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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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1432212435

Digital Object Identifier
doi:10.1214/14-AAP1041

Mathematical Reviews number (MathSciNet)
MR3349000

Zentralblatt MATH identifier
1322.60210

Subjects
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]

Keywords
Percolation social networks random graph phase transition

Citation

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. https://projecteuclid.org/euclid.aoap/1432212435


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References

  • [1] Aizenman, M. and Lebowitz, J. L. (1988). Metastability effects in bootstrap percolation. J. Phys. A 21 3801–3813.
  • [2] Ball, P. (2014). Crowd-sourcing: Strength in numbers. Nature 506 422–423.
  • [3] Barabási, A. L., Jeong, H., Néda, Z., Ravasz, E., Schubert, A. and Vicsek, T. (2002). Evolution of the social network of scientific collaborations. Phys. A 311 590–614.
  • [4] Bettencourt, L., Lobo, J., Helbing, D., Kühnert, C. and West, G. B. (2007). Growth, innovation, scaling, and the pace of life in cities. Proc. Natl. Acad. Sci. USA 104 7301.
  • [5] Bettencourt, L. M. A., Cintrón-Arias, A., Kaiser, D. I. and Castillo-Chávez, C. (2006). The power of a good idea: Quantitative modeling of the spread of ideas from epidemiological models. Phys. A 364 513–536.
  • [6] Bettencourt, L. M. A., Kaiser, D. I. and Kaur, J. (2009). Scientific discovery and topological transitions in collaboration networks. J. Informetr. 3 210–221.
  • [7] Bettencourt, L. M. A., Kaiser, D. I., Kaur, J., Castillo-Chávez, C. and Wojick, D. E. (2008). Population modeling of the emergence and development of scientific fields. Scientometrics 75 495–518.
  • [8] Bettencourt, L. M. A., Lobo, J., Strumsky, D. and West, G. B. (2010). Urban scaling and its deviations: Revealing the structure of wealth, innovation and crime across cities. PLoS ONE 5 e13541.
  • [9] Chai, S. and Fleming, L. (2011). Emergence of Breakthroughs. In DIME-DRUID ACADEMY Winter Conference 1–47. DRUID-DIME Academy, Aalborg, Denmark.
  • [10] Chen, C., Chen, Y., Horowitz, M., Hou, H., Liu, Z. and Pellegrino, D. (2009). Towards an explanatory and computational theory of scientific discovery. J. Informetr. 3 191–209.
  • [11] Chung, F. and Lu, L. (2002). The average distances in random graphs with given expected degrees. Proc. Natl. Acad. Sci. USA 99 15879–15882 (electronic).
  • [12] CIOinsight (2004). Web extra: Who’s on first?, CIOinsight (2004), 1–2. Available at http://www.cioinsight.com/c/a/Past-News/Web-Extra-Whos-on-First/.
  • [13] Coppersmith, D., Gamarnik, D. and Sviridenko, M. (2002). The diameter of a long-range percolation graph. Random Structures Algorithms 21 1–13.
  • [14] Cowan, R. and Jonard, N. (2007). Structural holes, innovation and the distribution of ideas. J. Econ. Interac. Coord. 2 93–110.
  • [15] D’Souza, R. M. and Mitzenmacher, M. (2010). Local cluster aggregation models of explosive percolation. Phys. Rev. Lett. 104 195702.
  • [16] Duch, J., Waitzman, J. S. and Amaral, L. A. N. (2010). Quantifying the performance of individual players in a team activity. PLoS ONE 5 e10937.
  • [17] Erdős, P. and Rényi, A. (1961). On the strength of connectedness of a random graph. Acta Math. Acad. Sci. Hung. 12 261–267.
  • [18] Gerstein, M. and Douglas, S. M. (2007). RNAi development. PLoS Comput. Biol. 3 e80.
  • [19] Gowers, T. and Nielsen, M. (2009). Massively collaborative mathematics. Nature 461 879–881.
  • [20] Gravner, J. and Holroyd, A. E. (2008). Slow convergence in bootstrap percolation. Ann. Appl. Probab. 18 909–928.
  • [21] Gravner, J. and Sivakoff, D. (2013). Nucleation scaling in jigsaw percolation. Preprint. Available at arXiv:1310.2194.
  • [22] Grimmett, G. (1999). Percolation, 2nd ed. Springer, Berlin.
  • [23] Holroyd, A. E. (2003). Sharp metastability threshold for two-dimensional bootstrap percolation. Probab. Theory Related Fields 125 195–224.
  • [24] Introne, J., Laubacher, R., Olson, G. and Malone, T. (2011). The Climate CoLab: Large scale model-based collaborative planning. In International Conference on Collaboration Technologies and Systems (CTS), May 2011 40–47. MIT Center for Collective Intelligence, Cambridge, MA.
  • [25] Johnson, S. (2010). Where Good Ideas Come from: The Natural History of Innovation. Riverhead Hardcover, New York.
  • [26] Lakhani, K. R., Garvin, D. A. and Lonstein, E. (2010). TopCoder (A): Developing software through crowdsourcing. Harvard Business School General Management Unit 610–032 1–18.
  • [27] Lambiotte, R. and Panzarasa, P. (2009). Communities, knowledge creation, and information diffusion. J. Informetr. 3 180–190.
  • [28] Liljeros, F., Edling, C. R., Amaral, L. A. N., Stanley, H. E. and Aberg, Y. (2001). The web of human sexual contacts. Nature 411 907–908.
  • [29] Molloy, M. and Reed, B. (1995). A critical point for random graphs with a given degree sequence. Random Structures Algorithms 6 161–180.
  • [30] Moore, K. and Neely, P. (2011). From social networks to collaboration networks: The next evolution of social media for business. Forbes September 15 1–3.
  • [31] Newman, M. (2001). Scientific collaboration networks. I and II. Phys. Rev. E 64 016131, 016132.
  • [32] Newman, M. E. J. (2001). The structure of scientific collaboration networks. Proc. Natl. Acad. Sci. USA 98 404–409 (electronic).
  • [33] Redner, S. (1998). How popular is your paper? An empirical study of the citation distribution. Eur. Phys. J. B 4 131–134.
  • [34] Schläpfer, M., Bettencourt, L. M. A., Raschke, M., Claxton, R., Smoreda, Z., West, G. B. and Ratti, C. (2014). The scaling of human interactions with city size. Journal of The Royal Society Interface 11 98.
  • [35] Schulman, L. S. (1983). Ong range percolation in one dimension. J. Phys. A 16 L639–L641.
  • [36] Slivken, E. (2013). Jigsaw percolation of Erdös–Rènyi random graphs. Preprint. Available at http://www.math.washington.edu/~slivken/jigsawER.pdf.
  • [37] Sood, V., Mathieu, M., Shreim, A., Grassberger, P. and Paczuski, M. (2010). Interacting branching process as a simple model of innovation. Phys. Rev. Lett. 105 178701.
  • [38] Uzzi, B. (2008). A social network’s changing statistical properties and the quality of human innovation. J. Phys. A 41 224023, 12.
  • [39] Uzzi, B. and Spiro, J. (2005). Collaboration and creativity: The small world problem. Am. J. Sociol. 111 447–504.
  • [40] van den Esker, H., van der Hofstad, R., Hooghiemstra, G. and Znamenski, D. (2005). Distances in random graphs with infinite mean degrees. Extremes 8 111–141.