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.

We use Lemma 1.1 to calculate the clustering coefficient C G of G with help of C G . Lemma 1.2. Let G = (V, E) be a network as above with n nodes, (u, v) ∈ E and G = (V, E ∪ {(u, v)}). Let k := |N (u, v)| 1 be the number of common neighbours of u and v. Then the difference in clustering is as follows: Proof. Let u, v ∈ V with d u , d v > 1 and set k := |N (u, v)| as the number of common neighbours. Then, it holds The difference in clustering by inserting e = (u, v) can be calculated as Now, consider u, v ∈ V with degree 1 and a common neighbour w. Then the difference in clustering is: .
, and C(v) all equal zero and k = 1, as they have a common neighbour.

Robustness simulation
We simulated habitat loss and subsequent metapopulation dynamics as proposed by Heer et al. (under review) on the landscape-based habitat networks to evaluate the increase of metapopulation robustness on those networks. Here, we briefly summarize how the simulation was modelled to make it easier for readers to follow our findings.

Simulation overview
To evaluate the robustness of a habitat network against habitat loss, we first simulated the habitat loss by randomly removing habitat patches from the network. Habitat patches on the remaining network were assumed to be fully colonised. Then, metapopulation dynamics consisting of local extinctions and subsequent recolonisation from neighbouring patches were simulated until a stationary distribution was reached. This process of simulated habitat loss and subsequent metapopulation dynamics was then repeated for different degrees of habitat loss to obtain a robustness curve describing the fraction of colonised habitat patches in dependence on the fraction of lost habitat patches. Based on this robustness curve, we used the 'area under the curve' (AUC) as a measure to quantify metapopulation robustness: the higher the fraction of colonised habitat patches across fractions of lost habitat patches, the higher the AUC, and thus the estimated metapopulation robustness. For each network, simulations were replicated ten times to average over the sources of randomness affecting habitat loss and metapopulation dynamics.

Habitat loss
We assumed a random habitat loss scenario, which removed each habitat patch with equal probability p.

Metapopulation dynamics
For a given level of habitat loss, metapopulation dynamics were simulated on the remaining habitat network, by considering local extinctions in habitat patches and the recolonization of habitat patches. We used the size of cliques to measure, how well a patch is connected within its neighbourhood, as the survival of a population in a habitat patch depends on its potential to exchange individuals with neighbouring patches. Denoting by c(v) the size of the largest clique that contains the node v we assumed that the population in v goes extinct with probability where a > 1 is a species-specific parameter governing the local-extinction risk of a species. We can think of these risks decreasing with increasing c(v) more slowly for habitat specialists (small values of a) and more rapidly for habitat generalists (large values of a). We investigated species with three different levels of local-extinction risks -low (a = 2), medium (a = 5), and high (a = 9).
Empty habitat patches can be recolonised from connected colonised patches. Recolonisation was modelled with the help of a Gaussian dispersal kernel and we assumed that an empty habitat patch v becomes recolonised from a colonised patch w with probability where m vw = exp(− 1 2 d 2 vw /σ 2 ) is the dispersal kernel, V the set of all network nodes, d vw the distance between habitat patches v and w in terms of dispersal costs and σ > 0 a species-specific dispersal parameter governing the dispersal range of a species. We can think of these dispersal ranges as being low for poor dispersers (small values of σ) and high for good dispersers (large values for σ). Similar to a, we investigate values of σ ∈ [2, 5, 9] to account for the different dispersal capacities of different species.
These local extinctions in and recolonizations of habitat patches were simulated alternately until a stationary frequency of colonized patches was reached.

Network
NetworkX algorithm Parameter sparse Parameter dense Regular nx.random_regular_graph d = 4 d = 58 Random nx.erdos_renyi_graph p = 0.04 p = 0.75 Small-world nx.newman_watts_strogatz_graph k = 2, p = 0.6 k = 39, p = 0.5 Table 1: Parameters to create standard networks. d is the degree of each node, p denotes the percentage of links present in the network and k is the degree of each node in the small-world network before rewiring.