[Paper Review] Localized attack on clustering networks

Gaogao Dong, Huifang Hao+4 more|arXiv (Cornell University)|Oct 15, 2016

Complex Network Analysis Techniques12 references3 citations

TL;DR

This paper investigates the robustness of clustering networks under localized attacks, analyzing how clustering coefficient and coupling strength affect percolation thresholds and giant component size. Using analytical and numerical methods, it shows that localized attack is more destructive than random attack, with higher clustering reducing robustness and stronger coupling inducing a first-order phase transition.

ABSTRACT

Clustering network is one of which complex network attracting plenty of scholars to discuss and study the structures and cascading process. We primarily analyzed the effect of clustering coefficient to other various of the single clustering network under localized attack. These network models including double clustering network and star-like NON with clustering and random regular (RR) NON of ER networks with clustering are made up of at least two networks among which exist interdependent relation among whose degree of dependence is measured by coupling strength. We show both analytically and numerically, how the coupling strength and clustering coefficient effect the percolation threshold, size of giant component, critical coupling point where the behavior of phase transition changes from second order to first order with the increase of coupling strength between the networks. Last, we study the two types of clustering network: one type is same with double clustering network in which each subnetwork satisfies identical degree distribution and the other is that their subnetwork satisfies different degree distribution. The former type is treated both analytically and numerically while the latter is treated only numerically. In each section, we compared two results obtained from localized attack and random attack according to Shao et al:[22].

Motivation & Objective

• To study the impact of localized attacks on clustering networks, particularly how clustering coefficient and coupling strength affect network robustness.
• To compare localized attack with random attack in interdependent clustering networks, assessing differences in percolation behavior and phase transitions.
• To examine the role of degree distribution (fixed vs. double Poisson) in determining network resilience under localized failure.
• To determine the critical coupling strength $ q_c $ at which phase transitions shift from second to first order.
• To generalize findings from single networks to interdependent network-of-networks (NON) models with clustering.

Proposed method

• Uses a double Poisson distribution (DPD) to model node degrees in single clustering networks, incorporating both single links and triangles.
• Employs generating functions $ G_0(x,y) = e^{\langle s\rangle(x-1)} e^{\langle t\rangle(y-1)} $ to analytically derive the size of the giant component and percolation thresholds.
• Applies shell decomposition and recursive removal of nodes within a localized region to simulate localized attacks, tracking the cascade of failures.
• Compares results under localized attack (LA) and random attack (RA) across multiple network models: single clustering, interdependent networks, star-like NON, and random regular NON.
• Introduces coupling strength $ q $ to model interdependence between networks, with $ q=1 $ indicating full interdependence.
• Uses numerical simulations with $ N=10,000 $ nodes to validate theoretical predictions, especially for fixed degree distribution (FDD) cases.

Experimental results

Research questions

• RQ1 How does the clustering coefficient $ c $ affect the percolation threshold $ p_c $ under localized attack?
• RQ2 What is the critical coupling strength $ q_c $ at which the phase transition shifts from second to first order in interdependent clustering networks?
• RQ3 How does localized attack compare to random attack in terms of network robustness and cascade dynamics?
• RQ4 How do different degree distributions (FDD vs. DPD) influence the robustness of interdependent clustering networks?
• RQ5 How does increasing the average degree $ \langle k\rangle $ or the number of clustered networks $ n $ affect the system's resilience?

Key findings

• Localized attack causes a more abrupt drop in the size of the giant component compared to random attack, indicating higher destructive power.
• The critical percolation threshold $ p_c $ increases with clustering coefficient $ c $, meaning higher clustering reduces network robustness.
• For strong coupling ($ q=0.5 $), the phase transition becomes first-order, while for weak coupling ($ q=0.2 $), it remains second-order, with a critical point $ q_c $ marking the transition.
• The critical coupling strength $ q_c $ increases with average degree $ \langle k\rangle $, indicating that higher connectivity enhances robustness against interdependence-induced failures.
• For fixed degree distribution (FDD), the network is more robust than for double Poisson distribution (DPD), especially under strong coupling ($ q=0.8 $).
• In star-like NON with clustering, increasing the number of clustered networks $ n $ reduces $ q_c $, making the system more vulnerable to cascading failures.