Idea, disease, computer virus spreads via connections in the network
Nodes are linked if one would "infect" the other
Will an infection take hold?
How many nodes/people will it reach?
When do we get diffusion?
What is the extent of diffusion?
How does it depend on the particulars of the process as well as the network?
Who is likely to be infected earliest?
Reach of contagion is determined by the component structure
Some players or nodes are immune, some links fail to transmit...
What do components look like of those who are susceptible and given links that work
So understanding what the component structure is will help us understand both the probability of starting and the eventual reach conditional on that
Get nontrivial diffusion if someone in the giant component is infected/adopts
Size of the giant component determines likelihood of diffusion and its extent
Random network models allow for giant component calculations
what we can do
Simple example of such a calculation
Work with Erdos-Renyi random network
How big is the giant component??
How big is the giant component when there is one?
Size of the giant component when $1/n < p < log(n)/n$
[know that if p << 1/n all isolated, and if log(n)/n << p then all path connected]