Diffusion

# Notes for Notes for Lectures (CC2012)

### Diffusion of Innovations

The general process of diffusion is relevant to many disciplines. In the social sciences the original use is with respect to the spread of ideas and practices within and across social space and time. We usually trace the concept back to the work of French criminologist G. Tarde in the 1890s. Tarde wrote about social imitation and "the group mind" and was influential on Durkheim and LeBon. Contemporary work on diffusion has its starting point in the work of E. Rogers in the 1960s.

### Basic Model

Suppose a certain fraction, $f$, of a homogeneous and constantly mixing population, P, has a particular trait or practice. Each contact between a practitioner and non-practitioner has a fixed probability, p, of converting the latter. Of all the random interactions, the proportion between practitioners is $f^{2}$, the proportion between non-practitioners is $(1-f)^{2}$ and the number that can result in a conversion is $2 \times f \times (1-f)$ since each practitioner can run into a non-practitioner and each non-practicioner can run into a practitioner.

Imagine the population below. We have 9 actors, 3 of whom have adopted blue-ness. If all the actors roam about randomly, how many encounters of various kinds will we have?

A given red actor will meet fellow red actors in 5/8 of her encounters and blue actors in 3/8. A blue actor will meet other blue actors 2/8 of the time, and a red actor 6/8 of the time. Since 6 of 9 are red actors we have

(1)
\begin{align} red-red-encounters = \frac {6} {9} \times \frac{5}{8} = \frac {30} {72} \end{align}
(2)
\begin{align} red-blue-encounters = \frac {6} {9} \times \frac{3}{8} + \frac {3} {9} \times \frac{6}{8} = \frac {36} {72} \end{align}
(3)
\begin{align} blue-blue-encounters = \frac {3} {9} \times \frac{2}{8} = \frac {6} {72} \end{align}

Now suppose that 10% of the time, a red blue encounter results in converting a red actor to a blue actor.

How should we think about the spread of blue-ness? The red-red and the blue-blue encounters do nothing. But half of all the interactions are red-blue; if 10% of these result in conversion then for each red actor any given encounter has a probability of $0.5 \times 0.1 = 0.05$ or one chance in twenty of resulting in a conversion.

What Happens When an Actor Converts?

We'd have to re-do these calculations because the density of blue goes up when an actor converts — the odds of a conversion increase.

### The Elements of Diffusion

History. 19th century interest in spread of culture and civilization. Imitation as basis of social learning. With emergence of disease theory we have question of contagion and spread of disease through space. Development of historical linguistics raised question of how words, sounds, and linguistic structures moved. See transcultural diffusion, diffusion of innovation, lexical diffusion.

### Underlying Processes: giving, giving off, taking

When we talk about diffusion, it is important to slow down and think through the actually underlying process — how do we think the diffusing thing is moving from place to place.

Contagion. An entity with the trait or practice can pass it to one without it through direct contact.

Spatial diffusion 1. An entity with the trait deposits it on objects she touches or in places she spends time and other entities can acquire the trait by touching or visiting.

Spatial diffusion 2. Trait emanates from a source spreading outward and potentially converting any entities it contacts.

Imitation. Entities can copy what others around them do.

### Examples

Chains
Network spread. Network can mean attenuation or amplification