A small network of nodes is built as a symmetric double star: two hub nodes are…
PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Complex and collective systems · Complex and collective systems
Many Parts, One Behaviour, Ch Epidemics on networks
Problem
A small network of N=18 nodes is built as a symmetric double star: two hub nodes are joined to each other by an edge, and each hub is also joined to 8 further leaf nodes that have no other connections. Every leaf therefore has degree 1 and each hub has degree 9.
((a)) Using the degree sequence and the giant-component argument of the random-graph chapter, applied here as the tree-like bond-percolation threshold, compute Tc=⟨k⟩/(⟨k2⟩−⟨k⟩). This is the SIR threshold of a large configuration-model network with the same proportions of degrees, a comparison value rather than a threshold of the pictured 18-node graph.
((b)) Using heterogeneous mean-field theory (annealed degree classes), compute the SIS threshold λc=⟨k⟩/⟨k2⟩.
((c)) Using the symmetric eigenvalue problem for the adjacency matrix A (exploiting the network's symmetry to reduce the eigenvector equation to leaf and hub components), find the leading eigenvalue κ1, and hence give the quenched SIS threshold λc=1/κ1.
((d)) State, in one sentence, which of the two SIS thresholds in (b) and (c) is larger and why degree-class theory, which knows only the degrees, misjudges this network.