Robustness, the ability to withstand failures and perturbations, is a critical attribute of many complex systems including complex networks. The study of robustness in complex networks is important for many fields. In ecology, robustness is an important attribute of ecosystems, and can give insight into the reaction to disturbances such as the extinction of species. For biologists, network robustness can help the study of diseases and mutations, and how to recover from some mutations. In economics, network robustness principles can help understanding of the stability and risks of banking systems. And in engineering, network robustness can help to evaluate the resilience of infrastructure networks such as the Internet or power grids.
Percolation theory
The focus of robustness in complex networks is the response of the network to the removal of nodes or links. The mathematical model of such a process can be thought of as an inverse percolation process. Percolation theory models the process of randomly placing pebbles on an n-dimensional lattice with probability p, and predicts the sudden formation of a single large cluster at a critical probability p c {\displaystyle p_{c}} . In percolation theory this cluster is named the percolating cluster. This phenomenon is quantified in percolation theory by a number of quantities, for example the average cluster size ⟨ s ⟩ {\displaystyle \langle s\rangle } . This quantity represents the average size of all finite clusters and is given by the following equation.
⟨ s ⟩ ∼ | p − p c | γ p {\displaystyle {\begin{aligned}\langle s\rangle \sim \left|p-p_{c}\right|^{\gamma _{p}}\end{aligned}}}
We can see the average cluster size suddenly diverges around the critical probability, indicating the formation of a single large cluster. It is also important to note that the exponent γ p {\displaystyle \gamma _{p}} is universal for all lattices, while p c {\displaystyle p_{c}} is not. This is important as it indicates a universal phase transition behavior, at a point dependent on the topology. The problem of robustness in complex networks can be seen as starting with the percolating cluster, and removing a critical fraction of the pebbles for the cluster to break down. Analogous to the formation of the percolation cluster in percolation theory, the breaking down of a complex network happens abruptly during a phase transition at some critical fraction of nodes removed.
Critical threshold for random failures The mathematical derivation for the threshold at which a complex network will lose its giant component is based on the Molloy–Reed criterion.
κ ≡ ⟨ k 2 ⟩ ⟨ k ⟩ > 2 {\displaystyle {\begin{aligned}\kappa \equiv {\frac {\langle k^{2}\rangle }{\langle k\rangle }}>2\end{aligned}}}
The Molloy–Reed criterion is derived from the basic principle that in order for a giant component to exist, on average each node in the network must have at least two links. This is analogous to each person holding two others' hands in order to form a chain. Using this criterion and an involved mathematical proof, one can derive a critical threshold for the fraction of nodes needed to be removed for the breakdown of the giant component of a complex network.
f c = 1 − 1 ⟨ k 2 ⟩ ⟨ k ⟩ − 1 {\displaystyle {\begin{aligned}f_{c}=1-{\frac {1}{{\frac {\langle k^{2}\rangle }{\langle k\rangle }}-1}}\end{aligned}}}
An important property of this finding is that the critical threshold is only dependent on the first and second moment of the degree distribution and is valid for an arbitrary degree distribution.
Random network
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