Bass diffusion of innovation on a network

Field: energy · Zoo name: bass_diffusion · Module: modelly.zoo.bass_diffusion.model

Households adopt an innovation, rooftop solar panels for instance. In each step of length dt years a household that has not adopted does so with probability (innovation + imitation * share) * dt, where innovation (Bass’s p) is the pull of advertising and imitation (Bass’s q) the pull of the households it knows. share is the share of adopters among everybody (mixing="well-mixed", Bass’s original model) or among the household’s neighbours in a random network of mean degree degree (mixing="network").

How it is written

Households are Household agents with an adopted Column; a step is one ColumnTable.draw for the whole population. The network is a one-layer MultilayerNetwork (see Networks), so the share of adopting neighbours of every household is one sparse product, adjacency() @ adopted, divided by the degrees.

from modelly.zoo.bass_diffusion.model import BassDiffusion, BassScenario, bass_curve

model = BassDiffusion(BassScenario(rng=42, mixing="network", degree=8))
model.run()
frame = model.datacollector.get_model_vars_dataframe()  # time (years), adoption share
bass_curve(frame["time"], 0.03, 0.38)  # the well-mixed Bass solution

Parameter	Default	Meaning
households	5000	number of households
innovation	0.03	Bass p, adoption rate per year from outside influence
imitation	0.38	Bass q, adoption rate per year when every neighbour has adopted
dt	0.1	step length, years
mixing	"well-mixed"	"well-mixed" or "network"
degree	10.0	mean number of neighbours in the random network
steps	200	run length of run()

Validation

In the well-mixed limit the expected adoption share follows the Bass equation dF/dt = (p + q F)(1 - F), solved by F(t) = (1 - exp(-(p + q) t)) / (1 + (q / p) exp(-(p + q) t)) (bass_curve). tests/zoo/test_bass_diffusion.py checks, with three seeds, that the simulated share is within the Euler discretisation error (the step recursion against the exact solution) plus five standard deviations of the ODE at every step. The standard deviation comes from the linear noise approximation: each step adds binomial noise of variance (1 - F) h (1 - h) / N, with h = (p + q F) dt, and a deviation grows by 1 + f'(F) per step. It also checks that a dense network (50 neighbours) stays within the same bound, that a sparse one (4 neighbours) lags the well-mixed curve at mid-diffusion, because non-adopters cluster together, and that the same seed gives the same curve.

References

• Bass, F. M. (1969). A new product growth for model consumer durables. Management Science, 15(5), 215-227.

Start a project from it with modelly zoo copy bass_diffusion my_diffusion.