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Long ties accelerate noisy threshold-based contagions


In widely used models of biological contagion, interventions that randomly rewire edges (generally making them ‘longer’) accelerate spread. However, recent work has argued that highly clustered, rather than random, networks facilitate the spread of threshold-based contagions, such as those motivated by myopic best response for adoption of new innovations, norms and products in games of strategic complement. Here we show that minor modifications to this model reverse this result, thereby harmonizing qualitative facts about how network structure affects contagion. We analyse the rate of spread over circular lattices with rewired edges and show that having a small probability of adoption below the threshold probability is enough to ensure that random rewiring accelerates the spread of a noisy threshold-based contagion. This conclusion is verified in simulations of empirical networks and remains valid with partial but frequent enough rewiring and when adoption decisions are reversible but infrequently so, as well as in high-dimensional lattice structures.

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Fig. 1: Illustration of activation functions and local network structure.
Fig. 2: Network structures combining lattices and random graphs.
Fig. 3: Spread time of noisy complex contagion over rewired \({{{{{\mathcal{C}}}}}}_{{2}}\) graphs.
Fig. 4: Noisy threshold-based contagions on empirical networks and their structural variations.

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Data availability

The simulations on the empirical networks use data that were publicly released8,33,34,40. The data from ref. 8 are available at The data from ref. 34 are available in the associated journal replication package. The data from ref. 33 are available at The data from ref. 40 are available at

Code availability

The code for the reported simulations can be accessed from


  1. Leskovec, J., Adamic, L. A. & Huberman, B. A. The dynamics of viral marketing. ACM Trans. Web 1, 5 (2007).

    Article  Google Scholar 

  2. Kempe, D., Kleinberg, J. & Tardos, É. Maximizing the spread of influence through a social network. In Proc. 9th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (eds Getoor, L. et al.) 137–146 (ACM, 2003).

  3. Hinz, O., Skiera, B., Barrot, C. & Becker, J. U. Seeding strategies for viral marketing: an empirical comparison. J. Mark. 75, 55–71 (2011).

    Article  Google Scholar 

  4. Libai, B., Muller, E. & Peres, R. Decomposing the value of word-of-mouth seeding programs: acceleration versus expansion. J. Mark. Res. 50, 161–176 (2013).

    Article  Google Scholar 

  5. Beaman, L., BenYishay, A., Magruder, J. & Mobarak, A. M. Can network theory-based targeting increase technology adoption?. Am. Econ. Rev. 111, 1918–1943 (2021).

    Article  Google Scholar 

  6. Cohen, R., Havlin, S. & Ben-Avraham, D. Efficient immunization strategies for computer networks and populations. Phys. Rev. Lett. 91, 247901 (2003).

    Article  PubMed  Google Scholar 

  7. Preciado, V. M., Zargham, M., Enyioha, C., Jadbabaie, A. & Pappas, G. J. Optimal resource allocation for network protection against spreading processes. IEEE Trans. Control Netw. Syst. 1, 99–108 (2014).

    Article  Google Scholar 

  8. Chami, G. F., Ahnert, S. E., Kabatereine, N. B. & Tukahebwa, E. M. Social network fragmentation and community health. Proc. Natl Acad. Sci. USA 114, E7425–E7431 (2017).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  9. Chaoji, V., Ranu, S., Rastogi, R. & Bhatt, R. Recommendations to boost content spread in social networks. In Proc. 21st International Conference on World Wide Web (eds Mille, A. et al.) 529–538 (ACM, 2012).

  10. Valente, T. W. Network interventions. Science 337, 49–53 (2012).

    Article  CAS  PubMed  Google Scholar 

  11. Carrell, S. E., Sacerdote, B. I. & West, J. E. From natural variation to optimal policy? The importance of endogenous peer group formation. Econometrica 81, 855–882 (2013).

    Article  Google Scholar 

  12. Cerdeiro, D. A., Dziubiński, M. & Goyal, S. Individual security, contagion, and network design. J. Econ. Theory 170, 182–226 (2017).

    Article  Google Scholar 

  13. Dodds, P. S. & Watts, D. J. A generalized model of social and biological contagion. J. Theor. Biol. 232, 587–604 (2005).

    Article  CAS  PubMed  Google Scholar 

  14. Watts, D. J. & Strogatz, S. H. Collective dynamics of ‘small-world’ networks. Nature 393, 440–442 (1998).

    Article  CAS  PubMed  Google Scholar 

  15. Hébert-Dufresne, L., Noël, P.-A., Marceau, V., Allard, A. & Dubé, L. J. Propagation dynamics on networks featuring complex topologies. Phys. Rev. E 82, 036115 (2010).

  16. Granovetter, M. S. The strength of weak ties. Am. J. Sociol. 78, 1360–1380 (1973).

    Article  Google Scholar 

  17. Aral, S. & Van Alstyne, M. The diversity-bandwidth trade-off. Am. J. Sociol. 117, 90–171 (2011).

    Article  Google Scholar 

  18. Jahani, E., Fraiberger, S. P., Bailey, M. & Eckles, D. Long ties, disruptive life events, and economic prosperity. Proc. Natl Acad. Sci. USA 120, e2211062120 (2023).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  19. Gee, L. K., Jones, J. J., Fariss, C. J., Burke, M. & Fowler, J. H. The paradox of weak ties in 55 countries. J. Econ. Behav. Organ. 133, 362–372 (2017).

    Article  Google Scholar 

  20. Rajkumar, K., Saint-Jacques, G., Bojinov, I., Brynjolfsson, E. & Aral, S. A causal test of the strength of weak ties. Science 377, 1304–1310 (2022).

