Heterogeneous Protection in Regular and Complete Bi-partite Networks

We examine the influence of heterogeneous curing rates for a SIS model, used for malware spreading on the Internet, information dissemination in unreliable networks, and propagation of failures in networks. The topology structures considered are the regular graph which represents the homogenous network structures and the complete bi-partite graph which represents the hierarchical network structures. We find the threshold in a regular graph with m different curing rates. Further, we consider a complete bi-partite graph with 2 curing rates and find the threshold for any distribution of curing rates among nodes. In addition, we consider the optimization problem and show that the minimum sum of the curing rates that satisfies the threshold equation is equal to the number of links in the graph. The optimization problem is simplified by assuming fixed curing rates δ1, δ2 and optimization of the distribution of curing rates among different sets of nodes.