We saw how a community in a graph can evolve. To track the nature of evolution, we need to measure how the community changes over time. So here are three cases. One, two and three of a community changing between two observation points. The goal is to figure out whether these are normal fluctuations in the network or are more drastic changes occurring. Look at them for a second. Just visually, the first case seems to show just minor changes. Whereas case two shows a merger, and case three shows a split. Now, to come up with a quantitative measure of change over time, we need to take two observations from two consecutive time points and fuse the graph. If you do it for case one, you'll find one new node, one living node, and the rest will come on over time. For case two, you will see that two previous communities, colored differently, are internally connected the same way as before. But some members of the two communities have created new crosslinks. Now can you tell me what you observe in case three? Well I see one join node, color purple. Apart from it, there are just two edges connecting the two groups. Now with these observations, we can now compute the autocorrelation between the graphs across time t and t plus 1. This is just a measure of the number of common nodes divided by the number of nodes in the combined graph. If the community does not change at all, this number is 1. If a community has only a few connection, the number is lower. After computing autocorrelation over every pair of time steps, we can then compute stationarity, which measures the overall change in the autocorrelation over a period of time. So if we measure over 100 time steps, we will add the 99 correlation values from the steps and then divide it by 99. This will tell us what fraction of members remain unchanged on an average over these 100 time steps. Therefore, the 1 minus zeta tells us the average ratio of members that are changed in a time step. Let's take three cases. In the first plot, the size of the graph is small and nothing much is happening here. A note occasionally joins or leaves, keeping the stationarity pretty flat. In the second case, the graph is small, but there are a lot of changes, especially at time step seven a whole bunch of purple nodes have joined and then they went away. The size of the graph clearly reflects this with a purple spike that you see on the size versus the time graph. This spike on the time series by the way is called a burst. The third plot shows a large graph with many nodes joining and leaving. The stationarity of this graph will be quite low given the abrupt changes we observe over time.