Comparative Resilience Notions and Vertex Attack Tolerance of Scale-Free Networks

We are concerned with an appropriate mathematical measure of resilience in the face of targeted node attacks for arbitrary degree networks, and subsequently comparing the resilience of different scale-free network models with the proposed measure. We strongly motivate our resilience measure termed \emph{vertex attack tolerance} (VAT), which is denoted mathematically as $\tau(G) = \min_{S \subset V} \frac{|S|}{|V-S-C_{max}(V-S)|+1}$, where $C_{max}(V-S)$ is the largest connected component in $V-S$. We attempt a thorough comparison of VAT with several existing resilience notions: conductance, vertex expansion, integrity, toughness, tenacity and scattering number. Our comparisons indicate that for artbitrary degree distributions VAT is the only measure that fully captures both the major \emph{bottlenecks} of a network and the resulting \emph{component size distribution} upon targeted node attacks (both captured in a manner proportional to the size of the attack set). For the case of $d$-regular graphs, we prove that $\tau(G) \le d\Phi(G)$, where $\Phi(G)$ is the conductance of the graph $G$. Conductance and expansion are well-studied measures of robustness and bottlenecks in the case of regular graphs but fail to capture resilience in the case of highly heterogeneous degree graphs. Regarding comparison of different scale-free graph models, our experimental results indicate that PLOD graphs with degree distributions identical to BA graphs of the same size exhibit consistently better vertex attack tolerance than the BA type graphs, although both graph types appear asymptotically resilient for BA generative parameter $m = 2$. BA graphs with $m = 1$ also appear to lack resilience, not only exhibiting very low VAT values, but also great transparency in the identification of the vulnerable node sets, namely the hubs, consistent with well known previous work.