A percolation model of innovation diffusion Koen Frenken a , Luis R. - - PowerPoint PPT Presentation

Introduction Standard percolation Percolation with learning Network structure

A percolation model of innovation diffusion

Koen Frenkena, Luis R. Izquierdob and Paolo Zeppinia,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 pr (minimum quality qr), 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 < pr (q > qr).

◮ 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, qr ∼ U[0, 1] (pr ∼ 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 qr < q or pr > 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):

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 initial-quality

n-of-adopters linear trend

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):

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

n-of-adopters price

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2000 4000 6000 8000 10000

price n-of-adopters n-of-adopters linear trend

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: qt = 1 − 1 − q0 nα

t

, (1) with qt the product quality at time t, q0 the initial quality, nt the number of adopters at time t and α a learning coefficient. In the price space one defines a learning curve: pt = p0 nα

t

. (2) The price pt 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):

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 initial-quality

n-of-adopters learning rate:

20 40 60 80 100 120 140 160 180 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 elapsed-time-to-equilibrium 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):

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

n-of-adopters

initial-quality

a = 0 a = 0.05 a = 0.1 a = 0.15 a = 0.2 a = 0.25 a = 0.3

200 400 600 800 1000 1200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

time initial-quality a = 0 a = 0.05 a = 0.1 a = 0.15 a = 0.2 a = 0.25 a = 0.3

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:

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

n-of-adopters

initial-quality

linear 4-grid 8-grid 4-ring 8-ring

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

• rders are in larger number (when the degree is 4,

r-neighbours are 4r 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):

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

n-of-adopters

initial-quality

a = 0 a = 0.05 a = 0.1 a = 0.15 a = 0.2 a = 0.25 a = 0.3

5 10 15 20 25 30 35 40 45 50 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

time initial-quality a = 0 a = 0.05 a = 0.1 a = 0.15 a = 0.2 a = 0.25 a = 0.3

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

Percolation in Small-World networks

The effect of the rewiring probability p on diffusion:

0.2 0.4 0.6 0.8 1 0.0001 0.001 0.01 0.1 1

p L(p) / L(0) C(p) / C(0)

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 n-of-adopters quality Ring (p = 0) Small World p = 0.001 Small World p = 0.01 Small World p = 0.1 Fully Random Network (p = 1)

Left: clustering coefficient C(p) and average path length L(p) in Small-World networks (from Watts and Strogatz, 1998). Right: simulated number of adopters as a function of product quality for different values of p (averages over 10 runs).

p.zeppini@tue.nl TU/e A percolation model of innovation diffusion

Introduction Standard percolation Percolation with learning Network structure

Regular and random networks (no learning)

Add random network and Small-World, both of degree 4:

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

n-of-adopters quality

linear 4-grid 8-grid 4-ring 8-ring Random SW p=0.01

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 random networks

◮ The random network has a lower percolation threshold than

regular networks. Just above this threshold the random network is more efficient than the grid: here the degree variance has a positive effect

◮ The transition is smoother for random networks, because of

the degree dispersion across nodes.

◮ But for relatively large quality (far above the threshold) the

random network is less efficient than the grid: this is due

◮ above threshold grids exploit the topological “spreading” ◮ unconnected components in random networks.

◮ Small-worlds do not do well: a low average path length is

not enough for percolation. Clustering is useless.

p.zeppini@tue.nl TU/e A percolation model of innovation diffusion