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The sharp threshold for jigsaw percolation in random graphs

  • Oliver Cooley (a1), Tobias Kapetanopoulos (a2) and Tamás Makai (a3)


We analyse the jigsaw percolation process, which may be seen as a measure of whether two graphs on the same vertex set are ‘jointly connected’. Bollobás, Riordan, Slivken, and Smith (2017) proved that, when the two graphs are independent binomial random graphs, whether the jigsaw process percolates undergoes a phase transition when the product of the two probabilities is $\Theta({1}/{(n\ln n)})$ . We show that this threshold is sharp, and that it lies at ${1}/{(4n\ln n)}$ .


Corresponding author

*Postal address: Institute of Discrete Mathematics, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria. Email address: Supported by the Austrian Science Fund (FWF) & German Research Foundation (DFG) under grant I3747.
**Postal address: Mathematics Institute, Goethe University, 10 Robert Mayer Street, Frankfurt 60325, Germany. Email address: Supported by a Stiftung Polytechnische Gesellschaft PhD grant.
***Postal address: School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS. Email address: Supported by the Austrian Science Fund (FWF) under grant P26826 and the EPSRC under grant EP/N004221/1.


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[1] Andrews, G. E., Askey, R. and Roy, R. (1999). Special Functions (Encyclopedia Math. Appl. 71). Cambridge University Press.
[2] Barabási, A.-L. and Albert, R. (1999). Emergence of scaling in random networks. Science 286 , 509512.
[3] Bollobás, B. and Thomason, A. (1985). Random Graphs of Small Order (North-Holland Math. Stud. 118). North-Holland, Amsterdam, pp. 47–97.
[4] Bollobás, B., Cooley, O., Kang, M. and Koch, C. (2017). Jigsaw percolation on random hypergraphs. J. Appl. Prob. 54 , 12611277.
[5] Bollobás, B., Riordan, O., Slivken, E., and Smith, P. (2017). The threshold for jigsaw percolation on random graphs. Electron. J. Combinatorics 24, 14pp.
[6] Bollobás, B., Riordan, O., Spencer, J. and Tusnády, G. (2001). The degree sequence of a scale-free random graph process. Random Structures Algorithms 18 , 279290.
[7] Brummitt, C. D., Chatterjee, S., Dey, P. S., and Sivakoff, D. (2015). Jigsaw percolation: What social networks can collaboratively solve a puzzle? Ann. Appl. Prob. 25 , 20132038.
[8] Buldyrev, S. V., Parshani, R., Paul, G., Stanley, H. E. and Havlin, S. (2010). Catastrophic cascade of failures in interdependent networks. Nature 464 , 10251028.
[9] Cooley, O. and Gutiérrez, A. (2017). Multi-coloured jigsaw percolation on random graphs. Preprint. Available at
[10] Erdös, P. and Rényi, A. (1959). On random graphs. I. Publ. Math. Debrecen 6 , 290297.
[11] Gravner, J. and Sivakoff, D. (2017). Nucleation scaling in jigsaw percolation. Ann. Appl. Prob. 27 , 395438.
[12] Janson, S., uczak, T., and Ruciski, A. (2000). Random Graphs (Wiley-Interscience Series in Discrete Mathematics and Optimization). Wiley-Interscience, New York.
[13] Krioukov, D., Papadopoulos, F., Kitsak, M., Vahdat, A. and Boguñá, M. (2010). Hyperbolic geometry of complex networks. Phys. Rev. E (3) 82, 18pp.
[14] Robbins, H. (1955). A remark on Stirling’s formula. Amer. Math. Monthly 62 , 2629.


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The sharp threshold for jigsaw percolation in random graphs

  • Oliver Cooley (a1), Tobias Kapetanopoulos (a2) and Tamás Makai (a3)


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