The Annals of Applied Probability

Jigsaw percolation: What social networks can collaboratively solve a puzzle?

Charles D. Brummitt, Shirshendu Chatterjee, Partha S. Dey, and David Sivakoff

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We introduce a new kind of percolation on finite graphs called jigsaw percolation. This model attempts to capture networks of people who innovate by merging ideas and who solve problems by piecing together solutions. Each person in a social network has a unique piece of a jigsaw puzzle. Acquainted people with compatible puzzle pieces merge their puzzle pieces. More generally, groups of people with merged puzzle pieces merge if the groups know one another and have a pair of compatible puzzle pieces. The social network solves the puzzle if it eventually merges all the puzzle pieces. For an Erdős–Rényi social network with $n$ vertices and edge probability $p_{n}$, we define the critical value $p_{c}(n)$ for a connected puzzle graph to be the $p_{n}$ for which the chance of solving the puzzle equals $1/2$. We prove that for the $n$-cycle (ring) puzzle, $p_{c}(n)=\Theta(1/\log n)$, and for an arbitrary connected puzzle graph with bounded maximum degree, $p_{c}(n)=O(1/\log n)$ and $\omega(1/n^{b})$ for any $b>0$. Surprisingly, with probability tending to 1 as the network size increases to infinity, social networks with a power-law degree distribution cannot solve any bounded-degree puzzle. This model suggests a mechanism for recent empirical claims that innovation increases with social density, and it might begin to show what social networks stifle creativity and what networks collectively innovate.

Article information

Ann. Appl. Probab., Volume 25, Number 4 (2015), 2013-2038.

Received: September 2012
Revised: May 2014
First available in Project Euclid: 21 May 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 91D30: Social networks
Secondary: 05C80: Random graphs [See also 60B20]

Percolation social networks random graph phase transition


Brummitt, Charles D.; Chatterjee, Shirshendu; Dey, Partha S.; Sivakoff, David. Jigsaw percolation: What social networks can collaboratively solve a puzzle?. Ann. Appl. Probab. 25 (2015), no. 4, 2013--2038. doi:10.1214/14-AAP1041.

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