Fragmentation properties of two-dimensional Proximity Graphs considering random failures and targeted attacks

One particular proximity graph type that will be considered subsequently is the relative neighborhood graph (RNG) Toussaint (1980). In order to determine whether in the construction procedure for an instance of a RNG two nodes $u,v\in V$ need to be connected to each other, it is necessary to check if there is a third node $w\in V\setminus\{u,v\}$ with ${\rm dist}(u,w)\leq{\rm dist}(v,u)$ and ${\rm dist}(v,w)\leq{\rm dist}(v,u)$. If such a node $w$ does not exist, $u$ and $v$ will get linked. In geometrical terms, for each pair $u$ and $v$ of points, the respective distance $\rm{dist}(u,v)$ can be used to construct the lune ${\rm lune}(u,v)$. The lune is given by the intersection of two circles with equal radius $\rm{dist}(u,v)$, centered at $u$ and $v$, respectively. If no other point $w\in V\setminus\{u,v\}$ lies within ${\rm lune}(u,v)$, i.e. if the lune is empty, both nodes are connected by means of an edge. To facilitate intuition, an example of a RNG for a small set of $N=5$ nodes is sketched in Fig. 1(a). A larger example that illustrates the principal structure of a RNG is shown in Fig. 2(a).

Another proximity graph that will be considered in this article is the Gabriel graph (GG) Gabriel and Sokal (1969); Bertin et al. (2002). To determine whether in the construction procedure for an instance of a GG two nodes $u,v\in V$ need to be connected, $\rm{circ}(u,v)$, i.e. the smallest of all possible circles which embeds both nodes is considered, which has a diameter $\rm{dist}(u,v)$. These two nodes will be connected unless there is another node $w$ which is located within the area enclosed by ${\rm circ}(u,v)$. To facilitate intuition, the linking rule for the GG is illustrated in Fig. 1(b). A larger example that illustrates the principal structure of a GG is shown in Fig. 2(b). Further, note that the GG is a super-graph of the RNG. This is due to the circumstance that $\rm{circ}(u,v)$, which is relevant in the construction procedure of a GG instance for a given set of nodes encloses a subarea of $\rm{lune}(u,v)$, being relevant in the construction procedure of the corresponding RNG instance (compare the grey shaded surfaces in Figs. 1(a,b)). Therefore, all edges contained in the RNG are also included in the GG. Note that this can also be seen in Figs. 2(a,b).

The construction of the Delaunay triangulation (DT; also a type of proximity graph) Sibson (1978) is quite similar. Two nodes $u,v\in V$ will be connected if any circle exists which embeds $u$ as well as $v$ but no further nodes. To facilitate intuition, the result of this linking-rule is shown in Fig. 1(c). A larger example that illustrates the principal structure of a DT is shown in Fig. 2(c). From the definition of these linking-rules, since the GG graph also involves the construction of a circle, it is evident that an instance of a DT for a given set of nodes must be a super-graph of the corresponding GG instance. As a consequence, being a sub-graph of the GG, the RNG is also a sub-graph of the DT. This can be observed in Figs. 1(a-c) (Figs. 2(a-c)), where the RNG, GG and DT are illustrated for the same set of $N=5$ ($100$) nodes.

The fourth network topology that will be considered is the minimum radius graph (MR). In the construction procedure of an instance of a MR, two nodes $u,v\in V$ will be joined by an edge, if ${\rm dist}(u,v)\leq r$. Therein, the “connectivity radius” $r$ specifies the smallest possible value which ensures that all nodes are connected to one another, possibly in a multi-hop manner. It becomes evident from Figs. 1(d) and 2(d) that, in contrast to the previous graphs, the MR might feature crossing edges.