    Article  CAS  PubMed  Google Scholar 

  21. Galeotti, A., Goyal, S., Jackson, M. O., Vega-Redondo, F. & Yariv, L. Network games. Rev. Econ. Stud. 77, 218–244 (2010).

    Article  Google Scholar 

  22. Blume, L. E. The statistical mechanics of strategic interaction. Games Econ. Behav. 5, 387–424 (1993).

    Article  Google Scholar 

  23. Morris, S. Contagion. Rev. Econ. Stud. 67, 57–78 (2000).

    Article  Google Scholar 

  24. Young, H. P. The dynamics of social innovation. Proc. Natl Acad. Sci. USA 108, 21285–21291 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  25. Granovetter, M. Threshold models of collective behavior. Am. J. Sociol. 83, 1420–1443 (1978).

    Article  Google Scholar 

  26. Centola, D. & Macy, M. Complex contagions and the weakness of long ties. Am. J. Sociol. 113, 702–734 (2007).

    Article  Google Scholar 

  27. Montanari, A. & Saberi, A. The spread of innovations in social networks. Proc. Natl Acad. Sci. USA 107, 20196–20201 (2010).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  28. Guilbeault, D. & Centola, D. Topological measures for identifying and predicting the spread of complex contagions. Nat. Commun. 12, 4430 (2021).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  29. Bakshy, E., Rosenn, I., Marlow, C. & Adamic, L. The role of social networks in information diffusion. In Proc. 21st International Conference on World Wide Web (eds Mille, A. et al.) 519–528 (ACM, 2012).

  30. Bakshy, E., Eckles, D., Yan, R. & Rosenn, I. Social influence in social advertising: evidence from field experiments. In Proc. 13th ACM Conference on Electronic Commerce (eds Faltings, B. et al.) 146–161 (ACM, 2012).

  31. Centola, D. The spread of behavior in an online social network experiment. Science 329, 1194–1197 (2010).

    Article  CAS  PubMed  Google Scholar 

  32. Jackson, M. & Rogers, B. The economics of small worlds. J. Eur. Econ. Assoc. 3, 617–627 (2005).

    Article  Google Scholar 

  33. Traud, A. L., Mucha, P. J. & Porter, M. A. Social structure of Facebook networks. Phys. A 391, 4165–4180 (2012).

    Article  Google Scholar 

  34. Ugander, J., Backstrom, L., Marlow, C. & Kleinberg, J. Structural diversity in social contagion. Proc. Natl Acad. Sci. USA 109, 5962–5966 (2012).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  35. Park, P. S., Blumenstock, J. E. & Macy, M. W. The strength of long-range ties in population-scale social networks. Science 362, 1410–1413 (2018).

    Article  CAS  PubMed  Google Scholar 

  36. Iacopini, I., Petri, G., Barrat, A. & Latora, V. Simplicial models of social contagion. Nat. Commun. 10, 2485 (2019).

    Article  PubMed  PubMed Central  Google Scholar 

  37. Ferraz de Arruda, G., Petri, G., Rodriguez, P. M. & Moreno, Y. Multistability, intermittency, and hybrid transitions in social contagion models on hypergraphs. Nat. Commun. 14, 1375 (2023).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  38. Akbarpour, M. & Jackson, M. O. Diffusion in networks and the virtue of burstiness. Proc. Natl Acad. Sci. USA 115, E6996–E7004 (2018).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  39. Banerjee, A., Chandrasekhar, A. G., Duflo, E. & Jackson, M. O. The diffusion of microfinance. Science 341, 1236498 (2013).

    Article  PubMed  Google Scholar 

  40. Cai, J., De Janvry, A. & Sadoulet, E. Social networks and the decision to insure. Am. Econ. J. Appl. Econ. 7, 81–108 (2015).

    Article  Google Scholar 

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E.M. was partially supported by a National Science Foundation (NSF) grant no. CCF 1665252, Department of Defense Office of Naval Research (ONR) grant no. N00014-17-1-2598, NSF grant no. DMS-1737944, Vannevar Bush Faculty Fellowship (ONR-N00014-20-1-2826), Simons Investigator award (no. 622132), ARO Multidisciplinary University Initiative W911NF1910217 and NSF award no. CCF 1918421. M.A.R. was partially supported by the NSF (SaTC-2318844), a Pitt Momentum Funds award and a Pitt Cyber Accelerator grant. This research was supported in part by the University of Pittsburgh Center for Research Computing, research resource identifier SCR_022735, through the resources provided. Specifically, this work used the H2P cluster, which is supported by NSF award no. OAC-2117681. During his postdoctoral work at the Massachusetts Institute of Technology, M.A.R. was supported by an Amazon Research Award to D.E. S.S. was partially supported by the NSF (DMS CAREER 2239234), ONR (N00014-23-1-2489) and Air Force Office of Scientific Research (FA9950-23-1-0429). The funders had no role in study design, data collection and analysis, decision to publish or preparation of the manuscript. We thank C. Hurtado, Y. Long and C. S. Reid for research assistance. We thank S. Aral, S. Morris and D. G. Rand for helpful comments. We also thank R. Cohen and J. Moody for their reviews.

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D.E., E.M., M.A.R. and S.S. conceived the research and contributed to the analysis. M.A.R. led the writing of the paper, with input from all authors. All authors approved the final paper.

Corresponding authors

Correspondence to Dean Eckles or M. Amin Rahimian.

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Competing interests

Meta (which operates Facebook) has sponsored a conference co-organized by D.E. and has funded some of his other research. M.A.R. has served on the advisory committee of a vaccine confidence fund created by Meta and Merck; some of his research has also been funded by Meta. The other authors declare no competing interests.

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Eckles, D., Mossel, E., Rahimian, M.A. et al. Long ties accelerate noisy threshold-based contagions. Nat Hum Behav (2024).

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