Introduction to the Modeling and Analysis of Complex Systems

18.6. MEAN-FIELD APPROXIMATION ON SCALE-FREE NETWORKS 425the network, and thus 〈k〉 = 2m and k min = m [57]. Then, from Eq. (18.42) 1 , we obtainq n = 12m∞∑k ′ · 2m 2 k ′−3 k ′ q n p i, (18.55)k ′ q n p i + p rk ′ =m1 = mp i∞∑k ′ =m1k ′ (k ′ q n p i + p r ) . (18.56)The summation can be approximated by a continuous integral, which results in∫ ∞dk ′1 ≈ mp im k ′ (k ′ q n p i + p r )= mp ∫ ∞(i 1p r m k − 1′ k ′ + p r /(q n p i )= mp [ (ik ′∞logp r k ′ + p r /(q n p i ))]mq n ≈= mp ip rlogp r(18.57))dk ′ (18.58)(18.59)(1 + p )r, (18.60)mq n p i(e prmp i − 1)mp i. (18.61)We can apply this result (together with P (k) = 2m 2 k −3 ) to r(k) in Eq. (18.49) to check thestability of this non-zero equilibrium state:r(k) = −k= −(e pr(e prp rp i +mp i − 1)mp ikkp rmp i − 1)m + mp ik 2pr(ep rpr(ep rk 2 · 2m 2 k −3 p i+ 1 − p r (18.62)p i + p r 2mmp i −1)mpimp i −1)m+ k + 1 − p r (18.63)This is rather complex, and it would be difficult to solve it in terms of p i . But still, we canshow something very interesting using this formula. If we lower the infection probability p idown to zero, this occurs asymptotically:lim r(k) = −p i →0([e prmp ikp r]→ ∞) +− 1 mm [p i → 0]([ k 2 )pre mp i →∞]−1 m+ k + 1 − p r (18.64)= 1 − p r (18.65)1 Here we assume that Barabási-Albert networks are non-assortative.