A configuration-model network has a power-law degree density on

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 configuration-model network has a power-law degree density p(k)∝k−γp(k) \propto k^{-\gamma} on kmin⁡≤k≤kmax⁡k_{\min} \le k \le k_{\max}, with γ=3.0\gamma = 3.0 (the marginal case, between the γ=2.5\gamma = 2.5 and γ=3.5\gamma = 3.5 examples worked for the vanishing epidemic threshold) and kmin⁡=3k_{\min} = 3. Treat kk as continuous, so that ⟨kj⟩=∫kmin⁡kmax⁡kj k−γ dk/∫kmin⁡kmax⁡k−γ dk\langle k^j\rangle = \int_{k_{\min}}^{k_{\max}} k^j\,k^{-\gamma}\,\mathrm{d}k \big/ \int_{k_{\min}}^{k_{\max}} k^{-\gamma}\,\mathrm{d}k, and take the network to be locally tree-like so that the bond-percolation threshold Tc=⟨k⟩/(⟨k2⟩−⟨k⟩)T_{\rm c} = \langle k\rangle/(\langle k^2\rangle - \langle k\rangle) applies. ((a)) Find TcT_{\rm c} for cutoffs kmax⁡=102k_{\max} = 10^2, 10310^3 and 10410^4. ((b)) For γ=2.5\gamma = 2.5, kmin⁡=2k_{\min} = 2, TcT_{\rm c} has been shown to fall by about a factor of three per decade of cutoff, while for γ=3.5\gamma = 3.5, kmin⁡=2k_{\min} = 2 it settles to a constant. State, with a one-sentence reason drawn from the moment integrals, which of these two patterns the γ=3.0\gamma = 3.0 sequence resembles more closely, and why γ=3\gamma = 3 is the boundary between them.

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