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Figure 5.5: Destroying networks by removing the fastest converging nodes existing bridges are actually used to connect the different clusters, nor how many different paths run through each bridge.
However, networks 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. We address two different questions. First, what topological properties do these few nodes possess? Second, and more importantly, how effective is BridgeFinder in detecting those very nodes?
We created 500 connected networks of each type. Each network consists of 250 nodes placed in a physical area of 100 x 100 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 created multiple obstacles with dimensions of 25 x 25, 50 x 50 and 30 x 30 units (see figure 4.2 for an example setup). Due to the obstacles and the random placement of nodes, the resulting networks are not always connected.
Therefore, during the generating process, we discarded non-connected networks, and new ones were generated until 500 connected instances of each type were acquired.
Evaluation Results Figure 5.5 shows the important role the nodes identified by BridgeFinder play in keeping the network connected. One by one, we removed the fastest 3.5% converging nodes. The x-axis shows the percent 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. Hence, isolating only a small number of nodes is not a globally critical event, and we do not consider it as a problem.
Figure 5.5 shows averaged results for all four different network types. Removing
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Figure 5.6: Intersecting the nodes with best centrality measures with 2% of the fastest converging nodes
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.
Note that the standard unit disc 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 possibilities. 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 unit disc graphs, there are no “special” nodes. Therefore, they all have very similar convergence speeds. Removing the fastest converging nodes is equivalent to removing randomly chosen nodes. Figure 5.5 just confirms the well-known fact that unit disc graphs are resilient against attacks.
Analysis of Results Recall the two global measures we defined in Section 4:
average betweennessandsquare closeness. They are tools from graph theory for describing the importance of nodes for distributing information within a network.
To answer the above question, we evaluate to what extent the top nodes identified by BridgeFinder overlap with the best nodes identified by the two global measures.
Betweenness and Closeness Sets For each of our test networks, we compute two sets:AandB. SetAconsists of the highest 5% (i.e., 13 of all 250 nodes) average betweenness nodes, and setBcontains the highest 5% square closeness nodes. The nodes in the union ofAandB,A∪B, have either the highest average betweenness or the highest square distance coefficients in the network, or in most cases, both.
Note thatAandBare not necessarily disjoint. On the contrary, averaged over all
generated networks,Aand Boverlapped to over 80%. The overlapping factor is almost identical for the four different network types.
Simulation Results The setA∪Bcontains 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. Figure 5.6 shows the results from the interSection of 2% (equivalent to 5) of the fastest converging nodes withA∪B. In 85% of the one-bridge networks, all five nodes have either high closeness or high betweenness, i.e., are inA∪B. Three of those five nodes have the same features in about 80% of the two-bridge and unit disc networks, and in about 60% of the three-bridge networks.
Thus, the top nodes identified by BridgeFinder lay exactly on the key positions in their networks. It is not surprising (as shown in Figure 5.5) that removing even a few such nodes can damage a significant portion of the underlying networks.
Next, we intersectA∪Bwith 5% (13 nodes) of the fastest converging nodes. Figure 5.7 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.
Our empirical results show that with very high probability, BridgeFinder identifies the same nodes one would obtain by using global network measures. In both cases, those are the few nodes keeping the network together.
Still, there are two very important differences between the two approaches. First, our algorithm is faster in several orders of magnitude than computing global network measures and furthermore produces very modest computational costs per node.
Second, and more importantly, BridgeFinder is a distributed approach. That makes it a useful tool that can be applied to a vast variety of real world networks.