It has been proved that social networks are based on a scale-free model: there are few hubs with lots of connections and several small nodes with few links. However, I wonder what is the level of global connectivity inside a well-known network like LinkedIn.
In other words I’d like to know if there are small or large isles made up of a group of isolated nodes; it’s not so easy to answer this question and maybe the only valid result may come from a direct analysis of network data, however it’s possible to assume some hypothesis in order to simplify our problem:
- The distribution of hubs in a scale-free network is homogeneous in a context defined by geography, jobs, common interests, etc.
- We exclude every smaller isle (they can be considered as isolated nodes). The critical dimension can be estimated according to the total number of nodes. A lower value always assures a better precision, but it can be more difficult to accomplish the analysis.
- There is at least one hub per isle. This hypothesis is not easy to accept at all, however, it’s not unduly simplistic to presuppose that all sufficiently larger isles are scale-free networks themselves; thus, according to our first hypothesis, the probability to find a hub in each aisle is rather high.
If the previous pre-conditions are accepted, we can check if all main hubs are connected: in the positive case, the probability of the presence of large isles is next to zero. Of course, I’m going to check it out…
Any comment or criticism is well-accepted!