In order to construct the RNG and GG, we made use of the sub-graph hierarchy ${\rm RNG}\subset{\rm GG}\subset{\rm DT}$. I.e., for a given set of nodes we first obtained the DT by means of the Qhull computational geometry library Barber (1995) (the DT for a set of $N$ points can be computed in time $O(N\log(N))$ Preparata and Shamos (1985); Barber (1995)) and then pruned the resulting edge-set $E$ until the linking requirements of GG or RNG are met. Here, we amend the naive implementation of this two-step procedure Toussaint (1980), yielding an algorithm with running time $O(N^{2})$, by means of the “cell-list” method Melchert (2013), resulting in a sub-quadratic running time. In this regard, note that Ref. Jaromczyk and Toussaint (1992) provides an overview of several algorithmic approaches for the construction of RNGs and GGs. Finally, note that RNGs and GGs can be found as the limiting cases of a parameters family of proximity graphs, termed $\beta$-skeletons Kirkpatrick and Radke (1985).

At this point, note that due to a yet unmentioned property of minimum weight spanning trees (MST; i.e. a spanning tree in which the sum of Euclidean edge lengths is minimal, see Ref. Cormen et al. (2001)) we can set the “connectivity radius” of MRs, i.e. a geometric random graph, in context to proximity graphs. Bear in mind that the longest edge present in any instance of a MR specifies the smallest possible edge length which ensures that all nodes are connected to one another. Exactly this edge length characterizes the longest edge in the MST of the corresponding node-set. For a given set of nodes, a MST is a spanning sub-graph of the RNG Toussaint (1980); Melchert (2013). Thus, considering MSTs, the previously mentioned sub-graph hierarchy can be extended to ${\rm MST}\subset{\rm RNG}\subset{\rm GG}\subset{\rm DT}$. This allows for a fast construction of a MR instance for a given set of points via a convenient three-step procedure: (i) compute the DT for the given set of points, (ii) filter the edge-set of the DT to determine the corresponding MST, and, (iii) use the length of the longest MST edge as “connectivity radius” to construct the respective MR. Therein, the overall running time is dominated by step (iii), which, in its most naive implementation has computational cost $O(N^{2})$. Note that during the latter step, the previously mentioned “cell-list” method can be used to achieve an improved running time.

Subsequently we will introduce the node-removal strategies that will be considered in the numerical simulations carried out to characterize the fragmentation process for the above graph types.

To emphasize structural differences between the graph types of the sub-graph hierarchy ${\rm RNG}\subset{\rm GG}\subset{\rm DT}$ we first consider the respective average node degree. Therefore, the scaling behavior of the effective, i.e. system-size dependent, average degree $d_{\rm eff}(N)$ is considered and analyzed using a fit to the function $d_{\rm eff}(N)=d-aN^{-b/2}$. For the three graph types ${\rm RNG}$, ${\rm GG}$ and ${\rm DT}$ the fits yield asymptotic degrees $d$ and scaling exponents $b$, where $d_{\rm RNG}=2.557(1)$ and $b_{\rm RNG}=0.99(4)$ (with a reduced chi-square $\chi^{2}_{\rm red}=0.87$; note that both, the asymptotic average degree and the scaling exponent compare well to the estimates reported in Ref. Melchert (2013)), $d_{\rm GG}=3.999(1)$ and $b_{\rm GG}=1.00(1)$ (reduced chi-square $\chi^{2}_{\rm red}=0.70$), $d_{\rm DT}=6.0001(1)$ and $b_{\rm DT}=1.76(1)$ (for a reduced chi-square $\chi^{2}_{\rm red}=1.87$; note that the average degree of the $\rm DT$ is known to be $d_{\rm DT}=6$). In Fig. 3 the correction to scaling, i.e. $d-d_{\rm eff}(N)\propto N^{-b/2}$, is shown for the three types of proximity graphs. It is interesting to note that RNG and GG exhibit a similar scaling, involving a correction of the form $N^{-1/2}$, whereas the scaling behavior for the average degree for the DT graphs is governed by a significantly larger exponent. Also, note that instances of the three types of proximity graphs are planar, i.e. there are no crossing edges. While the bounding cycles of the finite faces for the instances of RNGs and GGs might consist of an even or odd number of edges, all inner faces for instances of DTs are bounded by three edges.

