A constellation of small communication satellites relays signals through neighbours…

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 constellation of

N = 800

small communication satellites relays signals through neighbours according to an Erdos-Renyi random graph with mean degree

\langle k\rangle = 12

. Each satellite carries a beacon oscillator; the coupling that would be needed to entrain the beacons if every satellite could talk to every other is

K_0 = 9\,\mathrm{rad\,s^{-1}}

. ((a)) Estimate the leading eigenvalue

\kappa_1

of the constellation's link network and the coupling

K_{\rm c}

at which the beacons begin to entrain on the real network. Use

\kappa_1\approx\langle k\rangle+1

for an Erdos-Renyi graph and

K_{\rm c}\approx K_0/\kappa_1

. ((b)) Micrometeoroid strikes disable satellites at random. Using the random-failure percolation threshold, find the fraction of satellites that must be disabled before the relay network's giant component disappears.

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