Limited Improvement of Connectivity in Scale-Free Networks by Increasing the Power-Law Exponent

TL;DR

It is shown that smaller variance of degree distributions leads to stronger robustness and longer average length of the shortest loops, which means the existing of large holes in randomized SF networks.

Abstract

It has been well-known that many real networks are scale-free (SF) but extremely vulnerable against attacks. We investigate the robustness of connectivity and the lengths of the shortest loops in randomized SF networks with realistic exponents $2.0 < γ\leq 4.0$. We show that smaller variance of degree distributions leads to stronger robustness and longer average length of the shortest loops, which means the existing of large holes. These results will provide important insights toward enhancing the robustness by changing degree distributions.

Figures (34)

• Figure 1: Schematic illustration of related works. The blue region indicates this study, which focuses on SF networks with tunable exponents $2 < \gamma \leq 4$. The orange region represents the previous research in the wide class of randomized networks. It has been revealed that the robustness against malicious attacks becomes stronger with large holes as $P(k)$ is narrower Chujyo2021LoopEnhancementchujyo2022addingChujyo2023OptimalRobustnesskawato2025larger.
• Figure 2: Degree distributions $P(k) \sim k^{-\gamma}$ in generated SF networks with power-law exponents (a) $\gamma=2.1$, (b) $\gamma=2.5$, (c) $\gamma=3.0$, and (d) $\gamma=4.0$ for $N=10^3$ and $m=2$. Dashed lines guide the slope of power-law exponent $\gamma$ in the log-log plot. The shaded areas show the standard deviations in log-log scales.
• Figure 3: Monotone decreasing of (a) the maximum degree $k_{max}$ and (b) the variance $\sigma^2$ of degree distribution $P(k)$ with the power-law exponent $\gamma$ for $N=10^3$ and $m=2$.
• Figure 4: The relative size $S(q)/N$ of the largest connected component (LCC) against different attacks in randomized SF networks with the power-law exponents (a) $\gamma = 2.1$, (b) $\gamma = 2.5$, (c) $\gamma = 3.0$, and (d) $\gamma = 4.0$ for $N=10^3$ and $m=2$. Blue, red, and green curves correspond to recalculated degrees, betweenness centralities, and BP attacks, respectively. In comparing the areas under curves, red (BP attacks) and green (betweenness centralities) curves show more destructive with smaller areas than blue curves (degrees attacks).
• Figure 5: More detailed results for the robustness against recalculated (a) degrees, (b) betweenness centralities, and (c) belief propagation (BP) attacks for $N=10^3$ and $m=2$. The areas under colored curves represent the robustness index $R$ in SF networks with power-law exponents from $\gamma = 2.1$ (dark purple) to $\gamma = 4.0$ (red). As $\gamma$ increases, the areas under curves become larger from dark purple to red lines.