for a metapopulation of $N$ patches and tested metapopulation viability with respect to different connectivity matrices [@Vuilleumier06; @Bode08; @Vuilleumier10]. The model, which is attractive in its simplicity, implements dispersal through the connectivity matrix $D$. Initially all $N$ patches are populated. At each time step two events occur in succession: first, populated patches go extinct at per patch probability $e$. Subsequently, empty patches can be colonised with probability $c$ by each incoming dispersal connection from a populated patch. Newly populated patches cannot give rise to colonisation of other patches at the same time step they have been colonised.

In order to estimate the extinction risk of metapopulations the model is iterated $T$ times. If any populated patch is left after the $T^{\mbox{th}}$ iteration, the metapopulation is termed *viable* and *extinct* otherwise. As @Vuilleumier06 we chose the parameters $e=0.5$ and $T=1,000$, and discuss the probability of extinction of metapopulations consisting of $N=100$ patches as a function of the colonisation probability $c$.

\[sec:symmetry-def\]Symmetry of dispersal patterns
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For characterisation of the symmetry properties of dispersal patterns the connectivity matrix $D$ is divided into its symmetric and anti-symmetric contributions, $S$ and $A$, by defining the matrix elements

\[eq:def-a-s\] $$\begin{aligned}
    s_{ij} & := & \min\left(d_{ij},d_{ji}\right)\\
    