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

Abstract

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.

Pricing of cyber insurance premiums using a Markov-based dynamic model with clustering structure

“…Hence, the cardinality of N ( v ), also known as the degree of node v , expresses the number of neighbors of node v and can be written as | N ( v )| = k v , where k v is the degree of node v . A clique of three nodes { u , v , w }, where ( u , v ), ( u , w ), ( v , w ) ∈ E are links that connect all three nodes, is a triangle in a network G [ 38 ]. Let T ( v ) = |{( u , w ); w , u ∈ N ( v ), ( u , w ) ∈ E }| be the number of triangles formed with the center at node v .…”

An ensemble model to optimize modularity in dynamic bipartite networks

“…For community detection algorithms, the modularity score comes very little in bipartite networks, which leads to a very sparse relationship between entities of the same community. The clustering coefficient of small bipartite networks is also a not significant measure to find the strength of clusters (Heer et al 2020 ). Thus, an integrated model gets developed to optimize the results for both modularity and clustering.…”

“…The clustering coefficient is a measurement of the density of each vertex's 1.5-degree egocentric network. The clustering coefficient is high for dense linkages between nodes (Heer et al 2020 ; Chessa et al 2014 ). For example, in a friendship network, if all friends know each other very well, its clustering coefficient will be high.…”

Quantification of the robustness of functional neural networks: application to the characterization of Alzheimer’s disease continuum

“…Different approaches to evaluate network robustness have been developed in the literature (Jiang et al 2014, Aerts et al 2016, Bellingeri and Cassi 2018, Bellingeri et al 2019, Farooq et al 2020, Freitas et al 2020, Heer et al 2020, Rachdi et al 2020, Williams and Patterson 2020, Stacey et al 2021.…”

“…Williams and Patterson (2020) analyzed a global network of associations between coral species and Symbiodiniaceae; they used the concept of ecological robustness, defined by the amount of perturbation needed to decrease the number of nodes by 50%, and evaluated it using different node removal models. Heer et al (2020) used habitat networks to evaluate the habitat loss and to which extent it was possible to modify networks to obtain higher robustness; they suggested that the clustering coefficient was identified as a good indicator for network robustness and, specifically, they developed tools to identify which links should be added to the network to increase its robustness. Freitas et al (2020) presented a robustness analysis of an inter-cities mobility complex network, where cities are represented by nodes and the flow of people via terrestrial transports by links; they systematically isolated nodes either random or guided by specific strategies, and assessed the impact of these perturbations with different metrics.…”