Introduction to the Modeling and Analysis of Complex Systems

18.5. MEAN-FIELD APPROXIMATION ON RANDOM NETWORKS 417here we assume synchronous, simultaneous updating, in order to make the mean-fieldapproximation more similar to the approximation we applied to CA.For the mean-field approximation, we need to represent the state of the system by amacroscopic variable, i.e., the probability (= density, fraction) of the infected nodes in thenetwork (say, q) in this case, and then describe the temporal dynamics of this variableby assuming that this probability applies to everywhere in the network homogeneously(i.e., the “mean field”). In the following sections, we will discuss how to apply the meanfieldapproximation to two different network topologies: random networks and scale-freenetworks.18.5 Mean-Field Approximation on Random NetworksIf we can assume that the network topology is random with connection probability p e ,then infection occurs with a joint probability of three events: that a node is connected toanother neighbor node (p e ), that the neighbor node is infected by the disease (q), and thatthe disease is actually transmitted to the node (p i ). Therefore, 1 − p e qp i is the probabilitythat the node is not infected by another node. In order for a susceptible node to remainsusceptible to the next time step, it must avoid infection in this way n − 1 times, i.e., forall other nodes in the network. The probability for this to occur is thus (1 − p e qp i ) n−1 .Using this result, all possible scenarios of state transitions can be summarized as shownin Table 18.1.Table 18.1: Possible scenarios of state transitions in the network SIS model.Current state Next state Probability of this transition0 (susceptible) 0 (susceptible) (1 − q)(1 − p e qp i ) n−10 (susceptible) 1 (infected) (1 − q) (1 − (1 − p e qp i ) n−1 )1 (infected) 0 (susceptible) qp r1 (infected) 1 (infected) q(1 − p r )We can combine the probabilities of the transitions that turn the next state into 1, towrite the following difference equation for q t (whose subscript is omitted on the right hand