Who is More at Risk in Heterogenous Networks?

Network-based epidemics models try to characterize the impact of network topology, which represents contagion pathways, on the spread of infection. Although these models explicitly consider the dynamics of individuals in the given network (i.e., the state of the system is $\mathbf{x}(t)=[x_{1}(t), x_{2}(t), \ldots, x_{N}(t)]^{T})$, analysis has focused on characterizing the vulnerability of the entire population rather than the vulnerability of the individuals in the population. We focus on characterizing the vulnerability of the $i\mathrm{th}$ individual in the network by studying the marginal probability of infection, $P(x_{i}=1)$, of the scaled SIS process. Studying the vulnerability of individuals is important because it may be tempting to assume that $P(x_{i}=1)$ is related to the degree of the $i\mathrm{th}$. node. Since infection rate is usually assumed to be dependent on the number of infected neighbors, then it seems reasonable that nodes with more connections (i.e., higher degree) would be more at risk. We show that this is not always true. Further, with a closed-form approximation of $P(x_{i}=1)$, as solving for the exact probability requires the summation of $2^{N}$ terms, we characterize the conditions for when degree distribution is a good indicator of how susceptible an individual is to infection.