Introduction to the Modeling and Analysis of Complex Systems

18.6. MEAN-FIELD APPROXIMATION ON SCALE-FREE NETWORKS 423q n is the probability that a neighbor node is infected by the disease. Thus q n p i correspondsto the “something” part in Eq. (18.29). It can take any value between 0 and 1, but for ourpurpose of studying the epidemic threshold of infection probability, we can assume thatq n p i is small. Therefore, using the approximation (1 − x) k ≈ 1 − kx for small x again, theiterative map above becomesq t+1 (k) = (1 − q(k)) (1 − (1 − kq n p i )) + q(k)(1 − p r ) (18.37)= (1 − q(k))kq n p i + q(k) − q(k)p r = f(q(k)). (18.38)Then we can calculate the equilibrium state of the nodes with degree k as follows:q eq (k) = (1 − q eq (k))kq n p i + q eq (k) − q eq (k)p r (18.39)q eq (k)p r = kq n p i − kq n p i q eq (k) (18.40)q eq (k) =kq np ikq n p i + p r(18.41)We can apply this back to the definition of q n to find the actual equilibrium state:q n = 1 ∑k ′ P (k ′ k ′ q n p i)(18.42)〈k〉k ′ qk ′ n p i + p rClearly, q n = 0 (i.e., q eq (k) = 0 for all k) satisfies this equation, so the extinction of thedisease is still an equilibrium state of this system. But what we are interested in is theequilibrium with q n ≠ 0 (i.e., q eq (k) > 0 for some k), where the disease continues to exist,and we want to know whether such a state is stable. To solve Eq. (18.42), we will need toassume a certain degree distribution P (k).For example, if we assume that the network is large and random, we can use thefollowing crude approximate degree distribution:{ 1 if k = 〈k〉P (k) ≈(18.43)0 otherwiseWe can use this approximation because, as the network size increases, the degree distributionof a random graph becomes more and more concentrated at the average degree(relative to the network size), due to the law of large numbers well known in statistics.Then Eq. (18.42) is solved as follows:q n =〈k〉q np i〈k〉q n p i + p r(18.44)〈k〉q n p i + p r = 〈k〉p i (18.45)q n = 1 −p r〈k〉p i(18.46)