A different small network of nodes has an asymmetric double-star shape: hub is joined…
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 different small network of N=20 nodes has an asymmetric double-star shape: hub A is joined to 15 leaf nodes and to hub B; hub B is joined to 3 leaf nodes and to hub A. Every leaf has degree 1, hub A has degree 16 and hub B has degree 4.
((a)) Using the degree sequence and the tree-like bond-percolation argument built on the giant-component condition of the random-graph chapter, compute Tc=⟨k⟩/(⟨k2⟩−⟨k⟩), the SIR threshold of a large configuration-model network with the same proportions of degrees (a comparison value, not a threshold of the pictured 20-node graph).
((b)) Compute the heterogeneous mean-field SIS threshold λc=⟨k⟩/⟨k2⟩.
((c)) Set up the adjacency eigenvector equation for leaves and hubs (as for the symmetric eigenvalue problem on A), reduce it to a quadratic in κ2, and hence find the quenched SIS threshold λc=1/κ1.
((d)) State in one sentence how the gap between (b) and (c) compares here with a symmetric double star of the same total leaf count split evenly between the hubs, and which of the two SIS theories overstates how easily the infection persists.