diffusion and cascading behavior in random networks marc lelarge (inria-ens) wids mit, may 31, 2011
TRANSCRIPT
![Page 1: Diffusion and Cascading Behavior in Random Networks Marc Lelarge (INRIA-ENS) WIDS MIT, May 31, 2011](https://reader038.vdocuments.site/reader038/viewer/2022102906/56649c8e5503460f94947108/html5/thumbnails/1.jpg)
Diffusion and Cascading Behavior in Random Networks
Marc Lelarge (INRIA-ENS)
WIDS MIT, May 31, 2011.
![Page 2: Diffusion and Cascading Behavior in Random Networks Marc Lelarge (INRIA-ENS) WIDS MIT, May 31, 2011](https://reader038.vdocuments.site/reader038/viewer/2022102906/56649c8e5503460f94947108/html5/thumbnails/2.jpg)
(1) Diffusion Model
(2) Resultsfrom a mathematical analysis.
inspired from game theory and statistical physics.
![Page 3: Diffusion and Cascading Behavior in Random Networks Marc Lelarge (INRIA-ENS) WIDS MIT, May 31, 2011](https://reader038.vdocuments.site/reader038/viewer/2022102906/56649c8e5503460f94947108/html5/thumbnails/3.jpg)
(0) Context
Crossing the Chasm(Moore 1991)
![Page 4: Diffusion and Cascading Behavior in Random Networks Marc Lelarge (INRIA-ENS) WIDS MIT, May 31, 2011](https://reader038.vdocuments.site/reader038/viewer/2022102906/56649c8e5503460f94947108/html5/thumbnails/4.jpg)
(1) Diffusion Model
(2) Results
![Page 5: Diffusion and Cascading Behavior in Random Networks Marc Lelarge (INRIA-ENS) WIDS MIT, May 31, 2011](https://reader038.vdocuments.site/reader038/viewer/2022102906/56649c8e5503460f94947108/html5/thumbnails/5.jpg)
(1) Coordination game…
• Both receive payoff q.
• Both receive payoff 1-q>q.
• Both receive nothing.
![Page 6: Diffusion and Cascading Behavior in Random Networks Marc Lelarge (INRIA-ENS) WIDS MIT, May 31, 2011](https://reader038.vdocuments.site/reader038/viewer/2022102906/56649c8e5503460f94947108/html5/thumbnails/6.jpg)
(1)…on a network.• Everybody start with
ICQ.• Total payoff = sum of
the payoffs with each neighbor.
• A seed of nodes switches to
(Morris 2000)
![Page 7: Diffusion and Cascading Behavior in Random Networks Marc Lelarge (INRIA-ENS) WIDS MIT, May 31, 2011](https://reader038.vdocuments.site/reader038/viewer/2022102906/56649c8e5503460f94947108/html5/thumbnails/7.jpg)
(1) Threshold Model
• State of agent i is represented by
• Switch from to if:
![Page 8: Diffusion and Cascading Behavior in Random Networks Marc Lelarge (INRIA-ENS) WIDS MIT, May 31, 2011](https://reader038.vdocuments.site/reader038/viewer/2022102906/56649c8e5503460f94947108/html5/thumbnails/8.jpg)
(1) Model for the network?
Statistical physics: bootstrap percolation.
![Page 9: Diffusion and Cascading Behavior in Random Networks Marc Lelarge (INRIA-ENS) WIDS MIT, May 31, 2011](https://reader038.vdocuments.site/reader038/viewer/2022102906/56649c8e5503460f94947108/html5/thumbnails/9.jpg)
(1) Model for the network?
![Page 10: Diffusion and Cascading Behavior in Random Networks Marc Lelarge (INRIA-ENS) WIDS MIT, May 31, 2011](https://reader038.vdocuments.site/reader038/viewer/2022102906/56649c8e5503460f94947108/html5/thumbnails/10.jpg)
(1) Random Graphs
• Random graphs with given degree sequence introduced by Molloy and Reed (1995).
• Examples:– Erdös-Réyni graphs, G(n,λ/n).– Graphs with power law degree distribution.
• We are interested in large population asymptotics.
• Average degree is λ.
