Comparing the sensitivity of social networks, web graphs, and random graphs with respect to vertex removal

The sensitivity of networks regarding the removal of vertices has been studied extensively within the last 15 years. A common approach to measure this sensitivity is (i) removing successively vertices by following a specific removal strategy and (ii) comparing the original and the modified network using a specific comparison method. In this paper we apply a wide range of removal strategies and comparison methods in order to study the sensitivity of medium-sized networks from real world and randomly generated networks. In the first part of our study we observe that social networks and web graphs differ in sensitivity. When removing vertices, social networks are robust, web graphs are not. This effect is conclusive with the work of Boldi et al. who analyzed very large networks. For similarly generated random graphs we find that the sensitivity highly depends on the comparison method. The choice of the removal strategy has surprisingly marginal impact on the sensitivity as long as we consider removal strategies implied by common centrality measures. However, it has a strong effect when removing the vertices in random order.

💡 Research Summary

The paper investigates how the removal of vertices affects the structural integrity of different types of networks. The authors adopt a two‑step experimental framework: (1) a removal strategy that determines the order in which vertices are deleted, and (2) a comparison method that quantifies the difference between the original graph G and the modified graph G_R,θ. Removal strategies are derived from classic centrality measures—degree (dc), betweenness (bc), closeness (cc), eigenvector (ec), and PageRank (pr)—as well as from a label‑propagation community‑detection algorithm (lp). For directed graphs, in‑ and out‑versions of degree and closeness are also considered. The modification level θ denotes the fraction of edges removed; vertices are deleted until θ·m edges have been eliminated.

Two families of comparison methods are employed. The first family is based on the neighbourhood function N_G(t), which counts node pairs at distance ≤ t. From N_G(t) the authors compute the harmonic diameter D_harm(G) and its relative change δ = D_harm(G_R,θ)/D_harm(G) − 1. This metric captures both connectivity and average path length. The second family treats the shortest‑path distribution H_SPG(t) as a probability mass function and compares the original and perturbed distributions using three stochastic quantifiers: Kullback‑Leibler divergence (kl), Jensen‑Shannon distance (jsd), and Hellinger distance (hd). Finally, when centrality measures themselves are used as comparison tools, the Spearman rank‑correlation ρ between the centrality vectors of G and G_R,θ is calculated.

The empirical study comprises six real‑world networks (three social: Hamsterster, Brightkite, Slashdot; three web graphs: Google, Stanford, NotreDame) and four random‑graph models (Erdős‑Rényi, Barabási‑Albert, Watts‑Strogatz, and the configuration model). For each graph, the authors apply removal strategies at six modification levels (θ ∈ {0.05, 0.10, 0.15, 0.20, 0.25, 0.30}) and evaluate all comparison metrics. Approximate neighbourhood functions are obtained with HyperANF when exact computation is infeasible; each approximation is repeated at least ten times to keep the relative standard deviation below 1.45 %.

Key findings are as follows. (1) Social networks exhibit low δ values (≈0.2–0.5) and modest stochastic‑quantifier scores, indicating that even aggressive centrality‑based vertex removal only mildly perturbs their structure. The lp strategy shows slightly larger effects, but overall the social graphs are robust. (2) Web graphs are markedly more sensitive: δ can reach values above 200 for certain strategies, and hd, jsd, and kl increase substantially. Betweenness (bc) and out‑degree (dc_out) are the most destructive removal orders, while lp also causes pronounced degradation. (3) For random graphs, the choice of comparison method dominates the observed sensitivity. The harmonic‑diameter change δ is relatively insensitive to whether vertices are removed by centrality or at random, but the stochastic quantifiers reveal a strong distinction: random removal yields much higher divergence values than centrality‑guided removal. This suggests that random graphs maintain their degree‑based structure under targeted attacks but are vulnerable to indiscriminate loss. (4) Across all graph types, the specific centrality measure used to define the removal order has surprisingly little impact; degree, betweenness, closeness, eigenvector, and PageRank all produce similar δ and quantifier profiles. This is attributed to the high correlation among these centrality scores in the studied networks. In contrast, a purely random removal order consistently generates the largest structural changes.

The authors also note that symmetrising directed web graphs reduces—but does not eliminate—the gap between social and web networks, confirming earlier observations by Boldi et al. (2011). Additional analyses of average distance change (δ_avg_dist) and reachable‑pair percentage (δ_reachable) corroborate the main trends observed with δ.

Overall, the paper confirms that medium‑sized social networks are structurally resilient to vertex removal, while web graphs are highly vulnerable, especially to attacks targeting high‑betweenness or high‑out‑degree nodes. Random graph models display a nuanced behavior: targeted attacks have limited impact on global metrics, yet random deletions can dramatically alter shortest‑path distributions. The study highlights the importance of selecting appropriate comparison metrics when assessing network robustness, as different metrics capture distinct aspects of structural change. These insights are valuable for designing more resilient network architectures and for understanding the potential impact of node failures or adversarial attacks in various real‑world contexts.