article appeared in American Sociological Review 58 (1993). The inconsistency itself shows within the reference list on the page 31 image, which gives AJS 78 for 1973 (Granovetter), AJS 83 for 1978, and AJS 99 for 1993–94 (Hedstrom 1994; McAdam and Paulsen 1993). AJS volume 58 cannot therefore date from 1993. ASR 58 does fit 1993, matching the Opp and Gern 1993 entry.
Second check: another model agreed
Report a problem with this finding
Reference or label Reference list, p. 31: Collins 1974, "Three Faces of Cruelty..." Theory and Society 12:631–58. Volume 12 does not fit a 1974 date.
- p. 31 References, Collins 1974
Theory and Society 12:631–58.
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: that Theory and Society began in 1974, so a 1974 article would be in volume 1 (reportedly pp. 415–40). The page 31 image confirms the entry reads '12:631–58' with the year 1974. Nothing in the files settles which element is wrong, so the authors should check it.
Second check: another model agreed
Report a problem with this finding
Reference or label Heath, Bell and Sternberg 2001 is listed as "Journal of Personality and Social Psychology 2:563–69".
- p. 31 References, Heath et al.
Journal of Personality and Social Psychology 2:563–69.
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: JPSP volumes in 2001 were numbered in the 80s (the article is in vol. 81), so volume 2 cannot be right for 2001. The page image shows "2:563–69".
Second check: another model agreed
Report a problem with this finding
Wording P. 716 says "Wmax increases exponentially with z while Wc increases exponentially with a", and p. 718 repeats "exponentially". The paper's own eqs. (2) and (4) give Wmax = z(z+2)/8 and Wc = a(a+1)/2, which are quadratic.
- p. 15 Text before eq. 5
Wmax increases exponentially with z while Wc increases exponentially with a.
- p. 17 Summary point 1
in turn increases exponentially with the required number of activated neighbors (a).
- p. 15 Para after fig. 2
Wmax increases exponentially with z while Wc increases exponentially with a
- p. 17 Review point 1
form a bridge in turn increases exponentially with the required number of activated neighbors (a)
Judge's note: Checked the equations on pp. 713–714 against the claims on pp. 716–718. Under the paper's own formulas, the growth is polynomial, not exponential.
Second check: another model agreed
Report a problem with this finding
10 minor (typos and the like; most checked by the judge only)
Reference or label The Backstrom et al. reference gives the venue as "12th ACM SIGKDD International Conference on Knowledge Discovery in Data".
- p. 30 References, Backstrom et al.
International Conference on Knowledge Discovery in Data
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: the conference is titled Knowledge Discovery and Data Mining. The page image shows the printed wording.
Second check: another model agreed
Report a problem with this finding
Wording p. 3: Granovetter (1973) "has informed hundreds of empirical studies in the four decades since its publication". From 1973 to this 2007 article is about 34 years, roughly three and a half decades, not four.
- p. 3 Text
studies in the four decades since its publication
- p. 3 Paragraph beginning "It has become a truism"
in the four decades since its publication
- p. 3
hundreds of empirical studies in the four decades since its publication
Judge's note: The page image shows 'four decades' next to Granovetter (1973). The reference list gives Granovetter 1973, AJS 78:1360–80, and the article is from 2007. The span is 34 years, so 'four decades' overstates it. This is a wording point; no result depends on it.
Second check: another model agreed
Report a problem with this finding
Wording Footnote 12, p. 16: "However, if a ≥ 1 and R = 0, the first tie that is randomly rewired will break the ring but cannot create a shortcut." The previous sentence says that for a = 1, z = 2 (R = 0) the rewired tie does create a bridge across the ring, so the condition should exclude a = 1 (a > 1).
- p. 16 Footnote 12
However, if a ≥ 1 and R p 0, the first tie that is randomly rewired will break the ring but cannot create a shortcut.
- p. 16 Footnote 12
if a ≥ 1 and R p 0, the first tie that is randomly rewired will break the ring but cannot create a shortcut
Judge's note: The page 16 image shows '≥' in the footnote. The footnote first presents a = 1, z = 2 as a case where the rewired tie creates a bridge across the ring, then states with 'However' that for a ≥ 1 no shortcut is possible. That includes the a = 1 case just described, so the two sentences contradict each other as printed. The contrast only works if the condition reads a > 1.
