A large random-contact network of people is modelled as an Erdos-Renyi graph with…

PHYSICS · Multi-part · 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

A large random-contact network of N=2000N = 2000 people is modelled as an Erdos-Renyi graph with mean degree c=12c = 12, so that its leading adjacency eigenvalue is κ1≈c+1\kappa_1 \approx c + 1. ((a)) A rumour spreads on this network exactly as an SIS infection, with per-contact transmission rate β=0.02 day−1\beta = 0.02\,\mathrm{day^{-1}} and recovery (forgetting) rate μ=0.05 day−1\mu = 0.05\,\mathrm{day^{-1}}. Using the quenched mean-field threshold λc=1/κ1\lambda_{\rm c} = 1/\kappa_1 for the effective spreading ratio λ=β/μ\lambda = \beta/\mu, decide whether the rumour is above or below threshold on this network. ((b)) The same 20002000 people each carry a phase oscillator (a habitual daily-routine clock), with natural frequencies drawn from a Lorentzian of half-width Δ=0.5 rad day−1\Delta = 0.5\,\mathrm{rad\,day^{-1}}, coupled with strength KK along the same contact edges. Using the well-mixed Kuramoto threshold K0=2/[πg(0)]K_0 = 2/[\pi g(0)] for this Lorentzian and the network correction Kc≈K0/κ1K_{\rm c} \approx K_0/\kappa_1, estimate the coupling strength at which the population of clocks starts to synchronise. ((c)) In one sentence, say what is common to the two thresholds and what approximation both rest on.

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