The pivotal quality of proximity graphs is connectivity, i.e., all nodes in the graph are connected to one another either directly or via intermediate nodes. These types of graphs are often robust, i.e., they are able to function well even if they are subject to limited removal of elementary building blocks, as may occur for random failures or targeted attacks. Here, we study how the structure of these graphs is affected when nodes get removed successively until an extensive fraction is removed such that the graphs fragment. We study different types of proximity graphs for various node-removal strategies. We use different types of observables to monitor the fragmentation process, simple ones like the number and sizes of connected components and more complex ones like the hop diameter and the backup capacity, which is needed to make a network N-1 resilient. The actual fragmentation turns out to be described by a second-order phase transition. Using finite-size scaling analyses we numerically assess the threshold fraction of removed nodes, which is characteristic for the particular graph type and node deletion scheme; this suffices to decompose the underlying graphs.

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We perform Monte Carlo simulations to determine the critical temperatures of Ising Ferromagnets (IFM) on different types of two-dimensional proximity graphs, in which the distribution of their underlying node sets has been changed systematically by means of a parameter σ. This allows us to interpolate between regular grids and proximity graphs based on complete random placement of nodes. Each edge of the planar proximity graphs carries a weighted ferromagnetic coupling. The coupling strengths are determined via the Euclidean distances between coupled spins. The simulations are carried out on graphs with N = 162 to N = 1282 nodes utilising the Wolff cluster algorithm and parallel tempering method in a wide temperature range around the critical point to measure the Binder cumulant in order to obtain the critical temperature for different values of σ. Interestingly, the critical temperatures depend partially non-monotonously on the disorder parameter σ, corresponding to a non-monotonous change of the graph structure. For completeness, we further verify using finite-size scaling methods that the IFM on proximity graphs is for all values of the disorder in the same universality class as the IFM on the two-dimensional square lattice.