Percolation and resilience analysis

Percolation takes its origins in the analysis of disordered systems, where matter is modeled as sites that may or may not be occupied with probability pn (node percolation) or sites between which edges are present or absent with probability pe. The question is then asked whether or not a Giant Connected Component exists, that is, whether the largest component is extensive, i.e., proportional to the original system size. Originally developed around regular systems, such as two-dimensional or three-dimensional lattices, network theory expanded the notion to random networks, where each “site” has potentially a random degree taken from a given degree distribution. Specifically, important differences appear when the degrees are drawn from a Poisson distribution (Erdos–Renyi, (ER) networks, where every two nodes are connected by an edge with probability p) or a scale free (SF) degree distribution, where node degrees are distributed according to a power law, P(x)∝x-β with usually 2<β<3. In the former case the network experiences a percolation transition when the average degreee, ⟨k⟩ crosses the threshold of 1, where below that value the GCC increases slowly with network size, N, and its relative fraction of the entire network goes to zero as N goes to infinity^21,23. SF networks, on the other hand, do not experience such a transition, and rather have a non-zero GCC for any nonzero average degree. It is often assumed for real-world networks that in order for a node to be functional, whether the node is a financial institution, a geographical location or an infrastructure component such as a power plant²⁹, it needs to be part of the GCC. While we do not suggest Regions, Provinces and Municipalities disconnected from the GCC cease to function, we do suggest their participation in economic activity is impaired. Since there can only be one GCC in a network^21,23, we assume only nodes connected to it operate at their fullest capacity. Thus, as described in section "Local impact and recovery", we define a critical level of q at which the reduction of the GCC is largest, thereby reflecting the most significant impact to the economic activity.