A Quick Framework for Evaluating Worst Robustness of Complex Networks

TL;DR

The paper tackles the problem of assessing worst-case network robustness by introducing Most Destruction Attack (MDA), a knowledge‑stacking approach that assembles the most destructive components of existing dismantling strategies. It formalizes Worst Robustness $R_W$ via the MDA curve and proposes a Two‑part Worst Robustness Evaluation Framework (WRE) that pairs MDA with a CNN‑SPP based Quick Evaluator to enable rapid, topology‑dependent predictions of $R_W$. Through analyses on model and empirical networks, the authors validate MDA's rationality and demonstrate high predictive accuracy of the Quick Evaluator, including good generalization to unseen networks. The work offers a practical, scalable method for robustness budgeting and defense planning, with extensions to optimal control robustness and critical node identification.

Abstract

Robustness is pivotal for comprehending, designing, optimizing, and rehabilitating networks, with simulation attacks being the prevailing evaluation method. Simulation attacks are often time-consuming or even impractical, however, a more crucial yet persistently overlooked drawback is that any attack strategy merely provides a potential paradigm of disintegration. The key concern is: in the worst-case scenario or facing the most severe attacks, what is the limit of robustness, referred to as ``Worst Robustness'', for a given system? Understanding a system's worst robustness is imperative for grasping its reliability limits, accurately evaluating protective capabilities, and determining associated design and security maintenance costs. To address these challenges, we introduce the concept of Most Destruction Attack (MDA) based on the idea of knowledge stacking. MDA is employed to assess the worst robustness of networks, followed by the application of an adapted CNN algorithm for rapid worst robustness prediction. We establish the logical validity of MDA and highlight the exceptional performance of the adapted CNN algorithm in predicting the worst robustness across diverse network topologies, encompassing both model and empirical networks.

Figures (8)

• Figure 1: The schematic of the Most Destruction Attack (MDA) and the worst robustness of the network. Here, $p$ and $G$ represent the node removal ratio and the relative size of the largest connected component, respectively. The MDA is obtained by extracting the most destructive portions from each attack paradigm and combining them. The MDA demonstrates the most severe damage that a network may undergo, guided by existing knowledge, and the worst robustness exhibited under such circumstances.
• Figure 2: Maximum rationality of three types of synthetic networks. Here, $q$ represents the number of centrality metrics considered to capture MDA, the $x$-axis denotes the average degree $\langle k\rangle$ of the network, and the $y$-axis represents the value of MR. Each point in the panel represents a network, with each parameter comprising 100 instances of networks with a node count of 5000.
• Figure 3: Architecture of the proposed quick evaluator based on CNN. It comprises five main components, namely the network's input section, convolutional set, SPP-net, two fully connected layers, and a filtering section. The convolutional set consists of eight convolutional blocks, each typically comprising one or two convolutional layers, a rectified linear unit (ReLU) activation function, and a max-pooling layer. The size of the $i$-th feature map (FM) FM$_i$ is denoted as $N_i= \lceil \frac{N}{2^{i}} \rceil$, for $i=1,2,...,8$, where $N$ represents the number of nodes in the input network.
• Figure 4: Comparison of simulated and the quick evaluator predicted MDA curves on four types of synthetic networks. Here each panel title specifies the network type and average degree, $p$ and $G_{MDA}$ denote the node removal ratio and the corresponding relative size of the GCC under MDA, respectively. Blue and red dots represent simulated and Quick Evaluator predicted results, while blue and red $R_W$ denote the worst robustness values obtained based on the response curves. It is noteworthy that each point represents the average result of 100 independent experiments, and given their standard deviations are very small, they are indistinguishable in these panels.
• Figure 5: Performance of the proposed quick evaluator in predicting empirical networks when the predicted network types differ from those used in training. Here each panel's title specifies the name of the empirical network, $p$ and $G_{MDA}$ denote the node removal ratio and the corresponding relative size of the GCC under MDA, respectively. Blue and red dots represent simulated and Quick Evaluator predicted results, while blue and red $R_W$ denote the worst robustness values obtained based on the response curves.