Wireless multihop networks are usually modeled with unit disk graphs (UDG) [114]. The con-struction principle is as follows: given N nodes and a distance threshold ϑ, distribute all N nodes uniformly at random within an area of a fixed size. Then, all nodes among which the geometric distance is less than ϑ are connected by edges. In other words, all nodes which lay close enough to each other are considered neighbors in the resulting wireless ad-hoc network.
This model is valid for wireless networks within a perfect surrounding environment. In practice the entities usually move in groups, have favorite spots and there are areas that they avoid.
We extend the UDG model by inserting obstacles of different sizes within the area where we place the nodes. These obstacles represent unpopular or unaccessible areas. The generated graphs represent real world ad-hoc wireless networks more precisely. The approach for more realistic network models is based on the work of Jardosh [115]. We call the network model ODG (obstacle disk graph), see Figure 5.3 for an illustration.
We evaluate BridgeFinder using the ODG model for four different network types with none (equivalent to the standard UDG model), one, two and threebridges.
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Figure 5.4:Destroying networks by removing the fastest converging nodes.
Note that nodes are distributed uniformly at random within the areas around the obstacles.
Therefore, there is no guarantee how many of the existing bridges are actually used to connect the different clusters, nor how many different paths run through each bridge. However, net-works generated in this manner are very vulnerable. The few nodes lying on the paths among clusters are exactly the few nodes keeping the network together.
There are two important questions that need to be addressed. What topological properties do these few nodes have? More importantly, how effective is indeed BridgeFinder in detecting those very nodes?
We evaluate our approach on 500 networks of each type. Each network consists of 250 nodes placed in a physical area of 100x100 units. The maximum edge length is set to 12 units, i.e., we place edges among all nodes within distance of 12 units from each other. We create multiple obstacles with dimensions of 25x25, 30x30 and 50x50 units (see Figure 5.3 for an example setup). Because of the obstacles and the random placement of nodes, the resulting networks are not always connected. Therefore, during the generating process we discard non-connected networks and generate new ones until 500 connected instances of each type were acquired.
Now that we have diverse test scenarios, we finally can address the central question of this Chapter: what role do the nodes identified by BridgeFinder play for keeping the network con-nected. For this purpose, one by one we remove the fastest 3.5% converging nodes. The results are displayed in Figure 5.4. The x-axis shows the percentage of removed nodes. The y-axis shows the fraction of partitioned networks from the 500 instances of each type. We consider a network partitioned if more than 25% of the remaining nodes are not able to communicate with the rest of the network.
Figure 5.4 shows averaged results for all four different network types. Removing only 2%
(equivalent to just 5) of the fastest converging nodes breaks 90%, 70% and 50% of the one, two and three bridge networks respectively. BridgeFinder finds with very high probability exactly the few nodes that keep the networks together.
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Figure 5.5:Intersecting best centrality measure nodes with 5% of the fastest converging nodes.
Note that the standard UDG graph does not partition. This is not surprising. In such graphs, there are no critical nodes at all. Depending on the density of the nodes, there are two pos-sibilities. Either each node is very strongly interconnected with the rest of the network, or all links are so sparse that each node is critical. Thus, in UDG graphs there are no “special” nodes.
They all have very similar convergence rates. Removing fastest converging nodes is equivalent to removing randomly chosen nodes. Figure 5.4 just confirms the well known fact that UDG graphs are resilient to random failures.
The next question we tackle is whether there is an analytical explanation for the above pre-sented empirical results? Recall the two global measures we defined in Section 5.2: average betweenness and square closeness. They describe the importance of nodes for distributing infor-mation within a network. To answer the above question we test to what extent the top nodes identified by BridgeFinder overlap with the best nodes identified by the two global measures.
For each of our test networks we compute two sets: A and B. Aconsists of 5% (i.e. 13 of all 250 nodes) of the highest average betweenness nodes and Bcontains the highest 5% square closeness nodes. The nodes in the union of Aand B, A∪B, have either the highest average betweenness, or the highest square distance coefficients in the network, or in most of the cases both. Note thatAandBare not necessarily disjoint. On the contrary, averaged over all generated networks, Aand B overlapped to over 80%. The overlapping factor is almost identical for the four different network types.
The set A∪B contains the nodes with the “best” topological properties in the network. The question is how many of those nodes are within the nodes identified by BridgeFinder. We intersect A∪B with 5% (13 nodes) of the fastest converging nodes. Figure 5.5 displays the results. At least half of the fastest converging nodes lay on key topological positions in their networks. This is the case in more than 80% of the two-bridge networks and in over 90% of the one-bridge networks.
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Our results clearly show that with very high probability BridgeFinder identifies the same nodes one would get by using global network measures. In both cases those are the few nodes keeping the network together, see Figure 5.4.
There is however a huge difference between the two approaches. We had to use global network knowledge in centralized algorithms to compute the above discussed topological prop-erties of the nodes. Such global knowledge is unavailable in distributed environments, rendering that kind of algorithms useless in practice.
The striking advantage of BridgeFinder: It relies only on information directly exchanged among the participating peers in a fully distributed manner. That makes it not only applica-ble in distributed environments, while still being compatiapplica-ble to centralized approaches, but the results it produces are also simultaneously available to all participating peers.
All this together makes BridgeFinder outstanding in the area of distributed algorithms for detecting communication bottlenecks, as we discuss in detail in Section 5.6.