# Mean of a measurement on periodic data: what is the use of the inverse of correlation length?

Correlation and autocorrelation is something that in my Bachelor's programme in physics has been somewhat overlooked, so I'm in trouble understanding their use in this paper (The prisoner’s dilemma on co-evolving networks under perfect rationality. C Bielya, K Dragositsa and S Thurnera. Physica D 228, 40-48, arXiv:physics/0504190 [physics.soc-ph]).

Basically, they have a dynamic network which shows periodic behaviour. They try to measure the degree distribution $P(k)$, but of course they have to average over time. So, they present the resulting $\langle P(k)\rangle$, specifying the following:

To improve the accuracy of the plot, degree distributions of networks at $10^3$ different times have been averaged. The correlation in the time series has been taken care of by using time intervals of inverse correlation length.

As I understand it, they want to avoid taking samples of the observable of interest at time intervals equal the period of the cyclic evolution of the system, because that would bias the result, based on when in the period you start sampling. If so, then I don't understand why taking the inverse of the correlation length is the preferred option.

The key is the fact that they have used different times within the same simulation to provide data. This is only a valid approach if the situation has changed sufficiently between any two data points that the dynamics is uncorrelated and the different data points can be considered as independent random variables. This means that you need to wait for some length of time between measurements, or your new data point would just be (some sort of) a rehash of the previous one. The correlation time tells you (roughly) when the wait is long enough: it measures the timescale of the decay of the correlations that would screw up the statistics.

For a more simplistic situation, consider a bee trapped in a room, which I've observed every second for the past half-hour; I'm interested in the probability of it being in different areas of the room. To get better statistics I need more measurements, but I gain nothing from increasing the observation frequency to two per second: the bee can hardly be very far from halfway between the previous and the next positions; I'd just as well guess it.

Well my guess is because the phenomena is periodic, it is more natural to frame the problem as a Fourier transform of the actual real-time problem hence the correlations are given in reciprocal units (i.e. frequency)