a_{ij} & := & d_{ij}-s_{ij}\quad.
  \end{aligned}$$

Based on these matrices the degree of symmetry $\gamma$ of dispersal patterns is defined as the ratio of symmetric connections among all connections: $$\gamma:=\frac{\sum_{i,j}s_{ij}}{\sum_{i,j}a_{ij}+s_{ij}}\quad.\label{eq:def-deg-asymm}$$ Note that $1-\gamma$ is related to the asymmetry $Z$ discussed in [@Bode08].

By means of Equation , the symmetry properties of dispersal patterns are put on a firm footing: Dispersal patterns are called *symmetric* if $\gamma=1$ and *anti-symmetric* if $\gamma=0$. Generally connectivity matrices $D$ with intermediate $\gamma$ are neither symmetric nor anti-symmetric. We term them *asymmetric* if $\gamma<1$ corresponding to dispersal directed at least to some degree.

\[sec:viab-metap-conn\]Viability of metapopulations connected through regular dispersal patterns
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![\[fig:Algorithm-examples\]Examples of connectivity matrices $D$ generated by the algorithm described in Section \[sec:viab-metap-conn\] and in \[sec:algor-gener-balanc\] for a reduced number of patches, $N=8$, and different combinations of $L/N$ and $\gamma$. Only non-zero entries are printed explicitly. For reasons of clarity symmetric connections are denoted by ’S’ and asymmetric connections by ’A’. The colours indicate separated closed cycles of dispersal that can be identified in the matrices. While the connectivity matrices with $L/N=1$ (upper row) are degenerate into $2$ ($\gamma=0.0$), $3$ ($\gamma=0.5$), and $4$ ($\gamma=1.0$) clusters respectively, the clusters of all three matrices generated with 