# Calculating the number of clients of an online social network.

I recently read a paper by Zhu, Li and Fu [1] on modelling online social networks.

They derive a population growth model of online social networks as a function of time.

They use

- population distribution – P(s,t) – the proportion of the number of locations with population size – s – over the total number of populated locations,
- the total number of populated locations l(t)
- the largest population size n(t).

These distributions P(s,t), l(t) and n(t) follow Power Laws (empirical). This conclusion was obtained by looking at statistics of three online social networks and choosing Power Laws as the relevant distribution functions.

This can now be inserted into the above GP(t)

and integrated out to give

There are quite a few potential problems with this model, amongst others the distribution fit. The fact that the distribution has 6 parameters is not impressive.

However, the result is quite surprising in that it seems to apply to all online social networks (up to the above parameters).

We plot some of the results.

The authors give us some parameters. The rest we calculate by normalising the formula (that is we need to determine a constant to compare oranges with oranges). This we do at 18 months.

Here are our parameters

First we look at GP(t): the number of clients grows surprisingly slowly as a function of time.

Then we look at the distribution of the largest population size: n(t)

Then we look at the total number of populated locations l(t)

Finally we look at P(s,t) – the proportion of the number of locations

# Conclusion

These models are quite fun to work with and are quite useful. However care needs to be taken in drawing conclusions.

The paper [1] referred to in this discussion is now about 4 years old.

# Reference

[1] https://link.springer.com/article/10.1186/2194-3206-1-14