A simple undirected network of nodes has a degree distribution that follows a…

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

Many Parts, One Behaviour, Ch Heavy tails: power laws and how to measure them

Problem

A simple undirected network of N=106N = 10^6 nodes has a degree distribution that follows a continuous power law p(k)=Ck−γp(k) = C k^{-\gamma} for kmin⁡≤k≤kmax⁡k_{\min} \le k \le k_{\max}, with γ=2.5\gamma = 2.5 and kmin⁡=1k_{\min} = 1. There is no degree above kmax⁡k_{\max}, which is set by the largest-value scaling for a sample of size NN. ((a)) Use the largest-value scaling relation for a heavy-tailed sample to estimate kmax⁡k_{\max} for this network. ((b)) Normalise p(k)p(k) over [kmin⁡,kmax⁡][k_{\min}, k_{\max}] (you may neglect any correction of order (kmin⁡/kmax⁡)γ−1(k_{\min}/k_{\max})^{\gamma-1}, which is negligible here) and compute the mean degree ⟨k⟩\langle k\rangle and second moment ⟨k2⟩\langle k^2\rangle. ((c)) Chapter 3's friendship paradox gives the mean degree of a neighbour as ⟨k2⟩/⟨k⟩\langle k^2\rangle/\langle k\rangle. Compute it for this network and say how it would change if NN were increased with γ\gamma held fixed.

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