Universal features of dynamic small-world networks

In a dynamic small-world contact network, an individual has fixed short range links within its local neighborhood and time-varying stochastic long range links outside of that neighborhood. The probability of a long range link occurring ($p$, in analogy with the standard small-world rewiring parameter) is a measure of the mobility of the population. In this study we investigate the epidemic to non-epidemic phase transition that occurs in a susceptible-infected-recovered (SIR) disease spreading model located on this type of dynamic network. We first derive the finite-valued critical mobility p$_{c}$ and find excellent agreement with numerical simulations. Close to p$_{c}$ the outbreak size scales as (p-p$_{c})^{\beta }$ since it is a continuous transition; we find that $\beta $ is near 2, but varies as a function of the infectivity and recovery rates. At the critical point our study shows that the distribution of outbreak sizes scales as $\sim $ N$^{-\alpha }$ with $\alpha $ = 1.5$\pm $0.03. We compare these critical exponents to those found in related small-world and dynamic small-world networks and comment on potential universality.