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Isolating the Small World Problem
We now isolate exactly what it is that constitutes the Small World Problem. Rather than just considering people living on Earth, we move to a more abstract realm by considering any set of "agents" in a graph which share symmetric "connections." (In Milgram's experiment, the agents represent people and the connections represent social connections.)
Why exactly should we be surprised to discover short paths of connections between random agents in a graph? Watts presents four properties of a graph that together lead us to expect not to find such short path lengths:
The world's population seems to fulfill all four of these conditions. There are billions of people on the planet. People generally cannot know more than a few thousand aquaintances, and there is no central person everybody knows. Finally, friendship circles are highly overlapping -- that is, connected people have similar sets of aquaintances.
With all four of these conditions, we intuitively expect that randomly selected agents share no aquaintances in common and that there is no short path between them. How then can we explain the fact that Milgram seemingly discovered short paths between random people? This is the essence of the Small World Problem.
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