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7.4. SCALE-FREE NETWORKS 181 systematically disabling hubs should quickly partition a network into several disjoint components, a highly undesirable situation. To illustrate these matters, Figure 7.12 shows what happens when we systematically remove vertices from a scale-free graph in comparison to removing the best-connected vertices from an ER random graph. We also show the effect of removing randomly selected vertices from a scale-free graph (which is very similar to randomly removing vertices from an ER graph). A scale-free network is thus seen to be sensitive to a targeted attack, but just as robust as an ER random graph in the case of. | 7.4. SCALE-FREE NETWORKS 181 systematically disabling hubs should quickly partition a network into several disjoint components a highly undesirable situation. To illustrate these matters Figure 7.12 shows what happens when we systematically remove vertices from a scale-free graph in comparison to removing the best-connected vertices from an ER random graph. We also show the effect of removing randomly selected vertices from a scale-free graph which is very similar to randomly removing vertices from an ER graph . A scale-free network is thus seen to be sensitive to a targeted attack but just as robust as an ER random graph in the case of a random attack. Figure 7.12 The fraction of vertices outside the giant component when removing hubs from a scale-free graph and those from an ER random graph. Related networks As we mentioned the Barabasi-Albert approach for constructing a scale-free graph has one important shortcoming when comparing it to real-world networks its relatively low clustering coefficient. A better understanding of real-world phenomena should normally be reflected by better models and in this sense a BA random graph is difficult to validate against many real-world data. Therefore researchers have been seeking solutions for constructing scale-free graphs that have a high clustering coefficient. As argued by Dorogovtsev et al. 2003 constructing such graphs is actually quite simple. The trick is to make sure that there are many triangles. This can be achieved for example by adding an edge to a triple at each step of the growing process. Recall that a triple was a subgraph with 3 vertices and 2 edges. Holme and Kim 2002 provide a scheme that combines scale-freeness and at the same time allows to tune to what extent clustering is to be provided. Their algorithm proceeds as follows
