A giant component is a connected set of nodes that consists of the majority of the network. In many networks, the formation of a giant component is unavoidable. I investigated the time taken for a giant component to develop in a social network. I varied the rate at which people made friends, and observed the effect this had on the development of the giant component.To facilitate my analysis, I first built a virtual world containing 1000 people. Each person was immortal, non-reproducing, and had no initial friends. Every day people made and strengthened friendships according to the following precedence:

• Completing triadic closures
• Promoting weak ties to strong ties
• Forming weak ties with a stranger

I varied the rate at which people made friends, and ran the simulation until a giant component formed. I recorded the the time taken for the component to form, and the corresponding rate of friendship. The following two graphs summarize the simulation summary:

The graphs above contain data from the simulation I described earlier (blue dots) and a new simulation (green dots). The new simulation is just like the first, but with a population that dies and asexually reproduces. I made this modification to make the simulation more realistic. However, as seen in the graphs, this modification had no noticeable effect. This is probably because the giant component forms so fast, that an insignificant number of people get to die and reproduce during the simulation run.

Conclusion:

If L = time taken for giant component to form, and T = the average time taken for a person to make a friend : L is directly proportional to T. For this experiment, the constant of proportionality was 2.5 :

L = 2.5 T

My simulation source code can be found here (Visual C++ 2005).