The Myth of the Robust-Yet-Fragile Nature of Scale-Free Networks: An Empirical Analysis

Abstract

In addition to their defining skewed degree distribution, the class of scale-free networks are generally described as robust-yet-fragile. This description suggests that, compared to random graphs of the same size, scale-free networks are more robust against random failures but more vulnerable to targeted attacks. Here, we report on experiments on a comprehensive collection of networks across different domains that assess the empirical prevalence of scale-free networks fitting this description. We find that robust-yet-fragile networks are a distinct minority, even among those networks that come closest to being classified as scale-free.

6Appendix

1.1 6.1Scale-Freeness Classification: Further Analysis

In the following table, we present the scale-freeness classification results of the remaining categories, namely biological, transportation, and auxiliary. We can observe that the evaluations of the Broido et al. and Voitalov et al. methods are not necessarily aligned, for example, for the auxiliary and transportation categories. In fact, these evaluations sometimes disagree strongly, for instance, in the case of biological networks. The table also presents the scale-freeness classification results for the social category without considering the Facebook100 networks. It shows that even without these Facebook100 networks, 60% of all the social networks in our collection are evaluated as not scale-free, and around 47% of them are evaluated as hardly power-law or not power-law. This indicates that the earlier observed evaluation of both methods that scale-free networks do not constitute the strong majority of social networks is mostly consistent. We note that 99.98% of Facebook100 networks are hardly power-law or not power-law according to the Voitalov et al. method, while the evaluation of Broido et al. is inconclusive for 57.14% and places the rest in the not scale-free category (Table 2).

Table 2. The columns indicate, respectively, the Weak (W), Super-Weak (SW), Not Scale-Free (NSF), and Inconclusive (INC) categories from the Broido et al. method. The rows indicate, respectively, the Divergent second moment Power-Law (DPL), Other Power-Law (OPL), Hardly Power-Law (HPL), and Not Power-Law (NPL) categories from the Voitalov et al. method. Each cell at the intersection of a row and a column indicates the percentage of networks at the intersection of the respective classes. Since we did not have any Strong, Super-Weak and Weak, or Weakest but not Weak networks in our collection, we utilized a more concise non-overlapping representation of scale-freeness classes, compared to the one originally defined by Broido et al.

Fig. 3.

The figure illustrates the comparison between the networks in our collection and size-matching random graphs in terms of their robustness against targeted attacks and random failures. Dark dashes depict the identity line. The first inset (marked with solid gray borders) zooms in on the area where most non-social networks are located. The second inset (marked with dark dotted borders) focuses on the region where networks are more robust against random failures but more vulnerable to targeted attacks (based on the current degree), compared to the baseline of size-matching random graphs. Note that in favor of improving the clarity of the visualization, we have refrained from displaying the extreme outliers, which constitute 3% of all networks in our collection and attain (under both considered vertex removal strategies) highly negative z-scores. (Color figure online)

1.2 6.2Robustness: Further Analysis

To create Fig. 3, we used the same procedure as for Fig. 2, with the only variation being the usage of adaptive targeted attacks, in which vertices with a higher current degree are removed first rather than those with a higher initial degree.

As shown in Fig. 3, all networks, including the Collins yeast interactome network, exhibit increased sensitivity to adaptive targeted attacks compared to random failures. This is evident from the fact that the markers for the networks across different categories are above the identity line, and the marker for the Collins yeast interactome network is very slightly above the identity line. We also observe that the networks from the social category continue to constitute a noticeable fraction of the most fragile networks in the entire collection.

Our analysis shows that 88% of the networks in our collection are more fragile than size-matching random graphs, while the remaining 12% are robust-yet-fragile. This is consistent with our previous results in Sect. 4, with all robust-yet-fragile networks except for three Kegg Metabolic networks being from the technological category, which comprises almost exclusively the networks in our collection best suited for classification as scale-free. The percentage of robust-yet-fragile networks among technological networks is 14.86%, similar to the findings of Sect. 4.

Based on the above two paragraphs, we can conclude that all the networks in our collection are more sensitive to adaptive targeted attacks than they are to random failures. Furthermore, those networks that can be characterized as robust-yet-fragile constitute only a distinct minority, even among those networks that are the best candidates for being classified as scale-free, which is in line with our conclusions in Sect. 4.

1.3 6.3The Curious Case of Collins Yeast Interactome

A protein complex is a group of proteins that work together to perform a specific biological function. In the Collins yeast interactome network, individual proteins in budding yeast are represented as vertices, and edges connect all proteins that are part of the same protein complex to each other [5]. Thus, the structure of this network is characterized by the presence of dense, interconnected clusters, with larger clusters comprising high-degree vertices of proteins that interact mostly with other proteins within the same complex. These clusters are interconnected mainly through connectivity hubs, which are proteins that do not necessarily interact with many proteins within their protein complex, but with potentially few proteins from different complexes. In Fig. 4 on the next page, we present a visualization of the Collins yeast interactome network, in which the discussed network structure is apparent.

This structure has important implications for the robustness of the network. To observe this, note that high-degree vertices are found within dense clusters of similarly high-degree vertices, so targeted attacks which remove high-degree vertices first would mainly focus on disintegrating these high-density clusters, while leaving the low-degree vertices that interconnect them intact. In contrast, random failures are more likely to hit these low-degree interconnecting vertices, resulting in a faster disintegration of the network into many smaller, disjoint clusters. This explains the higher susceptibility of the robustness of the Collins yeast interactome network to random failures compared to targeted attacks, as opposed to any other network in our collection.

Fig. 4.

The largest connected component of the Collins yeast interactome network. The size of each vertex is proportional to its degree in the network.