Maximising the clustering coefficient of networks and the effects on habitat network robustness.

The robustness of networks against node failure and the response of networks to node removal has been studied extensively for networks such as transportation networks, power grids, and food webs. In many cases, a network's clustering coefficient was identified as a good indicator for network robustness. In ecology, habitat networks constitute a powerful tool to represent metapopulations or -communities, where nodes represent habitat patches and links indicate how these are connected. Current climate and land-use changes result in decline of habitat area and its connectivity and are thus the main drivers for the ongoing biodiversity loss. Conservation efforts are therefore needed to improve the connectivity and mitigate effects of habitat loss. Habitat loss can easily be modelled with the help of habitat networks and the question arises how to modify networks to obtain higher robustness. Here, we develop tools to identify which links should be added to a network to increase the robustness. We introduce two different heuristics, Greedy and Lazy Greedy, to maximize the clustering coefficient if multiple links can be added. We test these approaches and compare the results to the optimal solution for different generic networks including a variety of standard networks as well as spatially explicit landscape based habitat networks. In a last step, we simulate the robustness of habitat networks before and after adding multiple links and investigate the increase in robustness depending on both the number of added links and the heuristic used. We found that using our heuristics to add links to sparse networks such as habitat networks has a greater impact on the clustering coefficient compared to randomly adding links. The Greedy algorithm delivered optimal results in almost all cases when adding two links to the network. Furthermore, the robustness of networks increased with the number of additional links added using the Greedy or Lazy Greedy algorithm.

Optimisation Model of Dispersal Simulations on a Dendritic Habitat Network.

To predict and mitigate biodiversity loss, a better understanding of species distribution and reliable dispersal models are required. A promising approach in dispersal simulation is the method of spatially explicit graph-based analysis. While graph theory is strongly connected to the field of optimisation in a variety of disciplines, the potential of optimisation has not yet been exploited in dispersal models. We introduce an optimisation model built on a graph-based dispersal simulation of an aquatic invertebrate species with a terrestrial life stage. The model simulates a directed dispersal process and investigates the fastest route to colonise predefined vacant habitat patches. The optimisation model run-time is in general an order of magnitude faster than the underlying simulation and provides the minimum time until the considered habitat patches are colonised under the given landscape structure. These results can then be used to estimate how fast newly formed habitat patches can be reached and colonised. Our model can in principle be adapted to other simulation models and can thus be seen as a pioneer of a new set of models that may support landscape conservation and restoration.

Hypervolume Subset Selection in Two Dimensions: Formulations and Algorithms.

The hypervolume subset selection problem consists of finding a subset, with a given cardinality k, of a set of nondominated points that maximizes the hypervolume indicator. This problem arises in selection procedures of evolutionary algorithms for multiobjective optimization, for which practically efficient algorithms are required. In this article, two new formulations are provided for the two-dimensional variant of this problem. The first is a (linear) integer programming formulation that can be solved by solving its linear programming relaxation. The second formulation is a k-link shortest path formulation on a special digraph with the Monge property that can be solved by dynamic programming in [Formula: see text] time. This improves upon the result of [Formula: see text] in Bader ( 2009 ), and slightly improves upon the result of [Formula: see text] in Bringmann et al. ( 2014b ), which was developed independently from this work using different techniques. Numerical results are shown for several values of n and k.