We study structural changes of adaptive networks in the coevolutionary 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 and 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 coexistence 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 $\lambda =0$ (not at ${\lambda }_c$), which means that they are incapable of detecting the transition.

L. Horstmeyer, C. Kuehn, and S. Thurner, Network topology near criticality in adaptive epidemics, Phys Rev E 98 (2018) 042313