Single and Multi Step Survival Rates

Finally, the robustness of links over time can be examined by calculating survival ratios of links, or edges in successive MST. The single-step survival ratio is the fraction of links found in two consecutive MST in common at times $t$ and $t-1$ and is defined by Onnela et al. [20] as:

$$\sigma (t) = \frac{1}{N-1}\vert E(t) \cap E(t-1)\vert$$ (2.30)

where $E(t)$ is the set of edges of the MST at time $t$, $\cap$ is the intersection operator, and $\vert\cdots\vert$ gives the number of elements in the set. A multi-step survival ratio can be used to study the longer-term evolution [20]:

$$\sigma(t,k)=\frac{1}{N-1}\vert E(t) \cap E(t-1) \cdots E(t-k+1) \cap E(t-k)\vert$$ (2.31)

in which only the connections that continue for the entire period without any interruption are counted.