Compounding Vulnerability: Hub Removal Triggers Cascade Phase Transitions While Degrading Percolation Robustness in Scale-Free Networks

TL;DR

The results establish that hub manipulation creates compounding vulnerability: both percolation and cascade metrics worsen simultaneously on static, unweighted networks.

Abstract

Hub removal in scale-free networks is known to degrade percolation robustness by raising the bond percolation threshold. We show that this same intervention simultaneously triggers a cascade phase transition under the Watts threshold model. In Barabási--Albert networks, removing the top 10\% of nodes by degree raises $p_c$ from 0.34 to 0.90, reducing robustness to random edge failure. Simultaneously, at cascade threshold $\varphi=0.22$, mean cascade size increases from $0.29\%$ to $20.6\%$, crossing from the subcritical to supercritical regime -- compounding, rather than trading off, the percolation degradation. Using a controlled experiment that independently varies hub presence and hub activation threshold, we demonstrate that hub-mediated cascade suppression is primarily dynamical: making hubs vulnerable without removing them produces 95\% cascades versus 19\% with removal. We operationalize Watts' (2002) stable-node mechanism as a quantitative condition $k > 1/\varphi$ and derive, under the configuration-model approximation, a closed-form expression for the post-removal cascade branching factor $z_1(\varphi, α, m)$. This predicts a newly opened interval where the pre-removal network is subcritical but the post-removal network is supercritical. We define $\varphi^*$ as the upper supercritical boundary, $\varphi^* = \sup\{\varphi : z_1(\varphi) \geq 1\}$. The $z_1$ derivation confirms that at $\varphi=0.22$, the pre-removal network is subcritical ($z_1=0.850$) while the post-removal network is supercritical ($z_1=1.195$). Our results establish that hub manipulation creates compounding vulnerability: both percolation and cascade metrics worsen simultaneously on static, unweighted networks. The effect vanishes in homogeneous networks (ER, WS).

Figures (5)

• Figure 1: Hub vulnerability experiment results at $\varphi=0.22$, BA($N=2{,}000$, $m=2$). The 95% cascade under condition B (threshold modification only, no topology change) versus 19% under condition C (hub removal) demonstrates that cascade suppression is primarily dynamical. Error bars show standard deviation over 200 trials.
• Figure 2: Cascade size distributions before (left) and after (right) hub removal at $\varphi=0.22$, BA($N=2{,}000$, $m=2$, 200 trials). Before removal, cascades are unimodal near zero. After removal, the distribution becomes bimodal, with modes at $<1\%$ and $>10\%$---the hallmark of a phase transition straddling the critical point.
• Figure 3: Mean cascade size (before vs. after hub removal) as a function of cascade threshold $\varphi$ for BA($N=2{,}000$, $m=2$). Each point averages 200 trials (10 independent network realizations $\times$ 20 seed nodes each), with seed nodes drawn from the surviving-node set so that seeds are comparable pre/post removal. Shaded regions indicate the three cascade regimes described in the text.
• Figure 4: Comparison of cascade response to hub removal across network topologies with matched $\langle k \rangle=4$. Only the BA network ($\kappa \approx 12$) exhibits a genuine cascade phase transition. ER and WS networks show mild worsening without phase transition, confirming that degree heterogeneity $\kappa > {\sim}10$ is required.
• Figure 5: Cascade branching factor $z_1$ before (solid) and after (dashed) hub removal as a function of cascade threshold $\varphi$. The critical line $z_1 = 1$ separates subcritical from supercritical regimes. At $\varphi=0.22$, the pre-removal network is subcritical ($z_1=0.850$) while the post-removal network is supercritical ($z_1=1.195$), explaining the observed cascade phase transition.