Further, for the minimum radius graph we found that the effective, average degree fits best to a logarithmic scaling function of the form $d_{\rm eff}(N)=\log(aN)$, see inset of Fig. 3 where $a=15.7(4)$ (reduced chi-square $\chi^{2}_{\rm red}=1.42$; however, note that the data can also be fit by a scaling function with a small power-law correction as above, where $d_{\rm MR}\approx 57$ and $b_{\rm MR}\approx 0.04$).

Regarding MRs, consider that the longest edge present in any instance of a MST (which specifies the length of the longest edge in the respective MR instance; see discussion above) can by no means exceed the length of the longest edge of any of its super-graphs. Due to the geometric restrictions imposed by going from an instance of a DT to a RNG, it is thus plausible that the maximal edge length found for any MR instance is much shorter than, say, for the corresponding DT instance. This holds in particular for the case of open boundary conditions, where the outer faces of the DT instances feature rather long edges, see Fig. 2(c). For a set of 500 instances of point sets consisting of $N=16384$ nodes (i.e. for systems of effective side length $L=128$) we found that the longest edge length ratio $r_{\rm max}/L$ for the four graph types read $r_{\rm max}^{\rm DT}/L=0.714(6)$, $r_{\rm max}^{\rm GG}/L=0.02927(8)$, $r_{\rm max}^{\rm RNG}/L=0.02335(8)$, and, $r_{\rm max}^{\rm MR}/L=0.01548(6)$. For the first three graph types, these values should be more or less independent of the system size. On the other hand, for the minimum radius graphs we found that the finite-size scaling behavior of the connectivity radius $r_{\rm max}^{\rm MR}(L)$ as function of the effective system length $L$ exhibits a logarithmic scaling of the form $r_{\rm max}^{\rm MR}(L)=a+b\log(L)$, where $a=1.334(8)$ and $b=0.133(2)$ (reduced chi-square $\chi^{2}_{\rm red}=0.59$), supporting the logarithmic scaling of the average degree. I.e., the respective “connectivity area” $A_{r}=\pi r_{\rm max}^{2}$, which, if centered at the position of a given node, specifies the area in which all its nearest neighbors can be found, should be almost equal to the previously discussed average degree $d_{\rm eff}$, because the density of nodes is unity. E.g., at $N=2304$ (i.e. $L=48$) we find $A_{r}=10.7(2)$ and $d_{\rm eff}=10.52(5)$.

For the fragmentation analysis we consider instances of the four different graph types, introduced in Sec. II, and successively remove nodes according to one of the node removal strategies, presented in Sec. III, until the initially connected graph decomposes into small clusters. So as to determine the critical fraction $p$ of nodes that need to be removed until the graph decomposes we perform a finite-size scaling (FSS) analysis for different observables that are commonly used in studies of percolation Stauffer and Aharony (1992) in Sec. IV.2.1. In addition, in Sec. IV.2.2 we consider the scaling behavior of the hop-diameter, i.e. the longest among all shortest paths measured in terms of node-to-node hops, which, e.g., is relevant in the context of broadcasting problems on networks Jennings and Okino (2002).

The actual most important application example regarding proximity graphs are wireless ad hoc networks. Nevertheless, there might be some other fields of application for them. Proximity graphs ensure connectivity and the total length of all involved edges is small in comparison to many other networks that feature this quality. For applications where edges are expensive and connectivity is crucial, the topology of proximity graphs might be a good candidate to install. Up to here, we have assumed that the capacities of the edges and nodes are infinitely large, so the network components do not overload, regardless how intensive they get strained. In real scenarios, if some nodes or edges malfunction, network components which are hardly used under normal circumstances might become essential at once. In consequence, since the hardly used components are not designed to handle such a burden, this might trigger a cascading breakdown of the whole network Bao et al. (2009); Crucitti et al. (2004b); Motter (2004); Motter and Lai (2002). Therefore, it is reasonable to equip all network components with sufficient capacity.

