Critical behavior of disease spread on dynamic small-world networks

Dynamic small-world networks combine fixed short-range links within a node's local neighborhood with time-varying stochastic long-range links outside of that neighborhood. The probability of a long-range link occurring, p, is a measure of the mobility, or equivalently, the randomness of the population and is analogous to the standard static small-world rewiring parameter. In this study, we investigate the non-epidemic to epidemic phase transition that occurs in a susceptible-infected-recovered (SIR) propagation model within this type of dynamic network. We first derive the finite-valued critical randomness pc and find excellent agreement with numerical simulations. Close to pc, the order parameter, which is given by the fraction of the population infected during the outbreak, scales as (p−pc)β since it is a continuous transition; we find that β is near 2, but varies as a function of pc. At the critical point, our study shows that the distribution of outbreak sizes follows a power law characterized by an exponent of ≈1.5 in agreement with the value found in related small-world models, leading us to conjecture that it may be a universal feature of these models