Second check: another model agreed
Report a problem with this finding
Wording Footnote 14, p. 17: "Qualitatively similar results are found for r = 1 and r ≥ 1." The second condition includes the first, so the intended comparison (probably r > 1) is not clear.
- p. 17 Footnote 14
Qualitatively similar results are found for r p 1 and
- p. 17 footnote 14
Qualitatively similar results are found for r = 1 and r ≥ 1.
Only the authors can say what is right here.
Judge's note: The page 17 image shows 'r = 1 and r ≥ 1'. As printed the statement is redundant. The intended range is probably r > 1, given that z = 8 corresponds to r = 1 and z = 48 to r = 3, but only the authors can say which range was meant.
Second check: another model agreed
Report a problem with this finding
Typo p. 4: the in-text citation reads "Stark and Verdes 2006", but the reference list (p. 33) gives "Stark, David, and B. Vedres. 2006". The co-author's name is misspelled in the text.
- p. 4 Text
Stark and Verdes 2006
- p. 33 References
Stark, David, and B. Vedres. 2006.
- p. 33 References, Stark entry
Stark, David, and B. Vedres. 2006
- p. 33
Stark, David, and B. Vedres. 2006. “Social Times of Network Spaces: Network Sequences and
Judge's note: The page images show 'Verdes' on p. 4 and 'Vedres' in the reference entry on p. 33. One reviewer also says the first page should be 1367 rather than the printed 1368, based on Crossref. Rests on outside knowledge: that claim depends on the Crossref record and cannot be checked from these files, so it is not part of this verdict.
Second check: code check
Report a problem with this finding
Typo The text on p. 706 reads "the seemingly irrational behavior of a maddening crowd (McPhail 1991)", while the cited title is "The Myth of the Madding Crowd".
- p. 5 Text
the seemingly irrational behavior of a maddening crowd (McPhail 1991)
- p. 32 References
The Myth of the Madding Crowd.
- p. 5 From Simple to Complex Contagions, para 1
irrational behavior of a maddening crowd
- p. 32 References
The Myth of the Madding Crowd
Judge's note: The text paraphrases rather than quotes the title. The wording nonetheless departs from the cited title, which sits directly beside it, and "madding" and "maddening" differ in meaning. This is a minor wording point.
Second check: none (judge only)
Report a problem with this finding
Typo The McAdam and Paulsen quotation on p. 707 reads "embedded in many relationships means than any major decision"; "that" is intended.
- p. 6 Quotation of McAdam and Paulsen
bedded in many relationships means than any major decision
- p. 6 Para 2
means than any major decision
- p. 6 para 1, McAdam and Paulsen quote
means than any major decision we are
Judge's note: The page image shows "than". I could not check whether the slip is already in the original source; if it is, a [sic] would be needed.
Second check: none (judge only)
Report a problem with this finding
Typo P. 730 reads "(CDC 1997, pp. 3-2)": a single page in chapter-page numbering is cited with the plural "pp.".
- p. 29 Text
(CDC 1997, pp. 3-2)
Judge's note: The page image shows "pp. 3-2". Since 3-2 denotes one page, "p. 3-2" is the expected form.
Second check: none (judge only)
Report a problem with this finding
Typo The Erdos and Renyi reference prints the journal as "Publicationes Mathematicae Debrencen".
- p. 31 References, Erdos and Renyi
Debrencen 6:290–97.
- p. 31 References, Erdos and Renyi 1959
Publicationes Mathematicae Debrencen 6:290–97
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: the journal is Publicationes Mathematicae Debrecen. The page image shows "Debrencen".
Second check: none (judge only)
Report a problem with this finding
Typo Reference list, p. 32: Mizruchi and Stearns 2001, "Getting Deals Done: Thus of Social Networks in Bank Decision-Making." The title is garbled ('Thus of').
- p. 32 References, Mizruchi and Stearns
Getting Deals Done: Thus of Social Networks in Bank Decision-Making.
- p. 32 References
Getting Deals Done: Thus of Social Networks in Bank Decision-Making
- p. 32 References, Mizruchi and Stearns 2001
Thus of Social
- p. 32 References
Getting Deals Done: Thus of Social
Only the authors can say what is right here.
Judge's note: The crop image confirms 'Thus of Social Networks' as printed, so this is not an extraction artefact, and the wording is ungrammatical. Rests on outside knowledge: the exact published wording ('The Use of Social Networks...') cannot be checked from these files.
Second check: none (judge only)