PHYSICS · Subjective · Fermi University Physics · Thermal, Statistical & Complex Systems · Complex and collective systems · Complex and collective systems
Many Parts, One Behaviour, Ch Synchronisation, cascades and the resilience of networks
Problem
In a simplified model of interbank lending, banks are nodes of a network and a loan is an edge. A bank fails and stops honouring its obligations, in Watts's threshold model, once a fraction θW or more of its counterparties (its network neighbours) have already failed. Take θW=0.25: a bank becomes vulnerable to a single failed counterparty if that one failure alone reaches or exceeds this fraction. Model the interbank network as an Erdos-Renyi random graph of mean degree c (the mean number of counterparties per bank), with Poisson degree distribution pk=cke−c/k!. (a) Say which degrees k make a bank vulnerable under this rule, and hence write ρk (equal to 1 for a vulnerable degree, 0 otherwise) explicitly. (b) Apply the Molloy-Reed argument to the subnetwork of vulnerable banks to derive the general cascade condition ∑kk(k−1)ρkpk>⟨k⟩, explaining in one sentence why it is the same criterion used for the giant component, restricted to vulnerable nodes. (c) Evaluate the condition for this Poisson degree distribution and find, to two decimal places, the range of c for which a global cascade can occur. State the approximation (configuration-model/Erdos-Renyi, locally tree-like, large N) on which the result rests.