Introduction to the Modeling and Analysis of Complex Systems

424CHAPTER 18. DYNAMICAL NETWORKS III: ANALYSIS OF NETWORK DYNAMICSWe can check the stability of this equilibrium state by differentiating Eq. (18.38) by q(k)and then applying the results above. In so doing, we should keep in mind that q n containsq(k) within it (see Eq. (18.36)). So the result should look like this:df(q(k))dq(k)= −kq n p i + (1 − q(k))kp idq ndq(k) + 1 − p r (18.47)= −kq n p i + (1 − q(k)) k2 P (k)p i〈k〉+ 1 − p r (18.48)At the equilibrium state Eq. (18.41), this becomesdf(q(k))dq(k)∣∣q(k)=kqnp ikqnp i +pr= −kq n p i +p r k 2 P (k)p i+ 1 − p r = r(k). (18.49)kq n p i + p r 〈k〉Then, by applying P (k) and q n with a constraint k → 〈k〉 for large random networks, weobtain(r(〈k〉) = −〈k〉 1 − p )rp i +〈k〉p ip r〈k〉( )2 p i〈k〉 1 − pr〈k〉p ip i + p 〈k〉 r+ 1 − p r (18.50)= −〈k〉p i + p r + p r + 1 − p r (18.51)= −〈k〉p i + p r + 1. (18.52)In order for the non-zero equilibrium state to be stable:− 1 < −〈k〉p i + p r + 1 < 1 (18.53)p r〈k〉 < p i < p r + 2〈k〉(18.54)Note that the lower bound indicated above is the same as the epidemic threshold weobtained on random networks before (Eq. (18.28)). So, it seems our analysis is consistentso far. And for your information, the upper bound indicated above is another criticalthreshold at which this non-zero equilibrium state loses stability and the system begins tooscillate.What happens if the network is scale-free? Here, let’s assume that the network isa Barabási-Albert scale-free network, whose degree distribution is known to be P (k) =2m 2 k −3 where m is the number of edges by which each newcomer node is attached to