Question

Network Structures

one asks whether most pairs of people in the world are connected by a path of at most six edges in the social network, where an edge joins any two people who know each other on a first-name basis. Now let us consider a variation on this question. For each person in the world, we ask them to rank the 30 people they know best, in descending order of how well they know them. (Let us suppose for purposes of this question that each person is able to think of 30 people to list.) We then construct two different social networks:

(a) The “close-friend” network: from each person we create a directed edge only to their ten closest friends on the list.

(b) The “distant-friend” network: from each person we create a directed edge only to the ten people listed in positions 21 through 30 on their list.

Let us think about how the small-world phenomenon might differ in these two networks. In particular, let C be the average number of people that a person can reach in six steps in the close-friend network, and let D be the average number of people that a person can reach in six steps in the distant-friend network (taking the average over all people in the world). When researchers have done empirical studies to compare these two types of networks (the exact details often differ from one study to another), they tend to find that one of C or D is consistently larger than the other. Which of the two quantities, C or D, do you expect to be larger?

Answer 1

Quantity D I am expecting to be larger than C. The reason is that in network D, the directed edge is made to distant-friend and the two friend are distant mostly because their interaction is very less and they belongs to different domains and hence they are most likely to belong to different cluster if we divide the network into clusters such that density of edges within cluster is much higher than density of edges among different cluster.

Since network in D is likely to explore more new people from different cluster while network in C is more likely to explore people from same cluster, hence the average number of person we can reach in six steps in distant-friend network i.e. D is more likely to be greater than C.

Please comment for any clarification.