A container-shipping network grows one port at a time

PHYSICS · Subjective · Fermi University Physics · Thermal, Statistical & Complex Systems · Complex and collective systems · Complex and collective systems

Many Parts, One Behaviour, Ch Growing networks: preferential attachment and scale-free degree distributions

Problem

A container-shipping network grows one port at a time. Each new port opens m=4m=4 shipping routes to existing ports, choosing each destination in proportion to the number of routes that port already has (the Barabasi-Albert rule, Πk=k/2mt\Pi_k = k/2mt with tt the number of existing ports). Using the mean-field ansatz Nk(t)=tpk(t)N_k(t)=tp_k(t), the rate equation for this model reduces, after substitution, to the stationary recursion pk(k+2)=(k−1)pk−1+2δkm.p_k(k+2) = (k-1)p_{k-1} + 2\delta_{km}. (a) Solve this recursion for pmp_m, then derive the closed-form telescoping product for pkp_k, k≥mk\ge m, in terms of factorials, and simplify it to a single formula for pkp_k. (b) Evaluate p4p_4, p5p_5 and p10p_{10}. (c) Verify, using the partial-fraction identity 1k(k+1)(k+2)=12[1k(k+1)−1(k+1)(k+2)]\frac{1}{k(k+1)(k+2)} = \frac12\left[\frac{1}{k(k+1)}-\frac{1}{(k+1)(k+2)}\right], that the distribution is correctly normalised for m=4m=4.

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