A building-monitoring wireless sensor network has nodes

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 building-monitoring wireless sensor network has N=200N=200 nodes. Each node's onboard clock is modelled as a phase oscillator that exchanges timing pulses with the neighbours within its radio range. The deployment's estimated connectivity gives a mean degree ⟨k⟩=6\langle k\rangle = 6, well approximated as an Erdos-Renyi random graph. The clocks' free-running frequencies scatter about a common target with a Lorentzian density of half-width Δ=2.0×10−3 rad s−1\Delta = 2.0\times10^{-3}\ \mathrm{rad\,s^{-1}} (a typical crystal-oscillator drift spread). The synchronisation protocol achieves an effective coupling K=1.5×10−3 rad s−1K = 1.5\times10^{-3}\ \mathrm{rad\,s^{-1}}. ((a)) Estimate the network's leading eigenvalue κ1\kappa_1 using the Erdos-Renyi approximation κ1≈⟨k⟩+1\kappa_1 \approx \langle k\rangle + 1. ((b)) Estimate the network synchronisation threshold KcK_{\rm c}. ((c)) State, with a reason, whether the protocol's coupling is sufficient to bring the clocks into synchrony, and by what factor it exceeds or falls short of the estimated threshold.

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