## Monday, 28 January 2013

### Complex Contagion in Complex Network

Following up on the previous post referring to rumor spreading in social networks, I would like to extend the topic to complex contagion in a complex network.

This phenomenon tells us you will take some action or adopt a changed behavior only if certain fraction of your friends do so. This is different from spread of diseases or rumors. Complex contagion examples include whether to join a protest, see a movie, adopt a new hairstyle or get a tattoo.

Damon Centola of Harvard University and Michael Macy of Cornell University suggests four properties that necessitates multiple exposure , namely strategic complementarity, legitimacy, credibility and emotional contagion. I'll fall back on a very recent experience of mine when the movie 'Django unchained' was released ( or when its blu-ray print became available?). One of my friends (the movie nerd?) told me it was a good one, then another couple of friends supported him. Within a week, six of my friends watched and recommended it. If this happens to you, I am sure you'll also utter the already famous quote from the movie :
"You had my curiosity but now you have my attention." and go and watch it (I am sorry, I couldn't resist).

#### Mathematical Model

The diffusion through all of the network or cascading effect as indicated above can be explained through a very simple mathematical model.
• Whether or not you'll adopt a behavior or choose one of two possible alternatives depends upon a payoff .
• It also depends upon the fraction of your friends/neighbors who have already chosen one.
Suppose, p fraction of your friends play cricket and other (1-p) fraction play football. The payoff (i.e. your willingness) for playing cricket is c and playing football is f.

Now, you would choose cricket over football if,
p.c > (1-p).f
In other words, p > f/(c+f).

So in the network, if q fraction of your neighbors play cricket where, q > f/(c+f), you will also play cricket and in doing so, influence your other neighbors who play football to play cricket. In time, this results in the cascading effect.

#### Complex Contagion : some interesting aspects

• Long ties that connect socially distant relation, drastically bring down the degree of separation and results in fast information diffusion. So, a trend in fashion arising in anywhere in the world, should also readily affect India, right? After all, it's a small world. Unfortunately, the answer is no. Long ties not necessarily speed up the complex contagion process and may have no effect in this. Having many common neighbors helps in this process more. You are much more likely to join a protest if three of your close friends join it, in contrast to when your foreign friend whom you met during a holiday or intern joins it.
• Social networks have community structures embedded in it. Communities provide an interesting study. Within a community, there are many mutual links and an opinion spreads faster  and more easily,  encompassing the whole community. New opinions coming from outside the community share a different response. Links are sparse between different communities and different opinion of a different community is met with considerable resistance and a community is more likely to hold on to its own within group opinion even though it may be inferior compared to the new idea. This might tell you why parental 'community' finds teenagers' take on life 'rebellious' !!
• Moreover, when having two ideas, an individual can choose both for an additional cost. Falling back to our previous example of playing cricket and football, one (the sporty one!) may choose to play both with the additional cost of his or her own time. These 'bilingual' nodes help in coexistence of both kinds of opinion, here cricket and football and prevents one taking out the other one.
• While trying to spread an idea throughout the network, if one chooses the seed nodes carefully and inflicts them with the idea by whatever means (like a free cricket bat and ball for all those who will start to play cricket first), they may initiate the cascading effect. This has huge implication in viral marketing.