To ensure that the network operates orderly under all circumstances when one component drops out, referred to as “$N-1$ resilience” (or $N-1$ stability, or $N-1$ criterion), it is necessary to know the most adverse scenario that can happen. When having a transport model in mind, where some quantities have to be transported between all pairs of nodes, Motter and Lai (2002); Motter (2004); Bao et al. (2009) the betweenness centrality Newman (2001) is a measure of the capacity each node or edge has to provide in a well functioning situation. When one node or edge fails, given that the network is still connected, the loads have to be redistributed, visible from a recalculation of the betweenness centrality. In some nodes or edges the centrality will increase Ouyang et al. (2014); Hartmann (2014), corresponding to a higher capacity these nodes or edges have to provide a priori. The value of the highest increment, which is called backup capacity Hartmann (2014) $\Delta b_{\rm node}$ or $\Delta b_{\rm edge}$, provides an estimate for the additional costs for each node or edge that must be invested to protect the network against cascading failures upon such an incident.

The betweenness centrality has been calculated based on Dijkstra’s algorithm Cormen et al. (2001), i.e. the edge lengths have been taken into account for calculating the shortest path. The resulting probability mass functions of $\Delta b_{\rm node}$ and $\Delta b_{\rm edge}$ are illustrated in Fig. 7 and 8, respectively, for different network ensembles.

For the case of $\Delta b_{\rm edge}$ however, using a breadth-first search, i.e., without taking the actual edge lengths into account, the probability mass functions look qualitatively almost identical to the case of $\Delta b_{\rm node}$ without notable distinctions. From the statistics mediated by the figures it is evident that the typical backup capacity of DT networks is smallest while of RNG networks it is the highest. This means that the structure of the network of RNG networks is more vulnerable such that one has to invest more into the capacity of the edges in order to ensure $N-1$ stability. This is not surprising, since the RNG is a sub-graph of the GG and DT and includes less edges. On the other hand, due to the lack of the additional edges of the RNG in comparison to the others, the investment to provide the backup capacity must be applied for less edges. Thus, it makes sense to ask for the total backup capacity, either per edge, if investments cost are dominated by the number of connections, or per unit length, if investments are dominated by the length of the edges. It becomes evident from Fig. 9(a) that for the former case, the typical total investment ($M\cdot\Delta b$) is relatively speaking still the same for all three ensembles. For the second case, i.e., taking also edge lengths into account (Fig. 9(b)), it turns out that the investment of the DT is about the same level as the RNG. The GG appears to be the most cost efficient graph if this scenario is at hand.

To compensate for the simple resilience effect created by simply exhibiting more edges, we also compared the different topologies of the proximity graphs featuring the same total edge length $\ell_{\rm tot}$. Therefore, we measured the scaling behavior of this quantity for the different proximity graph types (see Fig. 10). The figure provides the number of nodes which have to be added to the RNG and GG in order to get same total edge length as the respective DT. E.g., it is evident from the figure that a DT with $N=718$ nodes has the same total edge length $\ell_{\rm tot}=100$ on average as a GG with $N=2625$ and a RNG with $N=9783$ nodes. Since the fragmentation thresholds for the different node-removal strategies are known (Tab. 1), it can be calculated easily ($N\cdot p_{c}$) for each topology how many nodes must be removed until the respective network decomposes into small clusters. E.g., if $\ell_{\rm tot}=100$, the RNG will tolerate $2005$ randomly removed nodes. In contrast, the GG tolerates $958$ and the DT tolerates merely $359$ nodes that fail randomly.

As a consequence, implementing the topology of the RNG will be the most reasonable, if installing edges is much more expensive than adding further nodes. Certainly, the additional edges of the DT and GG increases the stability, but the benefit of those is small in comparison to the edges which are contained in the RNG anyway.