Introduction to the Modeling and Analysis of Complex Systems

420CHAPTER 18. DYNAMICAL NETWORKS III: ANALYSIS OF NETWORK DYNAMICS18.6 Mean-Field Approximation on Scale-Free NetworksWhat if the network topology is highly heterogeneous, like in scale-free networks, sothat the random network assumption is no longer applicable? A natural way to reconcilesuch heterogeneous topology and mean-field approximation is to adopt a specific degreedistribution P (k). It is still a non-spatial summary of connectivities within the network, butyou can capture some heterogeneous aspects of the topology in P (k).One additional complication the degree distribution brings in is that, because thenodes are now different from each other regarding their degrees, they can also be differentfrom each other regarding their state distributions too. In other words, it is nolonger reasonable to assume that we can represent the global state of the network by asingle “mean field” q. Instead, we will need to represent q as a function of degree k (i.e.,a bunch of mean fields, each for a specific k), because heavily connected nodes maybecome infected more often than poorly connected nodes do. So, here is the summary ofthe new quantities that we need to include in the approximation:• P (k): Probability of nodes with degree k• q(k): Probability for a node with degree k to be infectedLet’s consider how to revise Table 18.1 using P (k) and q(k). It is obvious that all the q’sshould be replaced by q(k). It is also apparent that the third and fourth row (probabilitiesof transitions for currently infected states) won’t change, because they are simply basedon the recovery probability p r . And the second row can be easily obtained once the firstrow is obtained. So, we can just focus on calculating the probability in the first row: Whatis the probability for a susceptible node with degree k to remain susceptible in the nexttime step?We can use the same strategy as what we did for the random network case. Thatis, we calculate the probability for the node to get infected from another node, and thencalculate one minus that probability raised to the power of the number of neighbors, toobtain the probability for the node to avoid any infection. For random networks, all othernodes were potential neighbors, so we had to raise (1 − p e qp i ) to the power of n − 1. Butthis is no longer necessary, because we are now calculating the probability for a nodewith the specific degree k. So, the probability that goes to the first row of the table shouldlook like:(1 − q(k)) (1 − something) k (18.29)Here, “something” is the joint probability of two events, that the neighbor node is infectedby the disease and that the disease is actually transmitted to the node in question. The