![Page 11: Diffusion and Cascading Behavior in Random Networks Marc Lelarge (INRIA-ENS) WIDS MIT, May 31, 2011](https://reader038.vdocuments.site/reader038/viewer/2022102906/56649c8e5503460f94947108/html5/thumbnails/11.jpg)
(1) Diffusion Model
(2) Results
q = relative threshold λ = average degree
![Page 12: Diffusion and Cascading Behavior in Random Networks Marc Lelarge (INRIA-ENS) WIDS MIT, May 31, 2011](https://reader038.vdocuments.site/reader038/viewer/2022102906/56649c8e5503460f94947108/html5/thumbnails/12.jpg)
(1) Diffusion Model
(2) Results
q = relative threshold λ = average degree
![Page 13: Diffusion and Cascading Behavior in Random Networks Marc Lelarge (INRIA-ENS) WIDS MIT, May 31, 2011](https://reader038.vdocuments.site/reader038/viewer/2022102906/56649c8e5503460f94947108/html5/thumbnails/13.jpg)
(2) Some experiments
Seed = one node, λ=3 and q=0.24 (source: the Technoverse blog)
![Page 14: Diffusion and Cascading Behavior in Random Networks Marc Lelarge (INRIA-ENS) WIDS MIT, May 31, 2011](https://reader038.vdocuments.site/reader038/viewer/2022102906/56649c8e5503460f94947108/html5/thumbnails/14.jpg)
(2) Some experiments
Seed = one node, λ=3 and 1/q>4 (source: the Technoverse blog)
![Page 15: Diffusion and Cascading Behavior in Random Networks Marc Lelarge (INRIA-ENS) WIDS MIT, May 31, 2011](https://reader038.vdocuments.site/reader038/viewer/2022102906/56649c8e5503460f94947108/html5/thumbnails/15.jpg)
(2) Some experiments
Seed = one node, λ=3 and q=0.24 (or 1/q>4) (source: the Technoverse blog)
![Page 16: Diffusion and Cascading Behavior in Random Networks Marc Lelarge (INRIA-ENS) WIDS MIT, May 31, 2011](https://reader038.vdocuments.site/reader038/viewer/2022102906/56649c8e5503460f94947108/html5/thumbnails/16.jpg)
(2) Contagion threshold
No cascade
Global cascadesIn accordance with (Watts 2002)
![Page 17: Diffusion and Cascading Behavior in Random Networks Marc Lelarge (INRIA-ENS) WIDS MIT, May 31, 2011](https://reader038.vdocuments.site/reader038/viewer/2022102906/56649c8e5503460f94947108/html5/thumbnails/17.jpg)
(2) A new Phase Transition
![Page 18: Diffusion and Cascading Behavior in Random Networks Marc Lelarge (INRIA-ENS) WIDS MIT, May 31, 2011](https://reader038.vdocuments.site/reader038/viewer/2022102906/56649c8e5503460f94947108/html5/thumbnails/18.jpg)
(2) Pivotal players
• Giant component of players requiring only one neighbor to switch.
Tipping point: Diffusion like standard epidemic
Chasm : Pivotal players = Early adopters
![Page 19: Diffusion and Cascading Behavior in Random Networks Marc Lelarge (INRIA-ENS) WIDS MIT, May 31, 2011](https://reader038.vdocuments.site/reader038/viewer/2022102906/56649c8e5503460f94947108/html5/thumbnails/19.jpg)
(2) q above contagion threshold
• New parameter: size of the seed as a fraction of the total population 0 < α < 1.
• Monotone dynamic → only one final state.
![Page 20: Diffusion and Cascading Behavior in Random Networks Marc Lelarge (INRIA-ENS) WIDS MIT, May 31, 2011](https://reader038.vdocuments.site/reader038/viewer/2022102906/56649c8e5503460f94947108/html5/thumbnails/20.jpg)
(2)Minimal size of the seed, q>1/4
Chasm : Connectivity hurts
Tipping point: Connectivity helps
![Page 21: Diffusion and Cascading Behavior in Random Networks Marc Lelarge (INRIA-ENS) WIDS MIT, May 31, 2011](https://reader038.vdocuments.site/reader038/viewer/2022102906/56649c8e5503460f94947108/html5/thumbnails/21.jpg)
Conclusion
• Simple tractable model:– Threshold rule introduces local dependencies– Random network : heterogeneity of population
• 2 regimes:– Low connectivity: tipping point– High connectivity: chasm
• More results in the paper: – heterogeneity of thresholds, active/inactive links,
equilibria of the game and coexistence.
![Page 22: Diffusion and Cascading Behavior in Random Networks Marc Lelarge (INRIA-ENS) WIDS MIT, May 31, 2011](https://reader038.vdocuments.site/reader038/viewer/2022102906/56649c8e5503460f94947108/html5/thumbnails/22.jpg)
Thanks!
- Diffusion and Cascading Behavior in Random Networks.Available at http://www.di.ens.fr/~lelarge