Network topology near criticality in adaptive epidemics

We study structural changes of adaptive networks in the co-evolutionary susceptible-infected-susceptible (SIS) network model along its phase transition. We clarify to what extent these changes can be used as early-warning signs for the transition at the critical infection rate $\lambda_c$ at which the network collapses and the system disintegrates. We analyze the interplay between topology and node-state dynamics near criticality. Several network measures exhibit clear maxima or minima close to the critical threshold that could potentially serve as early-warning signs. These measures include the $SI$ link density, triplet densities, clustering, assortativity and the eigenvalue gap. For the $SI$ link density and triplet densities the maximum is found to originate from the co-existence of two power laws. Other network quantities, such as the degree, the branching ratio, or the harmonic mean distance, show scaling with a singularity at $\lambda=0$ and not at $\lambda_c$, which means that they are incapable of detecting the transition.