Introduction Standard percolation Percolation with learning Network structure A percolation model of innovation diffusion Koen Frenken a , Luis R. Izquierdo b and Paolo Zeppini a,c a Eindhoven University of Technology b University of Burgos c University of Amsterdam Zurich, 11 September 2012 p.zeppini@tue.nl TU/e A percolation model of innovation diffusion
Introduction Standard percolation Percolation with learning Network structure Scope ◮ Diffusion in a structured market: how the percolation phase transition affects market demand and social welfare. ◮ How endogenous learning curves affect diffusion. ◮ How market network topology affects diffusion: ◮ two-dimensional regular lattices, the grid, ◮ one-dimensional regular lattices, the ring, ◮ random network (Erdos-Renyi) ◮ small-world networks ◮ what is more important for diffusion, among degree, average path length, clustering. p.zeppini@tue.nl TU/e A percolation model of innovation diffusion
Introduction Standard percolation Percolation with learning Network structure Literature ◮ Epidemic model of diffusion: SI , SIR , SIS (see review in Vega Redondo, 2007, chapter 3). ◮ Economic models of diffusion, examples: Geroski (2000) for S-shaped diffusion curves, Young (2009) for different mechanisms of social interactions. ◮ Econ. models of local strategic interaction: Blume (1995), Morris (2000), Jackson-Yariv (2007), Goyal-Kearns (2011). ◮ Network models of diffusion, analytical studies: Moore and Newman (2000), Newman et al. (2002), Watts (2002). ◮ Percolation models of social systems: Solomon et al (2000), Silverberg -Verspagen (2005), Frenken et al (2008), Honhish et al. (2008). p.zeppini@tue.nl TU/e A percolation model of innovation diffusion
Introduction Standard percolation Percolation with learning Network structure The percolation model of diffusion in a market ◮ n consumers are the nodes of a network of social relationships, which is given once and for all. ◮ a new product is launched, with price p ∈ [0 , 1] (quality q ). ◮ consumers have preferences expressed as reservation price p r ( minimum quality q r ), which are randomly distributed according to a given distribution (uniform). ◮ A consumer buys the product if 1) at least one neighbour buys and 2) her preference is met: p < p r ( q > q r ). ◮ The model is initialized with a small number of consumers getting the product for free (seeds). p.zeppini@tue.nl TU/e A percolation model of innovation diffusion
Introduction Standard percolation Percolation with learning Network structure The operational network Consumers preferences are uniformly distributed, q r ∼ U [0 , 1] ( p r ∼ U [0 , 1] ). Once the product quality q (price p ) is set, and preferences are drawn, a random operational network of “would-buy” consumers (those for which q r < q or p r > p ) forms. willing to buy unwilling to buy p.zeppini@tue.nl TU/e A percolation model of innovation diffusion
Introduction Standard percolation Percolation with learning Network structure Simulations Batch simulations: we run the model a number of times, and look at the following outcomes: ◮ final number of adopters n-of-adopters when diffusion ends ◮ the number of steps time required for diffusion to end The values reported are averages over different simulation runs. The standard deviation is larger near the transition threshold. In relative terms, for n-of-adopters the standard deviation is about 30% near the threshold, and much smaller ( 1% ) away from it. For time is larger in low diffusion scenarios ( ∼ 40% ). p.zeppini@tue.nl TU/e A percolation model of innovation diffusion
Introduction Standard percolation Percolation with learning Network structure The percolation critical transition The number of adopters (size of giant operational component) reported as function of product quality (10 batch simulations): 10000 9000 n-of-adopters linear trend 8000 7000 6000 5000 4000 3000 2000 1000 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 initial-quality An infinite grid has percolation threshold q = 0 . 593 . Here we have a grid of 10000 nodes wrapped in a thorus, with 10 seeds. p.zeppini@tue.nl TU/e A percolation model of innovation diffusion
Introduction Standard percolation Percolation with learning Network structure A critical transition in demand Consider model in product price space (50 batch simulations): 10000 1 n-of-adopters 9000 0.9 8000 0.8 linear trend 7000 0.7 n-of-adopters 6000 0.6 price 5000 0.5 4000 0.4 3000 0.3 2000 0.2 1000 0.1 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2000 4000 6000 8000 10000 price n-of-adopters The local interactions of a structured (networked) market reduce efficiency and consumer surplus wrt the ideal market. p.zeppini@tue.nl TU/e A percolation model of innovation diffusion
