Five acoustic buoys anchored in a bay relay status pings to their nearest neighbours…

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Complex and collective systems · Complex and collective systems

Many Parts, One Behaviour, Ch Networks as graphs: degree, distance and clustering

Problem

Five acoustic buoys anchored in a bay relay status pings to their nearest neighbours along five acoustic links: buoy 11 to buoy 22, buoy 11 to buoy 33, buoy 22 to buoy 33, buoy 22 to buoy 44, and buoy 44 to buoy 55. The network is simple, undirected and unweighted. ((a)) Write the 5×55\times5 adjacency matrix A\mathbf A. State the induction argument for why (Ar)ij(\mathbf A^{r})_{ij} counts the number of walks of length rr between buoys ii and jj, then use successive powers of A\mathbf A to find the smallest rr for which (Ar)15≠0(\mathbf A^{r})_{15}\neq0, and hence the shortest path length d15d_{15}. ((b)) Derive the general result that, on a network with degree distribution pkp_k and mean degree ⟨k⟩\langle k\rangle, a node reached by following a randomly chosen edge has degree distributed as kpk/⟨k⟩kp_k/\langle k\rangle, and that its mean degree is ⟨k⟩nbr=⟨k2⟩/⟨k⟩\langle k\rangle_{\rm nbr}=\langle k^2\rangle/\langle k\rangle. Treating the five buoys' own degrees as the distribution pkp_k, compute ⟨k⟩\langle k\rangle, ⟨k2⟩\langle k^2\rangle and ⟨k⟩nbr\langle k\rangle_{\rm nbr}, and confirm the neighbour mean exceeds ⟨k⟩\langle k\rangle.

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