The vertex attack tolerance of complex networks

Abstract

The purpose of this work is four-fold: (1) We propose a new measure of network resilience in the face of targeted node attacks, vertex attack tolerance, represented mathematically as τ (G) = minS⊂V |S| |V −S−Cmax(V −S)|+1 , and prove that for d-regular graphs τ (G) = Θ(Φ(G)) where Φ(G) denotes conductance, yielding spectral bounds as corollaries. (2) We systematically compare τ (G) to known resilience notions, including integrity, tenacity, and toughness, and evidence the dominant applicability of τ for arbitrary degree graphs. (3) We explore the computability of τ , first by establishing the hardness of approximating unsmoothened vertex attack tolerance τ (G) = minS⊂V |S| |V −S−Cmax(V −S)|under various plausible computational complexity assumptions, and then by presenting empirical results on the performance of a betweenness centrality based heuristic algorithm applied not only to τ but several other hard resilience measures as well. (4) Applying our algorithm, we find that the random scale-free network model is more resilient than the Barabasi−Albert preferential attachment model, with respect to all resilience measures considered.

Node-Based Resilience Measure Clustering with Applications to Noisy and Overlapping Communities in Complex Networks

“…Popular graph partitioning methods such as the Girvan-Newman algorithm [2], sparsest-cuts [3], spectral partitioning [4], and general conductance based methods (related to spectral methods via Cheeger's inequality [5]) may be viewed as solving an edge-based resilience problem on a graph while simultaneously outputting the components resulting from the removal of the critical edge set as the set of clusters. In contrast to these graph-theoretic edge-based resilience methods, in preliminary work [6], we introduced a node-based resilience clustering approach using vertex attack tolerance (VAT) [7][8][9][10] with some unique applicability for noisy datasets. A node-based resilience measure by definition must express the relative size of a most critical set of target vertices whose removal, upon an attack, would be detrimental to the remaining network and attempt to quantify the amount of resulting damage [10].…”

“…In contrast to these graph-theoretic edge-based resilience methods, in preliminary work [6], we introduced a node-based resilience clustering approach using vertex attack tolerance (VAT) [7][8][9][10] with some unique applicability for noisy datasets. A node-based resilience measure by definition must express the relative size of a most critical set of target vertices whose removal, upon an attack, would be detrimental to the remaining network and attempt to quantify the amount of resulting damage [10].…”

Comparing the speed and accuracy of approaches to betweenness centrality approximation

“…Computational aspects of many important node-based resilience measures are considered in [60], including vertex attack tolerance (VAT) [61,62], integrity [55], tenacity [63,64], toughness [65], and scattering number [66]. As all of these measures have associated computational hardness results, the performance of heuristics was considered on several representative networks in [60]. Amongst the algorithms considered in [60], a betweenness centrality-based heuristic called Greedy-BC exhibited high-quality performance, particularly on the most relevant measures such as integrity, VAT, and tenacity.…”

“…As all of these measures have associated computational hardness results, the performance of heuristics was considered on several representative networks in [60]. Amongst the algorithms considered in [60], a betweenness centrality-based heuristic called Greedy-BC exhibited high-quality performance, particularly on the most relevant measures such as integrity, VAT, and tenacity. Integrity quantifies resilience by measuring the largest connected component after removal of an attack set.…”

Biomarker discovery in inflammatory bowel diseases using network-based feature selection

“…We briefly describe below, the node resilience measures (specifically the three utilized in this work) before proceeding to describe how they are used to identify biomarkers. Node-based resilience measures quantify the resilience of a network in terms of the extent of damage (as measured by disruption of connectivity between otherwise connected components or clusters of nodes) caused to the network by the removal of a set of critical nodes (called the attack set) 32 . Because the nodes in the attack set are crucial for maintaining connectivity across the network, removal of the nodes in the attack set can be expected to partition the network into clusters that are isolated from (i.e., disconnected from) each other.…”