Introduction Standard percolation Percolation with learning Network structure Different network structures Left: a grid of degree 4. Centre: a ring of degree 4. Right: a random network (Erdos-Renyi type) of average degree 4. p.zeppini@tue.nl TU/e A percolation model of innovation diffusion
Introduction Standard percolation Percolation with learning Network structure Endogenous technological progress Assume that product quality increases during diffusion due to learning, with an endogenous mechanism where quality depends positively on the number of previous adopters: q t = 1 − 1 − q 0 , (1) n α t with q t the product quality at time t , q 0 the initial quality, n t the number of adopters at time t and α a learning coefficient. In the price space one defines a learning curve : p t = p 0 . (2) n α t The price p t goes down as more consumers adopt the product. p.zeppini@tue.nl TU/e A percolation model of innovation diffusion
Introduction Standard percolation Percolation with learning Network structure Grid (degree 4) Seven different learning rates α (10000 consumers, 10 seeds): n-of-adopters 180 10000 160 9000 140 8000 elapsed-time-to-equilibrium 7000 120 learning rate: 6000 100 5000 80 4000 60 3000 40 2000 1000 20 0 0 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 initial-quality initial-quality Left: number of adopters as a function of initial product quality. Right: time to equilibrium as a function of initial product quality. p.zeppini@tue.nl TU/e A percolation model of innovation diffusion
Introduction Standard percolation Percolation with learning Network structure Ring (degree 4) Seven different learning rates α (10000 consumers, 10 seeds): 1200 10000 n-of-adopters a = 0 9000 a = 0.05 a = 0 1000 8000 a = 0.1 a = 0.05 a = 0.15 a = 0.1 7000 800 a = 0.2 a = 0.15 6000 a = 0.25 a = 0.2 time a = 0.3 5000 a = 0.25 600 a = 0.3 4000 400 3000 2000 200 1000 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 initial-quality initial-quality Left: number of adopters as a function of initial product quality. Right: time to equilibrium as a function of initial product quality. p.zeppini@tue.nl TU/e A percolation model of innovation diffusion
Introduction Standard percolation Percolation with learning Network structure Four different regular networks (no learning) Two grids with degree k = 4 , k = 8 , and two rings k = 4 , k = 8 : 10000 n-of-adopters 9000 linear 8000 4-grid 7000 8-grid 6000 4-ring 5000 8-ring 4000 3000 2000 1000 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 initial-quality Number of adopters as a function of product quality. Batch simulations: values are averages over 10 different runs. p.zeppini@tue.nl TU/e A percolation model of innovation diffusion
Introduction Standard percolation Percolation with learning Network structure Messages from regular networks ◮ Grids are more efficient than rings: the grid with degree 4 is more efficient than the ring with degree 8. ◮ Caveat: what counts for diffusion in regular networks is not the degree, but the average path-length between nodes. ◮ Moreover, the grid “ spreads ”: neighbours of successive orders are in larger number (when the degree is 4, r -neighbours are 4 r in the grid, while the ring has 4 neighbours of any order). p.zeppini@tue.nl TU/e A percolation model of innovation diffusion
Introduction Standard percolation Percolation with learning Network structure Random Networks Left: a ring of 20 nodes with degree 4. Centre: a Small-World network of 50 nodes with average degree 4 and rewiring probability p = 0 . 1 . Right: a fully random network ( p = 1 ) of 50 nodes with average degree 4. p.zeppini@tue.nl TU/e A percolation model of innovation diffusion
Introduction Standard percolation Percolation with learning Network structure Percolation in Fully Random Network (avg degree 4) Seven different learning rates α (10000 consumers, 10 seeds): 50 10000 n-of-adopters a = 0 45 9000 a = 0.05 40 8000 a = 0.1 a = 0.15 35 7000 a = 0.2 30 6000 a = 0.25 time 25 a = 0.3 5000 a = 0 4000 a = 0.05 20 a = 0.1 3000 15 a = 0.15 2000 a = 0.2 10 a = 0.25 1000 5 a = 0.3 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 initial-quality initial-quality Left: number of adopters as a function of initial product quality. Right: time to equilibrium as a function of initial product quality. p.zeppini@tue.nl TU/e A percolation model of innovation diffusion
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